- Email: [email protected]
Fluidization at vacuum conditions. A generalized equation for the prediction of minimum fluidization velocity
Pergamon
Chemical Engineerinff Science, Vol, 51, No. 23, pp. 5149-5157, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00351-X 0009-2509/96 $15.00 + 0.00
F L U I D I Z A T I O N AT VACUUM CONDITIONS. A GENERALIZED E Q U A T I O N FOR THE P R E D I C T I O N OF M I N I M U M F L U I D I Z A T I O N VELOCITY M. F. L L O P Chemical Engineering Department, Universitat de Girona, Catalonia, Spain and F. MADRID, J. ARNALDOS* and J. CASAL* Chemical Engineering Department, Universitat Polit~cnica de Catalunya, Diagonal, 647, 08028-Barcelona, Catalonia, Spain (Received 4 January 1996; accepted 22 May 1996)
Abstract--The hydrodynamical behaviour of fluidized beds at reduced pressures is studied. The minimum fluidization velocity is strongly influenced by pressure, decreasing as pressure increases. The well-known equation of Wen and Yu and two equations proposed for the calculation of u,,I at reduced pressures are tested; the comparison with experimental data shows that they can be applied only over restricted pressure ranges. The relationship between flow rate, pressure and pressure drop at conditions ranging from high vacuum to high pressure is analysed for the different flow regimes (molecular, slip, laminar, transition and turbulent). A generalized equation is found which predicts fairly well the value of Uml at vacuum conditions, atmospheric pressure and high pressures. Copyright © 1996 Elsevier Science Ltd Keywords: Fluidization, vacuum, slip flow, minimum fluidization velocity, hydrodynamics.
INTRODUCTION A number of authors have studied the effect of pressure on the hydrodynamical behaviour of fluidized beds. However, most of them have operated at high pressure, i.e. at pressures higher than atmospheric pressure. Very few articles have been published on fluidization at reduced pressures [Kawamura and Suezawa, 1961 and Huschka and Popp, 1967 (cited by Kusakabe et al., 1989); Mitev and Russev, 1990; Leuenberger et al., 1991; Fletcher et al., 1993]. The literature on the behavior of fluidization at pressures less than atmospheric has been reviewed by Kusakabe et al. (1989) and Fletcher et al. (1993). The relatively small number of researchers interested in fluidization at reduced pressure is due to the fact that low-pressure fluidized beds seemed not to be very interesting from the point of view of practical applications. However, this type of fluidization has a promising potential field in the fine chemicals industry. Fine chemicals are often dried at the end of the manufacturating process, and as they are usually particulate materials, fluidized-bed drying seems to be a suitable procedure. But a problem arises from the fact that in a number of cases the moisture which must be removed by the drying operation is constituted of an organic solvent. Therefore, drying with hot air will *Corresponding authors.
easily lead to the formation of a mixture within the flammability limits; in these conditions, the risk of explosion is very high. This is the reason why fluidized-bed drying has been gradually put aside in many processes. Low-pressure fluidized-bed drying is therefore very interesting for such situations. Fluidized-bed features give optimum conditions for the treatment of granular materials, and low-pressure operation allows the control of the resulting atmosphere, giving the possibility of maintaining it outside the flammability conditions and ensuring a completely safe operation. Furthermore, low-pressure fluidization leads to the use of much lower operating temperatures; this is a very important aspect if a thermolabile substance must be dried, as the risk of degradation is much lower. Nevertheless, this type of fluidization is not well known, and in order to use it, a better understanding of its behaviour is required. Low-pressure fluidization follows flow patterns which can be very different from those usually found in fluidized beds, as the hydrodynamical regime is no longer laminar flow, but is replaced by slip flow. In these conditions, the usual equations for the calculation of minimum fluidization velocity do not apply any more; instead, specific equations for these types of flow must be applied (Kusakabe et al., 1989). Therefore, obtaining a generalized equation, which could be applied to any conditions from high-pressure
5149
M. F. LLOPet
5150
fluidization to vacuum fluidization, is very interesting not only because of its practical applications, but also because it would represent a complete understanding of this aspect of fluidization phenomena in its whole domain of existence. In this paper, such an expression is developed, and its validity is tested through the application to experimental data. The main hydrodynamical features of vacuum fluidization are also discussed. EXPERIMENTALSET-UP A diagram of the experimental unit can be seen in Fig. 1. The fluidization column, made of glass, had an inside diameter of 50 mm and a length of 600 mm. The distributor was a stainless-steel plate, 2 mm thick, tied by two Teflon flanges. The orifices (1 mm diameter) were located according to a hexagonal arrangement, with a density of 18 orifices/era 2. Below the plate a mesh (stainless steel) was placed in order to avoid the weeping of particles when the flow was stopped. A calming section packed with Raschig rings was provided under the distributor.
al.
The fluid used was air at 298 _+ 1 K and at absolute pressures ranging between 1 and 101 kPa. The flow at vacuum conditions was obtained by using a rotary vacuum pump. The vacuum in the fluidized bed was controlled with a membrane valve located between the pump suction and the top of the column, and with a pin valve connected to the atmosphere. The vacuum inside the column was measured by two systems. On the one hand, there was a vacuometer installed in the aspiration tube. On the other hand, a tap above the fluidized bed was coupled to a differential manometer, measuring the differential pressure with respect to the atmospheric pressure. The temperature inside the bed was measured with a thermocouple; a control loop connected to an electrical heating system allowed the operation at a constant temperature. Two other taps were located at 100 and 200 mm above the distributor, respectively; both were connected to differential manometers in order to measure the pressure drop through the whole bed and through a given height of the fluidized bed (see Fig. 1). The gas flow rate was measured at atmospheric pressure with a
14 I Flowmeters 2 Flow controlvalves 3 Air heater 4 Temper:~ure oontroler 5 111ermocouple 6 Fluidiz~tionoolumn 7 Calming section 8 Distributor 9 Solids feeder 10 Solids outtet 11 Mercury manometer 12 Water/Anilinemanometers 13 V~nzrn indicator
13
i-
14 Pressure
mntrol
15 Vacuumpump -9
_5 z
Ljj 11
12
12
2:
2
2
1
1
1
Fig. 1. Schematic diagram of the experimental unit.
valves
Fluidization at vacuum conditions system of three rotameters; pin valves were installed downstream of each rotameter to control the gas flow. The solid particles fluidized were silica sand with a density of 2650kg/m 3, a shape factor of 0.6 and an average diameter of 213, 450, 728 and 1460 lam, and millet with a density of 1600 kg/m 3, a shape factor of 0.9 and an average diameter of 1600 p.m. The experimental procedure followed to determine the value of urn: (minimum fluidization velocity) at the different operating conditions was to achieve complete fluidization of the bed and afterwards gradually decrease the gas flow rate. Minimum fluidization velocity was therefore measured at decreasing velocity. Thus, the possible error associated with any variation of bed voidage at incipient fluidization-which could exist when operating at increasing gas velocity--was avoided. The values obtained were reproducible and no difficulties were found in the measurements.
EXPERIMENTAL RESULTS
Bubbling fluidization, essentially similar to that found at atmospheric pressure, was obtained with all the particles used. Bubbling started from the onset of fluidization and increased with gas flow. Following the same behavior that is found at higher pressures, different pressure-drop-gas-velocity relationships were found according to the (increasing/decreasing) trend in the fluid velocity variation. Starting from a fixed bed, the pressure drop for the fixed bed at increasing velocity was higher, for a given bed mass, than the pressure drop measured at decreasing velocities; thus, typical fluidization--defluidization cycles were obtained. This was clearly due to the higher compaction of the initial fixed bed, caused by some vibrations or even by the introduction of the solid in the column. Instead, the fixed bed obtained at decreasing velocity from the fluidized condition always had a lower bed voidage, associated with the looser structure of the particle system. The more compact condition of the initial fixed bed was also shown by the overpressure required to break down the 'structure' linking the particles; this effect was not found with millet beds (round particles), but only with silica sand (sharp particles). The same behavior was obtained at atmospheric pressure. Therefore, minimum fluidization velocity had to be determined at decreasing gas velocity in order to obtain significant and reproducible values. Defluidization curves were obtained at different pressures. The effect of pressure was clear: as pressure decreased, the minimum fluidization velocity increased. This trend can be seen for five different particle systems in Fig. 2, where the variation of urn/has been plotted as a function of absolute pressure. For all the solids the minimum fluidization velocity decreases following an approximately linear trend as pressure increases. The slope of the curves varies slightly with the particle weight; the change in Urn: as pressure increases is greater for the heaviest particles.
5151
10, •
sik:a r~ll(I 1d-213 ~u11)
•
llllcaun¢l ((jm1~
•
,~ ..= (d-'~O ~n)
•
m,~t(.-~O00 ~)
•
=,VYCa=mind(d,-~ .rn)
,'...
• ,
=,
•
•
11n1)
•
nm
•
•
nm
0.1, •
0.01 •
•
000e
I•
100
P (kPe)
Fig. 2. Variation of urn:as a function of absolute pressure for five solids.
As a whole, the experimental results obtained showed a behavior similar to that obtained in fluidization at atmospheric pressure, with the aforementioned variation in minimum fluidization velocity. FLOW RATE, PRESSURE AND PRESSURE DROP IN PARTICLE BEDS AT DIFFERENT REGIMES
The behavior of a gas flowing through a pipe or through a bed of particles can change significantly as pressure is reduced, due to the increase in the mean free path of molecules. Depending on the pressure, the gas can be in a molecular state, in a viscous state or in an intermediate state (Roth, 1976). As a consequence, different flow regimes can exist, depending on the operating conditions (pressure and diameter of the duct). The range over which each flow regime is encountered can be specified as a function of the Knudsen number, defined as the ratio between the mean free path of molecules and the diameter of the duct: 2 Kn = - - . D
(1)
The different flow regimes which can exist are the following ones: Molecular flow: Kn>>1. The gas is in a molecular or rarefied state. The mean free path is much larger than the diameter of the duct; collisions are mostly with the wall rather than between gas molecules, and the concept of viscosity has no meaning. A fluidized bed could not be obtained in these conditions. Intermediate or slip flow: Kn ~ 1. The mean free path of the molecules is similar to the diameter of the duct; the flow of gas is established both by molecular phenomena and by the viscosity. A fluidized bed can operate at this flow regime.
M. F. LLOP et al.
5152
Laminar flow: Kn<<1. The gas is in a viscous state; gas flow is governed by the viscosity and the HagenPoiseuille law applies. As the Reynolds number increases, the transition flow is reached and, at high values of Re, the flow is turbulent. Fluidized beds usually operate in these three regimes (laminar, transition and turbulent). The flow rate of a gas in a pipe of a given diameter and with a fixed value of pressure drop as a driving force can be a function of pressure, i.e. a function of the mean free path of molecules, depending on the flow regime. At very low pressures the flow is molecular. The throughput of gas [QP] does not depend on pressure and has a constant value; from a value reached at 2 = 1.57D, as the mean free path increases the throughput of gas also decreases asymptotically toward the aforementioned constant value. As pressure increases, the mean free path decreases from 1.57D; viscous effects appear and the conductance starts to increase gradually with pressure. This is the range of conditions over which slip flow exists. Finally, in the laminar regime the throughput of gas increases linearly with pressure, as does the conductance. The transition from one flow regime to another will depend on the gas (the mean free path will change from one gas to another), on the pressure and on the diameter of the channel. For air, PD < 0.01 Pa m PD > 0.8 Pa m
molecular flow laminar flow.
The relationships between pressure, pressure drop and flow rate in the different regimes are analysed in detail in the next paragraphs.
As in real low-pressure fluidization processes this extremely low pressure cannot be reached in practice, it may be assumed that for practical purposes ~ = 0.8. In these conditions, eq. (2) becomes:
X/
M
dl~'
Slip and viscous refime In the viscous flow regime (Kn<
~zD~P dP QP = 128~ die"
(5)
Therefore, in the intermediate or slip flow regime
(Kn .~ 1) the relationship between flow rate and pressure can be obtained by adding eqs (4) and (5):
? QP =
/2nRT dP rtD4pdP D3 X[ M dlc t- 128~ die"
dP
dlc
Q
~ + - - nD~
?D~
This equation can be used to determine the pressure drop undergone by a gas flowing through a particulate bed in this flow regime. By considering the flow through the bed to be equivalent to the flow through a bundle of interstitial channels, the following values (Amaldos et al., 1985) can be taken: D2n (8)
Q = Uc--
The flow of a gas through a tube of uniform circular cross-section in the Knudsen regime (Kn>> 1) can be described by the following expression (Roth, 1976):
eP
= 6 X[ M
l+L24Lgfj
u~ =
TJ
(9)
I
(10)
COS I~
2eddp Dc=dh
(2)
The term in brackets takes a value which approaches 1 as the pressure approaches zero. However, this value decreases as the pressure increases; at values of absolute pressure near 7.6 x 10- 6 mmHg, this term reaches a value that is practically constant and equal to 0.8:
FMI'/2D~P l
'+L FJM ] mZDcPII
3(l-e)'
(11)
Equation (8) gives the flow rate as a function of interstitial velocity. Equation (9) gives the fluid velocity in the interstitial channels as a function of bed voidage and bed tortuosity. Equation (10) gives the length of these channels as a function of bed height and tortuosity, and eq. (11) gives the value of the hydraulic diameter of bed interstitial channels. By introducing these values in eq. (7), and taking into account the law of perfect gases (RT/M = P/p), the following expression is obtained: dP dl
,+ 1. 4L j -;-j
U
~
= 0.8 (P ~> 7.6 x 10-6 mmHg).
U 8 COS I~
l~ = - -
7
[-M 71/2 D~PI dlc "
(7)
V- -P-r 128
Low pressures
rM
(6)
This expression can be rearranged to give:
4
I+L J -7- [dP
(4)
(3)
16 2 e2ckd ~-~cos ~k( - T- e~ ) 4 ~ - ~
COS2~083t~2d2 72
p(1-e) 2
(12)
Fluidization at vacuum conditions This equation therefore allows the calculation of the pressure drop undergone by a gas flowing through a particulate bed over the range of conditions covering Knudsen flow, slip flow and viscous flow. Turbulent regime An expression covering a wider range of flow regimes can be obtained by using the equation of Burke and Plummer, which gives the relationship between pressure drop and velocity in a fluid flowing through a particulate bed in turbulent regime (Ergun, 1952):
(1 - - ~) 2 = 1.75-w-;--7,, pu . d---1 e~ q~a
dP
(13)
A general equation covering all the conditions and regimes ranging from Knudsen flow to turbulent flow can be obtained by adding the contributions of slip flow, viscous flow and turbulent flow; thus, adding the right-hand terms of eqs (12) and (13) the following expression is obtained:
where 2 is the mean free path of molecules, Uav = 8~kT mTt k m
dl
16 211/ e2~bd ~ cos2~b e3~b2d2 ~ c o s r(-f'-~--e) qTtpP'4 72 # ( l - - e ) 2
R M'
(19)
By introducing eqs (18) and (19) in eq. (17), and taking into account the law of perfect gases, we obtain the following relationship: p=0.998
2~P2~
2~P),
(20)
2 being kT 2 = 21/2 n~2-"""""~
(21)
and ~ being the molecular diameter of the gas. Introducing eq. (20) in eq. (16), the following relationship between Ar and Re,.: is found: Ar =
Remf
32 457z
u
{18)
and
--
dP
5153
2
cos 2 ~
A
~m: ¢ -d +
+ 1.75 ~
COS2~
72
e3 .4.2 m:V (1 - - emf )
(22)
Re2:
F.raf ~) Now, by introducing the following parameters,
+ 1.75 ~
pu 2 .
(14)
MINIMUM FLUIDIZATIONVELOCITY For a bed of particles in incipient fluidization conditions, the drag force exerted by the fluid on the whole system of particles must be equal to the weight of the bed. This can be expressed by the following relationship: dP AP d-I = ct = T = (1 - graf)(Ps -- P)g.
(15)
From eqs (14) and (15), and by introducing dimensionless numbers Ar and Re, the following expression is obtained relating the pressure drop, the fluid velocity and diverse parameters of the bed at the condition of incipient fluidization:
72 K 1 = cos2 ~ K2
16 z z # /--2 4-5 c°s ~l~'mf~)'d~/~P 1
+ 1.75 ~ Re2:. gmf t~
(16)
From the molecular theory, Dushman (Roth, 1976) proposed the following correlation for the gas viscosity as a function of gas density and the mean free path of molecules: /~ = 0.499 Uavp2
(17)
(23c)
as well as the two well-known parameters defined by Wen and Yu (1966) as a function of particle shape factor and bed voidage in minimum fluidization conditions, C1 =
1
~s~
C2 = 1 - era: 3 2
~,mf ~)
(24a) (24b)
eq. (22) can be expressed as Ar -
cos2~k e3f~b2 "[- 7 ~ ( 1 - e . : ~
(23b)
1
C3 = e2S¢
Remf
ar =
45n 32 cos 2 ff
(23a)
Remf
1 + 1.75 C1Re~: +-K2C3 K1C2 Knv
(25)
Knv being the Knudsen number of the particle. By introducing a new constant
Z
Knv + - - 1 K2C3 K1C2
(26)
eq. (25) can be written as 1.75C1 Re2: + ZRemf - Ar -- 0.
(27)
5154
M.F. LLOPet al. 10 • • • •
C1.139kUTt ¢1.156pm d .222 ?m
•
d - ,l~l lu~
-
-
d .1800 wn
E¢I.(29-a)
0.8
i
v.~.af 0.6,
•"" ~ o ••I-;...~.
0.1
0.4.
•
Wen-Yu(19~) C3 = 5.5 (muno)
.....
Ca - 6 (~lwp)
0.2-
1 02
0
~ 0.4
~ 0.6
3.01 0.8
0.1
,
,
i
1
10
100
$
P (kPl)
Fig. 3. Bed voidage at incipient fluidization as a function of particle shape factor.
Fig. 4. Variation of u,: vs absolute pressure: experimental data and the prediction of eq. (29a).
From this quadratic equation, the following expression is finally obtained for the Reynolds number at incipient fluidization:
The agreement between these two equations and the experimental data [from this work and from Kusakabe et al. (1989)] can be seen in Figs 4 and 5. Equation (29a) predicts fairly well the data for round particles (glass beads and millet) of diverse sizes over a wide range of pressures (from 1 to 101 kPa) (Fig. 5). Figure 5 also shows a good agreement--with a certain scattering in some cases--between experimental data corresponding to two types of particles (silica sand and silica gel) and the values predicted by eq. (29b). Equation (29) also predicts fairly well the minimum fluidization velocity at higher pressures. Figure 6 shows the comparison between the prediction of eq. (29b) and the experimental values published by Llop et al. (1991) with silica sand of four different diameters fluidized by air at pressures ranging between 101 and 1200 kPa at room temperature. As can be observed, the experimental values are in close agreement with eq. (29b) for all the particle diameters. Finally, Fig. 7 shows again this agreement for the whole set of experimental data and the corresponding values predicted by eq. (29a) or eq. (29b), respectively. For all the data the standard deviation is equal to 3.7%. The fairly good agreement obtained is due to the fact that eqs (29a) and (29b) can predict the variation
Re,,: =
~
+ 1.75ClJ
3.5C1
The constants K1, K2, CI, C2 and Ca are a function of particle shape; the values of Table 1 are proposed for two arbitrary categories of solid particles, defined as a function of the degree of roundness for practical purposes. K~ and K2 have been calculated by using the values of angle 0 proposed by Ergun (1952) for spherical particles and by Casal et al. (1985) for irregular partides. The values of constants C1 and C2 are those already proposed by Lucas et al. (1986). Finally, the parameter C3 has been determinated from the relationship between the shape factor (sphericity) of the partides and bed voidage at incipient fiuidization (Fig. 3), by using the procedure proposed by Lucas et aL (1986). By applying these values, eq. (28) is transformed into the following two expressions for the two particle categories: Round particles (~b > 0.8): Rein:
=V(
0.909
"~2
C.\~Knp-~- 0-.0309/
+
-(Kn, °.9°9 + 0.0309/"
0.0357Ar11/2
(29a)
Sharp particles (0.5 < q~ ~ 0.8): 1.9
Table 1. Values of constants K1, K2, CI, C 2 and C3
2
- (Xnp 1'9.0492).
(29b)
Type of particles
Kt
K2
C)
C2
C3
Round Sharp
150 180
9.2 11.05
16 10
11 7.5
5.5 6
Fluidization at vacuum conditions 10'1~
~
I~1~=~(¢/e • d . 2131utl • d . 460~n • d . 72Bitm
~
(Ku~ et ~.. lg8~) d . 3901~
•
- -
Eq. (29-b)
5155 1 |
•
111iSwork
O
Kusakabe et al., 1989
rl
Llop lit al., 1991
J
1.5'
Q
E Q
•
1-
•
. E
•
0.5-
0.01
Ol
l"
,~
1oo
P (kPa)
O-f m'-
0
Fig. 5. Variation of u,,s vs absolute pressure: experimental data and the prediction of eq. (29b).
v
01.5
1
1T5
i 2
Urnf exp (m/s)
Fig. 7. Comparison between experimental results and the prediction of u,,I by eqs (29a) and (29b).
g -
0.1
¸
~lm ~ • •
0.01
lO
(Mopot ~., 1991) d .177.250 ltm d . 250-500 Wn d - 500-750 Ira1 d - 750-12(~ lun
16o
Eq. (294))
1~
p (kP.)
Fig. 6. Fitting of eq. (29b) to experimental data at high pressures.
of umy over a very wide range of operating pressures. In Fig. 8 the trends of these equations are plotted as well as those corresponding to the equations proposed by Wen and Yu (1966), Kusakabe et al. (1989) and Fletcher et al. (1993). This plot shows clearly that the equations proposed by Wen and Yu, Kusakabe et al. and Fletcher et al. cannot be applied over a wide range of pressures (in fact, these authors did not intend to do that), but only over a restricted range. The equation of Fletcher et al. really predicts an approximately constant value of umr as pressure
changes [Fig. 8(a)]; these authors found a good agreement between their correlation and their experimental results because they operated at very low vacuum, near to atmospheric pressure. As can be seen, in this zone their correlation approaches the curves of eq. (29a) and of the equation of Kusakabe et al. For large particles, the equation proposed by Kusakabe et al. predicts practically constant values of urn/from P ~ 10 kPa [Fig. 8(b)], while for small particles it is essentially constant from P ~ 100 kPa. Finally, the Wen-Yu equation gives good predictions at atmospheric pressures and at pressurized conditions (P >i 100 kPa), but it gives a constant value at low pressures (vacuum conditions). The different trend followed over the diverse ranges of pressure by the minimum fluidization velocity can be predicted by eqs (29a) and (29b) because these equations take into account the effect of inertial forces, which becomes more and more important as the Reynolds number increases, as well as the flow features at high vacuum conditions; however, the correlations proposed by the aforementioned authors do not consider the inertial forces (Kusakabe et al. and Fletcher et al. equations) or the vacuum effect (Wen-Yu equation). In Fig. 9 the trend of the equation proposed can be seen in a plot of [ u m i P ] vs P. The curves correspond to a fixed pressure drop, as this variable is essentially constant in a fluidized bed. Equation (29b) has been plotted for sharp particles (silica sand) of two different sizes. The equation covers the different regimes: slip flow, laminar flow, transition flow; the range of conditions over which each regime exists has been indicated
5156
M.F. LLOPet al.
0.05
glass beads
d.,~, 0.o4.
..........
Kul~
.....
Fielciler el ~d., 1993
el ~., 1989
o,oa
for each particle diameter. In the molecular flow range the equation cannot be applied, as in this regime a bed of particles cannot be fluidized; in this zone the equation would predict a slightly decreasing value of [UmfP] as P decreases, instead of a constant value which should correspond to molecular flow.
CONCLUSIONS
0.02 .............
o.o~-
~'o
~o
(a)
~ooo
P (kPa)
0.4'
d~.,wn
....... K ~ . ~ . . I ~ 0
d . 4 5 o pm
w~vo.l~s
r~,~.~ , ~ 0.a.
0.2.
•. "" oJ
/o
~o
(b}
looo
p (k~)
Fig. 8. Variation of u,,I vs pressure: trend of different equations,
// ,,/ /
equation have been deduced for round particles and sharp particles, respectively [eqs (29a) and (29b)]. The agreement between the values predicted by these equations and the experimental data obtained in this work and from Kusakabe et al. is very good. Moreover, the equations have also been applied to high pressure conditions, and again the agreement with experimental data was fairly good. It can be considered, therefore, that eqs (29a) and (29b) are generalized equations for the calculation of u,,f.
i ~V,,~
Acknowledgement--This work was supported by the Commissionfor Universitiesand Research of the Generalitat (government)of Catalonia and by the Comisi6n Interministerial de Ciencia y Technologia.
106
~,,,~,r. ~--7 //
....... ~,.~"~'~'~"~,(~-,~,~j, - 10, ~ i
s~ rom i / / ! .... .~/ /
~.,.p
(m.s-,.pa)
..,~_~__,~____. / / , /
102
~ ...............
~
.~wcaw f ~ ,
'
Little attention has been paid to fluidization at reduced pressures; however, as it has a potential application field a research effort should be made in this field. The experimental work carried out has shown that it is possible to achieve fluidization in high vacuum conditions (slip flow). The hydrodynamical behaviour observed is similar to that found at atmospheric pressure: a fluidization-defluidization cycle exists and bubbling fluidization is obtained. The minimum fluidization velocity is a function of pressure: it decreases as pressure increases and this influence is more importaut as particle size increases. Some of the equations available in the literature, those of Wen and Yu, Kusakabe et al. and Fletcher et al. for the prediction of u,,I, have been tested. The comparison with the experimental results has shown that they can only be applied over restricted ranges of pressure, as they do not consider the effect of vacuum (equations of Wen and Yu and Fletcher et al.) or the influence of inertial forces (equation of Kusakabe et al.). The relationship between pressure drop and gas flow rate has been studied at different pressures and in different flow regimes (molecular, slip, laminar, transition and turbulent flows). Finally, an equation has been obtained which allows the prediction of minimum fluidization velocity in vacuum fluidized beds; two expressions of this
/
/
...... d NOTATION
~©~
Ar ,0, ,
~
~
0
8
P (¢a)
Fig. 9. Relationship between (umiP) and pressure and range of the different flow regimes according to the proposed equations.
C1 C2 C3 d dh D
Archimedes number [ = d 3P(Ps parameter defined by eq. (24a) parameter defined by eq. (24b) parameter defined by eq. (23c) particle diameter, m hydraulic diameter, m duct diameter, m
-
P)g//~2]
Fluidization at vacuum conditions Oc
g k Kn Knp KI K2 l lc L m M P AP
Q R
Re Remy T U Uav Uc
ttrnf Z
channel diameter, m acceleration due to gravity, m/s 2 Boltzmann's constant, J/K K n u d s e n n u m b e r ( = 2/D) K n u d s e n n u m b e r of the particle ( = 2/d) parameter defined by eq. (23a) parameter defined by eq. (23b) bed height, m channel length, m bed height in fluidization conditions, m mass of a molecule, kg molar mass of gas, kg/mol pressure, Pa pressure drop, Pa volumetric flow rate, m3/s gas constant, J/K mol Reynolds n u m b e r ( = upd/p) Reynolds n u m b e r in m i n i m u m fluidization conditions ( = Umypd/l~) absolute temperature, K superficial gas velocity, m/s average arithmetic value of the molecular collision velocity, m/s gas velocity in an interstitial channel, m/s m i n i m u m fluidization velocity, m/s constant defined by eq. (26)
Greek letters ct parameter defined by eq. (3) e bed voidage ~ms bed voidage in m i n i m u m fluidization conditions 2 mean free path of molecules, m /~ gas viscosity, Pa s molecular diameter of the gas, m
p p~ ~b
5157
gas density, kg/m 3 particle density, kg/m 3 shape factor interstitial angle REFERENCES
Arnaldos, J., Casal, J., Lucas, A. and Puigjaner, L., 1985, Magnetically stabilized fluidization: modelling and application to mixtures. Powder Technol. 44, 57~2. Casal, J., Lucas, A. and Arnaldos, J., 1985, A new method for the determination of shape factor and particle density. Chem. Engng J. 30, 155-158. Ergun, S., 1952, Fluid flow through packed columns. Chem. Engng Prog. 48, 89-94. Fletcher, J. V., Deo, M. D. and Hanson, F. V., 1993, Fluidization of a multi-sized Group B sand at reduced pressure. Powder Technol. 76, 141 147. Kusakabe, K., Kuriyama, T. and Morooka, S., 1989, Fluidization of fine particles at reduced pressure. Powder Technol. 58, 125-130. Leuenberger, H., Luy, B. and Hirschfeld, P., 1991, Esperimenti con un nuovo sistema a letto fluido sottovuoto. Tecnol. Chim. 11, 100-105. Llop, M. F., Coil, T., Perales, J. F., Arnaldos, J., Puigjaner, L. and Casal, J., 1991, Hydrodynamical features of pressurized fluidization, In Rbcent Progrbs en Gbnie des Procbdbs: La Fluidisation (Edited by Laguerie, C. and Guigon, P.), Vol. ll, pp. 32 38. Lavoisier Industrie, Paris. Lucas, A., Arnaldos, J., Casal, J. and Puigjaner, L., 1986, Improved equation for the calculation of minimum fluidization velocity. Ind. Engng Chem. Process Des. Dev. 25, 426-429. Mitev, D. T. and Russev, D. R., 1990, Some hydrodynamics problems of fluidized bed at gas pressure different from atmospheric pressure. J. Appl. Chem. U.S.S.R. 63, 459-461. Roth, A., 1976, Vacuum Technology. North-Holland, Amsterdam. Wen, C. Y. and Yu, Y. H., 1966, Mechanics of fluidization. Chem. Engng Proy. Syrup. Ser. 62, 100-111.
Chemical Engineerinff Science, Vol, 51, No. 23, pp. 5149-5157, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00351-X 0009-2509/96 $15.00 + 0.00
F L U I D I Z A T I O N AT VACUUM CONDITIONS. A GENERALIZED E Q U A T I O N FOR THE P R E D I C T I O N OF M I N I M U M F L U I D I Z A T I O N VELOCITY M. F. L L O P Chemical Engineering Department, Universitat de Girona, Catalonia, Spain and F. MADRID, J. ARNALDOS* and J. CASAL* Chemical Engineering Department, Universitat Polit~cnica de Catalunya, Diagonal, 647, 08028-Barcelona, Catalonia, Spain (Received 4 January 1996; accepted 22 May 1996)
Abstract--The hydrodynamical behaviour of fluidized beds at reduced pressures is studied. The minimum fluidization velocity is strongly influenced by pressure, decreasing as pressure increases. The well-known equation of Wen and Yu and two equations proposed for the calculation of u,,I at reduced pressures are tested; the comparison with experimental data shows that they can be applied only over restricted pressure ranges. The relationship between flow rate, pressure and pressure drop at conditions ranging from high vacuum to high pressure is analysed for the different flow regimes (molecular, slip, laminar, transition and turbulent). A generalized equation is found which predicts fairly well the value of Uml at vacuum conditions, atmospheric pressure and high pressures. Copyright © 1996 Elsevier Science Ltd Keywords: Fluidization, vacuum, slip flow, minimum fluidization velocity, hydrodynamics.
INTRODUCTION A number of authors have studied the effect of pressure on the hydrodynamical behaviour of fluidized beds. However, most of them have operated at high pressure, i.e. at pressures higher than atmospheric pressure. Very few articles have been published on fluidization at reduced pressures [Kawamura and Suezawa, 1961 and Huschka and Popp, 1967 (cited by Kusakabe et al., 1989); Mitev and Russev, 1990; Leuenberger et al., 1991; Fletcher et al., 1993]. The literature on the behavior of fluidization at pressures less than atmospheric has been reviewed by Kusakabe et al. (1989) and Fletcher et al. (1993). The relatively small number of researchers interested in fluidization at reduced pressure is due to the fact that low-pressure fluidized beds seemed not to be very interesting from the point of view of practical applications. However, this type of fluidization has a promising potential field in the fine chemicals industry. Fine chemicals are often dried at the end of the manufacturating process, and as they are usually particulate materials, fluidized-bed drying seems to be a suitable procedure. But a problem arises from the fact that in a number of cases the moisture which must be removed by the drying operation is constituted of an organic solvent. Therefore, drying with hot air will *Corresponding authors.
easily lead to the formation of a mixture within the flammability limits; in these conditions, the risk of explosion is very high. This is the reason why fluidized-bed drying has been gradually put aside in many processes. Low-pressure fluidized-bed drying is therefore very interesting for such situations. Fluidized-bed features give optimum conditions for the treatment of granular materials, and low-pressure operation allows the control of the resulting atmosphere, giving the possibility of maintaining it outside the flammability conditions and ensuring a completely safe operation. Furthermore, low-pressure fluidization leads to the use of much lower operating temperatures; this is a very important aspect if a thermolabile substance must be dried, as the risk of degradation is much lower. Nevertheless, this type of fluidization is not well known, and in order to use it, a better understanding of its behaviour is required. Low-pressure fluidization follows flow patterns which can be very different from those usually found in fluidized beds, as the hydrodynamical regime is no longer laminar flow, but is replaced by slip flow. In these conditions, the usual equations for the calculation of minimum fluidization velocity do not apply any more; instead, specific equations for these types of flow must be applied (Kusakabe et al., 1989). Therefore, obtaining a generalized equation, which could be applied to any conditions from high-pressure
5149
M. F. LLOPet
5150
fluidization to vacuum fluidization, is very interesting not only because of its practical applications, but also because it would represent a complete understanding of this aspect of fluidization phenomena in its whole domain of existence. In this paper, such an expression is developed, and its validity is tested through the application to experimental data. The main hydrodynamical features of vacuum fluidization are also discussed. EXPERIMENTALSET-UP A diagram of the experimental unit can be seen in Fig. 1. The fluidization column, made of glass, had an inside diameter of 50 mm and a length of 600 mm. The distributor was a stainless-steel plate, 2 mm thick, tied by two Teflon flanges. The orifices (1 mm diameter) were located according to a hexagonal arrangement, with a density of 18 orifices/era 2. Below the plate a mesh (stainless steel) was placed in order to avoid the weeping of particles when the flow was stopped. A calming section packed with Raschig rings was provided under the distributor.
al.
The fluid used was air at 298 _+ 1 K and at absolute pressures ranging between 1 and 101 kPa. The flow at vacuum conditions was obtained by using a rotary vacuum pump. The vacuum in the fluidized bed was controlled with a membrane valve located between the pump suction and the top of the column, and with a pin valve connected to the atmosphere. The vacuum inside the column was measured by two systems. On the one hand, there was a vacuometer installed in the aspiration tube. On the other hand, a tap above the fluidized bed was coupled to a differential manometer, measuring the differential pressure with respect to the atmospheric pressure. The temperature inside the bed was measured with a thermocouple; a control loop connected to an electrical heating system allowed the operation at a constant temperature. Two other taps were located at 100 and 200 mm above the distributor, respectively; both were connected to differential manometers in order to measure the pressure drop through the whole bed and through a given height of the fluidized bed (see Fig. 1). The gas flow rate was measured at atmospheric pressure with a
14 I Flowmeters 2 Flow controlvalves 3 Air heater 4 Temper:~ure oontroler 5 111ermocouple 6 Fluidiz~tionoolumn 7 Calming section 8 Distributor 9 Solids feeder 10 Solids outtet 11 Mercury manometer 12 Water/Anilinemanometers 13 V~nzrn indicator
13
i-
14 Pressure
mntrol
15 Vacuumpump -9
_5 z
Ljj 11
12
12
2:
2
2
1
1
1
Fig. 1. Schematic diagram of the experimental unit.
valves
Fluidization at vacuum conditions system of three rotameters; pin valves were installed downstream of each rotameter to control the gas flow. The solid particles fluidized were silica sand with a density of 2650kg/m 3, a shape factor of 0.6 and an average diameter of 213, 450, 728 and 1460 lam, and millet with a density of 1600 kg/m 3, a shape factor of 0.9 and an average diameter of 1600 p.m. The experimental procedure followed to determine the value of urn: (minimum fluidization velocity) at the different operating conditions was to achieve complete fluidization of the bed and afterwards gradually decrease the gas flow rate. Minimum fluidization velocity was therefore measured at decreasing velocity. Thus, the possible error associated with any variation of bed voidage at incipient fluidization-which could exist when operating at increasing gas velocity--was avoided. The values obtained were reproducible and no difficulties were found in the measurements.
EXPERIMENTAL RESULTS
Bubbling fluidization, essentially similar to that found at atmospheric pressure, was obtained with all the particles used. Bubbling started from the onset of fluidization and increased with gas flow. Following the same behavior that is found at higher pressures, different pressure-drop-gas-velocity relationships were found according to the (increasing/decreasing) trend in the fluid velocity variation. Starting from a fixed bed, the pressure drop for the fixed bed at increasing velocity was higher, for a given bed mass, than the pressure drop measured at decreasing velocities; thus, typical fluidization--defluidization cycles were obtained. This was clearly due to the higher compaction of the initial fixed bed, caused by some vibrations or even by the introduction of the solid in the column. Instead, the fixed bed obtained at decreasing velocity from the fluidized condition always had a lower bed voidage, associated with the looser structure of the particle system. The more compact condition of the initial fixed bed was also shown by the overpressure required to break down the 'structure' linking the particles; this effect was not found with millet beds (round particles), but only with silica sand (sharp particles). The same behavior was obtained at atmospheric pressure. Therefore, minimum fluidization velocity had to be determined at decreasing gas velocity in order to obtain significant and reproducible values. Defluidization curves were obtained at different pressures. The effect of pressure was clear: as pressure decreased, the minimum fluidization velocity increased. This trend can be seen for five different particle systems in Fig. 2, where the variation of urn/has been plotted as a function of absolute pressure. For all the solids the minimum fluidization velocity decreases following an approximately linear trend as pressure increases. The slope of the curves varies slightly with the particle weight; the change in Urn: as pressure increases is greater for the heaviest particles.
5151
10, •
sik:a r~ll(I 1d-213 ~u11)
•
llllcaun¢l ((jm1~
•
,~ ..= (d-'~O ~n)
•
m,~t(.-~O00 ~)
•
=,VYCa=mind(d,-~ .rn)
,'...
• ,
=,
•
•
11n1)
•
nm
•
•
nm
0.1, •
0.01 •
•
000e
I•
100
P (kPe)
Fig. 2. Variation of urn:as a function of absolute pressure for five solids.
As a whole, the experimental results obtained showed a behavior similar to that obtained in fluidization at atmospheric pressure, with the aforementioned variation in minimum fluidization velocity. FLOW RATE, PRESSURE AND PRESSURE DROP IN PARTICLE BEDS AT DIFFERENT REGIMES
The behavior of a gas flowing through a pipe or through a bed of particles can change significantly as pressure is reduced, due to the increase in the mean free path of molecules. Depending on the pressure, the gas can be in a molecular state, in a viscous state or in an intermediate state (Roth, 1976). As a consequence, different flow regimes can exist, depending on the operating conditions (pressure and diameter of the duct). The range over which each flow regime is encountered can be specified as a function of the Knudsen number, defined as the ratio between the mean free path of molecules and the diameter of the duct: 2 Kn = - - . D
(1)
The different flow regimes which can exist are the following ones: Molecular flow: Kn>>1. The gas is in a molecular or rarefied state. The mean free path is much larger than the diameter of the duct; collisions are mostly with the wall rather than between gas molecules, and the concept of viscosity has no meaning. A fluidized bed could not be obtained in these conditions. Intermediate or slip flow: Kn ~ 1. The mean free path of the molecules is similar to the diameter of the duct; the flow of gas is established both by molecular phenomena and by the viscosity. A fluidized bed can operate at this flow regime.
M. F. LLOP et al.
5152
Laminar flow: Kn<<1. The gas is in a viscous state; gas flow is governed by the viscosity and the HagenPoiseuille law applies. As the Reynolds number increases, the transition flow is reached and, at high values of Re, the flow is turbulent. Fluidized beds usually operate in these three regimes (laminar, transition and turbulent). The flow rate of a gas in a pipe of a given diameter and with a fixed value of pressure drop as a driving force can be a function of pressure, i.e. a function of the mean free path of molecules, depending on the flow regime. At very low pressures the flow is molecular. The throughput of gas [QP] does not depend on pressure and has a constant value; from a value reached at 2 = 1.57D, as the mean free path increases the throughput of gas also decreases asymptotically toward the aforementioned constant value. As pressure increases, the mean free path decreases from 1.57D; viscous effects appear and the conductance starts to increase gradually with pressure. This is the range of conditions over which slip flow exists. Finally, in the laminar regime the throughput of gas increases linearly with pressure, as does the conductance. The transition from one flow regime to another will depend on the gas (the mean free path will change from one gas to another), on the pressure and on the diameter of the channel. For air, PD < 0.01 Pa m PD > 0.8 Pa m
molecular flow laminar flow.
The relationships between pressure, pressure drop and flow rate in the different regimes are analysed in detail in the next paragraphs.
As in real low-pressure fluidization processes this extremely low pressure cannot be reached in practice, it may be assumed that for practical purposes ~ = 0.8. In these conditions, eq. (2) becomes:
X/
M
dl~'
Slip and viscous refime In the viscous flow regime (Kn<
~zD~P dP QP = 128~ die"
(5)
Therefore, in the intermediate or slip flow regime
(Kn .~ 1) the relationship between flow rate and pressure can be obtained by adding eqs (4) and (5):
? QP =
/2nRT dP rtD4pdP D3 X[ M dlc t- 128~ die"
dP
dlc
Q
~ + - - nD~
?D~
This equation can be used to determine the pressure drop undergone by a gas flowing through a particulate bed in this flow regime. By considering the flow through the bed to be equivalent to the flow through a bundle of interstitial channels, the following values (Amaldos et al., 1985) can be taken: D2n (8)
Q = Uc--
The flow of a gas through a tube of uniform circular cross-section in the Knudsen regime (Kn>> 1) can be described by the following expression (Roth, 1976):
eP
= 6 X[ M
l+L24Lgfj
u~ =
TJ
(9)
I
(10)
COS I~
2eddp Dc=dh
(2)
The term in brackets takes a value which approaches 1 as the pressure approaches zero. However, this value decreases as the pressure increases; at values of absolute pressure near 7.6 x 10- 6 mmHg, this term reaches a value that is practically constant and equal to 0.8:
FMI'/2D~P l
'+L FJM ] mZDcPII
3(l-e)'
(11)
Equation (8) gives the flow rate as a function of interstitial velocity. Equation (9) gives the fluid velocity in the interstitial channels as a function of bed voidage and bed tortuosity. Equation (10) gives the length of these channels as a function of bed height and tortuosity, and eq. (11) gives the value of the hydraulic diameter of bed interstitial channels. By introducing these values in eq. (7), and taking into account the law of perfect gases (RT/M = P/p), the following expression is obtained: dP dl
,+ 1. 4L j -;-j
U
~
= 0.8 (P ~> 7.6 x 10-6 mmHg).
U 8 COS I~
l~ = - -
7
[-M 71/2 D~PI dlc "
(7)
V- -P-r 128
Low pressures
rM
(6)
This expression can be rearranged to give:
4
I+L J -7- [dP
(4)
(3)
16 2 e2ckd ~-~cos ~k( - T- e~ ) 4 ~ - ~
COS2~083t~2d2 72
p(1-e) 2
(12)
Fluidization at vacuum conditions This equation therefore allows the calculation of the pressure drop undergone by a gas flowing through a particulate bed over the range of conditions covering Knudsen flow, slip flow and viscous flow. Turbulent regime An expression covering a wider range of flow regimes can be obtained by using the equation of Burke and Plummer, which gives the relationship between pressure drop and velocity in a fluid flowing through a particulate bed in turbulent regime (Ergun, 1952):
(1 - - ~) 2 = 1.75-w-;--7,, pu . d---1 e~ q~a
dP
(13)
A general equation covering all the conditions and regimes ranging from Knudsen flow to turbulent flow can be obtained by adding the contributions of slip flow, viscous flow and turbulent flow; thus, adding the right-hand terms of eqs (12) and (13) the following expression is obtained:
where 2 is the mean free path of molecules, Uav = 8~kT mTt k m
dl
16 211/ e2~bd ~ cos2~b e3~b2d2 ~ c o s r(-f'-~--e) qTtpP'4 72 # ( l - - e ) 2
R M'
(19)
By introducing eqs (18) and (19) in eq. (17), and taking into account the law of perfect gases, we obtain the following relationship: p=0.998
2~P2~
2~P),
(20)
2 being kT 2 = 21/2 n~2-"""""~
(21)
and ~ being the molecular diameter of the gas. Introducing eq. (20) in eq. (16), the following relationship between Ar and Re,.: is found: Ar =
Remf
32 457z
u
{18)
and
--
dP
5153
2
cos 2 ~
A
~m: ¢ -d +
+ 1.75 ~
COS2~
72
e3 .4.2 m:V (1 - - emf )
(22)
Re2:
F.raf ~) Now, by introducing the following parameters,
+ 1.75 ~
pu 2 .
(14)
MINIMUM FLUIDIZATIONVELOCITY For a bed of particles in incipient fluidization conditions, the drag force exerted by the fluid on the whole system of particles must be equal to the weight of the bed. This can be expressed by the following relationship: dP AP d-I = ct = T = (1 - graf)(Ps -- P)g.
(15)
From eqs (14) and (15), and by introducing dimensionless numbers Ar and Re, the following expression is obtained relating the pressure drop, the fluid velocity and diverse parameters of the bed at the condition of incipient fluidization:
72 K 1 = cos2 ~ K2
16 z z # /--2 4-5 c°s ~l~'mf~)'d~/~P 1
+ 1.75 ~ Re2:. gmf t~
(16)
From the molecular theory, Dushman (Roth, 1976) proposed the following correlation for the gas viscosity as a function of gas density and the mean free path of molecules: /~ = 0.499 Uavp2
(17)
(23c)
as well as the two well-known parameters defined by Wen and Yu (1966) as a function of particle shape factor and bed voidage in minimum fluidization conditions, C1 =
1
~s~
C2 = 1 - era: 3 2
~,mf ~)
(24a) (24b)
eq. (22) can be expressed as Ar -
cos2~k e3f~b2 "[- 7 ~ ( 1 - e . : ~
(23b)
1
C3 = e2S¢
Remf
ar =
45n 32 cos 2 ff
(23a)
Remf
1 + 1.75 C1Re~: +-K2C3 K1C2 Knv
(25)
Knv being the Knudsen number of the particle. By introducing a new constant
Z
Knv + - - 1 K2C3 K1C2
(26)
eq. (25) can be written as 1.75C1 Re2: + ZRemf - Ar -- 0.
(27)
5154
M.F. LLOPet al. 10 • • • •
C1.139kUTt ¢1.156pm d .222 ?m
•
d - ,l~l lu~
-
-
d .1800 wn
E¢I.(29-a)
0.8
i
v.~.af 0.6,
•"" ~ o ••I-;...~.
0.1
0.4.
•
Wen-Yu(19~) C3 = 5.5 (muno)
.....
Ca - 6 (~lwp)
0.2-
1 02
0
~ 0.4
~ 0.6
3.01 0.8
0.1
,
,
i
1
10
100
$
P (kPl)
Fig. 3. Bed voidage at incipient fluidization as a function of particle shape factor.
Fig. 4. Variation of u,: vs absolute pressure: experimental data and the prediction of eq. (29a).
From this quadratic equation, the following expression is finally obtained for the Reynolds number at incipient fluidization:
The agreement between these two equations and the experimental data [from this work and from Kusakabe et al. (1989)] can be seen in Figs 4 and 5. Equation (29a) predicts fairly well the data for round particles (glass beads and millet) of diverse sizes over a wide range of pressures (from 1 to 101 kPa) (Fig. 5). Figure 5 also shows a good agreement--with a certain scattering in some cases--between experimental data corresponding to two types of particles (silica sand and silica gel) and the values predicted by eq. (29b). Equation (29) also predicts fairly well the minimum fluidization velocity at higher pressures. Figure 6 shows the comparison between the prediction of eq. (29b) and the experimental values published by Llop et al. (1991) with silica sand of four different diameters fluidized by air at pressures ranging between 101 and 1200 kPa at room temperature. As can be observed, the experimental values are in close agreement with eq. (29b) for all the particle diameters. Finally, Fig. 7 shows again this agreement for the whole set of experimental data and the corresponding values predicted by eq. (29a) or eq. (29b), respectively. For all the data the standard deviation is equal to 3.7%. The fairly good agreement obtained is due to the fact that eqs (29a) and (29b) can predict the variation
Re,,: =
~
+ 1.75ClJ
3.5C1
The constants K1, K2, CI, C2 and Ca are a function of particle shape; the values of Table 1 are proposed for two arbitrary categories of solid particles, defined as a function of the degree of roundness for practical purposes. K~ and K2 have been calculated by using the values of angle 0 proposed by Ergun (1952) for spherical particles and by Casal et al. (1985) for irregular partides. The values of constants C1 and C2 are those already proposed by Lucas et al. (1986). Finally, the parameter C3 has been determinated from the relationship between the shape factor (sphericity) of the partides and bed voidage at incipient fiuidization (Fig. 3), by using the procedure proposed by Lucas et aL (1986). By applying these values, eq. (28) is transformed into the following two expressions for the two particle categories: Round particles (~b > 0.8): Rein:
=V(
0.909
"~2
C.\~Knp-~- 0-.0309/
+
-(Kn, °.9°9 + 0.0309/"
0.0357Ar11/2
(29a)
Sharp particles (0.5 < q~ ~ 0.8): 1.9
Table 1. Values of constants K1, K2, CI, C 2 and C3
2
- (Xnp 1'9.0492).
(29b)
Type of particles
Kt
K2
C)
C2
C3
Round Sharp
150 180
9.2 11.05
16 10
11 7.5
5.5 6
Fluidization at vacuum conditions 10'1~
~
I~1~=~(¢/e • d . 2131utl • d . 460~n • d . 72Bitm
~
(Ku~ et ~.. lg8~) d . 3901~
•
- -
Eq. (29-b)
5155 1 |
•
111iSwork
O
Kusakabe et al., 1989
rl
Llop lit al., 1991
J
1.5'
Q
E Q
•
1-
•
. E
•
0.5-
0.01
Ol
l"
,~
1oo
P (kPa)
O-f m'-
0
Fig. 5. Variation of u,,s vs absolute pressure: experimental data and the prediction of eq. (29b).
v
01.5
1
1T5
i 2
Urnf exp (m/s)
Fig. 7. Comparison between experimental results and the prediction of u,,I by eqs (29a) and (29b).
g -
0.1
¸
~lm ~ • •
0.01
lO
(Mopot ~., 1991) d .177.250 ltm d . 250-500 Wn d - 500-750 Ira1 d - 750-12(~ lun
16o
Eq. (294))
1~
p (kP.)
Fig. 6. Fitting of eq. (29b) to experimental data at high pressures.
of umy over a very wide range of operating pressures. In Fig. 8 the trends of these equations are plotted as well as those corresponding to the equations proposed by Wen and Yu (1966), Kusakabe et al. (1989) and Fletcher et al. (1993). This plot shows clearly that the equations proposed by Wen and Yu, Kusakabe et al. and Fletcher et al. cannot be applied over a wide range of pressures (in fact, these authors did not intend to do that), but only over a restricted range. The equation of Fletcher et al. really predicts an approximately constant value of umr as pressure
changes [Fig. 8(a)]; these authors found a good agreement between their correlation and their experimental results because they operated at very low vacuum, near to atmospheric pressure. As can be seen, in this zone their correlation approaches the curves of eq. (29a) and of the equation of Kusakabe et al. For large particles, the equation proposed by Kusakabe et al. predicts practically constant values of urn/from P ~ 10 kPa [Fig. 8(b)], while for small particles it is essentially constant from P ~ 100 kPa. Finally, the Wen-Yu equation gives good predictions at atmospheric pressures and at pressurized conditions (P >i 100 kPa), but it gives a constant value at low pressures (vacuum conditions). The different trend followed over the diverse ranges of pressure by the minimum fluidization velocity can be predicted by eqs (29a) and (29b) because these equations take into account the effect of inertial forces, which becomes more and more important as the Reynolds number increases, as well as the flow features at high vacuum conditions; however, the correlations proposed by the aforementioned authors do not consider the inertial forces (Kusakabe et al. and Fletcher et al. equations) or the vacuum effect (Wen-Yu equation). In Fig. 9 the trend of the equation proposed can be seen in a plot of [ u m i P ] vs P. The curves correspond to a fixed pressure drop, as this variable is essentially constant in a fluidized bed. Equation (29b) has been plotted for sharp particles (silica sand) of two different sizes. The equation covers the different regimes: slip flow, laminar flow, transition flow; the range of conditions over which each regime exists has been indicated
5156
M.F. LLOPet al.
0.05
glass beads
d.,~, 0.o4.
..........
Kul~
.....
Fielciler el ~d., 1993
el ~., 1989
o,oa
for each particle diameter. In the molecular flow range the equation cannot be applied, as in this regime a bed of particles cannot be fluidized; in this zone the equation would predict a slightly decreasing value of [UmfP] as P decreases, instead of a constant value which should correspond to molecular flow.
CONCLUSIONS
0.02 .............
o.o~-
~'o
~o
(a)
~ooo
P (kPa)
0.4'
d~.,wn
....... K ~ . ~ . . I ~ 0
d . 4 5 o pm
w~vo.l~s
r~,~.~ , ~ 0.a.
0.2.
•. "" oJ
/o
~o
(b}
looo
p (k~)
Fig. 8. Variation of u,,I vs pressure: trend of different equations,
// ,,/ /
equation have been deduced for round particles and sharp particles, respectively [eqs (29a) and (29b)]. The agreement between the values predicted by these equations and the experimental data obtained in this work and from Kusakabe et al. is very good. Moreover, the equations have also been applied to high pressure conditions, and again the agreement with experimental data was fairly good. It can be considered, therefore, that eqs (29a) and (29b) are generalized equations for the calculation of u,,f.
i ~V,,~
Acknowledgement--This work was supported by the Commissionfor Universitiesand Research of the Generalitat (government)of Catalonia and by the Comisi6n Interministerial de Ciencia y Technologia.
106
~,,,~,r. ~--7 //
....... ~,.~"~'~'~"~,(~-,~,~j, - 10, ~ i
s~ rom i / / ! .... .~/ /
~.,.p
(m.s-,.pa)
..,~_~__,~____. / / , /
102
~ ...............
~
.~wcaw f ~ ,
'
Little attention has been paid to fluidization at reduced pressures; however, as it has a potential application field a research effort should be made in this field. The experimental work carried out has shown that it is possible to achieve fluidization in high vacuum conditions (slip flow). The hydrodynamical behaviour observed is similar to that found at atmospheric pressure: a fluidization-defluidization cycle exists and bubbling fluidization is obtained. The minimum fluidization velocity is a function of pressure: it decreases as pressure increases and this influence is more importaut as particle size increases. Some of the equations available in the literature, those of Wen and Yu, Kusakabe et al. and Fletcher et al. for the prediction of u,,I, have been tested. The comparison with the experimental results has shown that they can only be applied over restricted ranges of pressure, as they do not consider the effect of vacuum (equations of Wen and Yu and Fletcher et al.) or the influence of inertial forces (equation of Kusakabe et al.). The relationship between pressure drop and gas flow rate has been studied at different pressures and in different flow regimes (molecular, slip, laminar, transition and turbulent flows). Finally, an equation has been obtained which allows the prediction of minimum fluidization velocity in vacuum fluidized beds; two expressions of this
/
/
...... d NOTATION
~©~
Ar ,0, ,
~
~
0
8
P (¢a)
Fig. 9. Relationship between (umiP) and pressure and range of the different flow regimes according to the proposed equations.
C1 C2 C3 d dh D
Archimedes number [ = d 3P(Ps parameter defined by eq. (24a) parameter defined by eq. (24b) parameter defined by eq. (23c) particle diameter, m hydraulic diameter, m duct diameter, m
-
P)g//~2]
Fluidization at vacuum conditions Oc
g k Kn Knp KI K2 l lc L m M P AP
Q R
Re Remy T U Uav Uc
ttrnf Z
channel diameter, m acceleration due to gravity, m/s 2 Boltzmann's constant, J/K K n u d s e n n u m b e r ( = 2/D) K n u d s e n n u m b e r of the particle ( = 2/d) parameter defined by eq. (23a) parameter defined by eq. (23b) bed height, m channel length, m bed height in fluidization conditions, m mass of a molecule, kg molar mass of gas, kg/mol pressure, Pa pressure drop, Pa volumetric flow rate, m3/s gas constant, J/K mol Reynolds n u m b e r ( = upd/p) Reynolds n u m b e r in m i n i m u m fluidization conditions ( = Umypd/l~) absolute temperature, K superficial gas velocity, m/s average arithmetic value of the molecular collision velocity, m/s gas velocity in an interstitial channel, m/s m i n i m u m fluidization velocity, m/s constant defined by eq. (26)
Greek letters ct parameter defined by eq. (3) e bed voidage ~ms bed voidage in m i n i m u m fluidization conditions 2 mean free path of molecules, m /~ gas viscosity, Pa s molecular diameter of the gas, m
p p~ ~b
5157
gas density, kg/m 3 particle density, kg/m 3 shape factor interstitial angle REFERENCES
Arnaldos, J., Casal, J., Lucas, A. and Puigjaner, L., 1985, Magnetically stabilized fluidization: modelling and application to mixtures. Powder Technol. 44, 57~2. Casal, J., Lucas, A. and Arnaldos, J., 1985, A new method for the determination of shape factor and particle density. Chem. Engng J. 30, 155-158. Ergun, S., 1952, Fluid flow through packed columns. Chem. Engng Prog. 48, 89-94. Fletcher, J. V., Deo, M. D. and Hanson, F. V., 1993, Fluidization of a multi-sized Group B sand at reduced pressure. Powder Technol. 76, 141 147. Kusakabe, K., Kuriyama, T. and Morooka, S., 1989, Fluidization of fine particles at reduced pressure. Powder Technol. 58, 125-130. Leuenberger, H., Luy, B. and Hirschfeld, P., 1991, Esperimenti con un nuovo sistema a letto fluido sottovuoto. Tecnol. Chim. 11, 100-105. Llop, M. F., Coil, T., Perales, J. F., Arnaldos, J., Puigjaner, L. and Casal, J., 1991, Hydrodynamical features of pressurized fluidization, In Rbcent Progrbs en Gbnie des Procbdbs: La Fluidisation (Edited by Laguerie, C. and Guigon, P.), Vol. ll, pp. 32 38. Lavoisier Industrie, Paris. Lucas, A., Arnaldos, J., Casal, J. and Puigjaner, L., 1986, Improved equation for the calculation of minimum fluidization velocity. Ind. Engng Chem. Process Des. Dev. 25, 426-429. Mitev, D. T. and Russev, D. R., 1990, Some hydrodynamics problems of fluidized bed at gas pressure different from atmospheric pressure. J. Appl. Chem. U.S.S.R. 63, 459-461. Roth, A., 1976, Vacuum Technology. North-Holland, Amsterdam. Wen, C. Y. and Yu, Y. H., 1966, Mechanics of fluidization. Chem. Engng Proy. Syrup. Ser. 62, 100-111.