idization process at a selected range of particle Reynolds number. The remarkable feature of this approach is to pro- vide a simple, yet practical method for ...
Chemical Engineering Science 58 (2003) 2035 – 2051
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The voidage function and e"ective drag force for #uidized beds Y. T. Makkawi, P. C. Wright∗ Department of Chemical Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK Received 14 June 2002; received in revised form 12 November 2002; accepted 13 December 2002
Abstract Here, an experimental investigation on the e"ective drag force in a conventional #uidized bed is presented. Two beds of di"erent particle size distribution belonging to group B and group B/D powders were #uidized in air in a 13:8 cm diameter column. The drag force on a particle has been calculated based on the measurement of particle velocity and concentration during pulse gas tests, using twin-plane electrical capacitance tomography. The validity of the voidage function “correction function”, (1 − s )n , for the reliable estimation of the e"ective drag force has been investigated. The parameter n shows substantial dependence on the relative particle Reynolds number (Rep∗ ), and the spatial variation of the e"ective static and hydrodynamic forces. It is also illustrated that, a simple correlation for the e"ective drag coe;cient as function of the particle Reynolds number (Rep ), expressed implicitly in terms of the interstitial gas velocity, can serve in estimating the e"ective drag force in a real #uidization process. Analysis shows that, the calculated drag force is comparable to the particle weight, which enables a better understanding of the particle dynamics, and the degree of spatial segregation in a multi-sized particle bed mixture. The analogy presented in this paper could be extended to obtain a generalized correlation for the e"ective drag coe;cient in a #uidized bed in terms of Rep and the particle physical properties. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Process tomography; Gas–solid #uidization; Correction function; E"ective drag force; Solid fraction measurement
1. Introduction The prediction of #uidization behavior requires an a priori knowledge of the e"ective forces acting on a moving particle. These forces are mainly the gravity, buoyancy, bubble lifting, particle contact forces, electrostatic forces and gas drag force. Apart from the gravity force, the prediction of the other forces and thus their resultant e"ect is di;cult. From a micro-scale point of view, the hydrodynamics surrounding a suspension of particles are highly complex, and several forces such as collision, cohesion and electrostatics may exist between particles. In addition, in dense non-uniformly distributed particle clusters, there is a steep variation in the interstitial gas velocity due to the restricted gas #ow passage (void fraction). Thus, resulting in a greater shear stress, which in turn tends to reduce the e"ective gas drag force. When numerically evaluating the e"ective drag force, these additional forces, and the possible resulting modiAcations in ∗ Corresponding author. Present address: Department of Chemical and Process Engineering, University of She;eld, Mappin Street, She;eld S1 3JD, UK. Tel.:+44-114-2227577; fax:+44-114-2227501. E-mail address: p.c.wright@she;eld.ac.uk (P. C. Wright).
the #ow Aeld, are usually incorporated in the e"ective drag force equation by the so-called “void function” or “correction function”, f(s ), such that Fd = Fdo f(s )m ;
(1)
where Fd is the drag force on a particle when taking into consideration the e"ect of neighboring particles, and Fdo is the drag force exerted on the same particle when falling freely under the e"ect of the gas force and gravity force only. The exponent m varies between 1 and 2 depending on the particle Reynolds number based on the interstitial gas velocity (Rep ), as will be discussed later. The function f(s ), which we shall refer to as the correction function is deAned as follows (Richardson & Zaki, 1954; Maude & Whitmore, 1958; Di Felice, 1994): f(s ) = (1 − s )−x ;
(2)
where s is the solid volume fraction and x is a parameter function of the particle properties and the particle Reynolds number based on the relative gas velocity (Rep∗ ). The origin of the correction function arises from the study reported by Richardson and Zaki (1954) for sedimentation and solid–liquid #uidization, where they described the
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(03)00054-X
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Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
steady-state bed expansion/contraction behavior in terms of the particle terminal fall velocity, ut and the gas superAcial velocity, U , such that U = ut f(s ):
(3)
Similarly, according to the generalized theory of sedimentation reported by Maude and Whitmore (1958), it was shown that the settling velocity of a single particle ut -o , relative to that in a concentrated suspension ut , or a bed, can be expressed in terms of the correction function such that ut = ut -o f(s ):
(4)
The correction function is mainly introduced to take into consideration the uncertainties in the interstitial gas velocity distribution, or the e"ects of inter-particle forces existing in a particle cluster. In sedimentation and particle suspension in #uids, this simple hypothesis has found great acceptance and shown reasonable agreement with experimental data. However, the value of the exponent x appearing in the correction function has been the subject of continuous discussion, especially among #uidization researchers (Di Felice, 1994; MostouA & Chaouki, 1999; Helland, Occelli, & Tadrist, 2000a). In many numerical studies, this exponent was assumed to be constant or a function of the particle physical properties. As admitted by Di Felice (1994), there is no reason for believing that the exponent x is the same for a varied bed concentration ranging from dense to dilute #uidization. In other words, it is hard to believe that the correction function remains the same for a wide range of gas velocity, even when considering the spatial variation of the hydrodynamic and static forces over the entire bed cross-section. This consideration prompted re-examination of the applicability of this approach in calculating the e"ective drag force in a #uidized bed. To our knowledge, no experimental information is available about the numerical value of the e"ective drag force, and its spatial distribution in a #uidized bed. In most of the numerical simulations on #uidized bed hydrodynamics, the gas drag force is quantiAed in terms of the correction function, or by using unreliable correlation expressions for the drag coe;cient. Therefore, we attempt to provide a descriptive, as well as numerical evaluation on the distribution of the e"ective drag exerted on a single particle during a #uidization process at a selected range of particle Reynolds number. The remarkable feature of this approach is to provide a simple, yet practical method for predicting the e"ective drag force once the superAcial gas velocity and solid fraction is given. For this purpose, pulsed gas and continuous #uidization experiments were conducted to measure the spatial variation of solid distribution. 2. Motivation of this study Many of the theoretical and experimental evaluations of drag force in a suspension were conducted during the period
from the mid-1950s to the late 1970s (Richardson & Zaki, 1954; Maude & Whitmore, 1958; Richardson & Jeronimo, 1979). Meanwhile, most of the recent investigations have focused on the theoretical validation of the correction function by examining the e"ect of the variation of the correction function exponent on the particle dynamics (e.g. Helland et al., 2000a; Helland, Occelli, & Tadrist, 2000b), or by comparison with experimental data (Gibilaro, Di Felice, & Walderam, 1985; Di Felice, 1994). However, experimental conArmations on the values of the correction function exponent in a typical #uidization process are rare in the literature. MostouA and Chaouki (1999) recently reported a similar study on liquid–solid #uidization. The particle velocity was measured using a radioactive particle tracing technique, and a correlation expression for the correction function exponent was proposed as a function of Archimedes number (Ar) and Reynolds number. The vast majority of researchers in gas– solid #uidization, especially those who are concerned with numerical simulation, adopted a constant value for the correction function exponent, or assumed it to be a function of the particle terminal velocity and incipient #uidization conditions (Richardson & Meikle, 1961; Mikami, Kamiya, & Horio, 1998; Helland et al., 2000b). A number of experimental studies have also reported that values of the correction function exponent can be greater than the theoretical ones. For example, Godard and Richardson (1968) found values of the exponent x between 4.6 and 8.9 for various materials #uidized with air. Meanwhile, Geldart and Wong (1984, 1985) reported values of x as high as 60 for various materials #uidized with di"erent gases. This great discrepancy makes it interesting to conduct further investigations. Therefore, in this study, we decided to examine the drag force exerted on particles in a real #uidization process, and attempt to determine an accurate correlation for the e"ective drag force. We also found it to be of interest to evaluate the correction function, and provide a correlation expression for the correction function exponent as a function of Reynolds number. For this purpose, a non-invasive imaging technique known as electrical capacitance tomography (ECT) was used to measure the solid concentration variation during the #uidization process. This technique is relatively new, but recently become a successful tool in quantitatively understanding the hydrodynamics in #uidized beds (Makkawi & Wright, 2002a, b; Malcus, Chaplin, & Pugsley, 2000). 3. Theory In gas–solid #uidization processes, it is important to know the relationship between the gas velocity and the drag force. This drag force is the main acting source of the rapid bed expansion and extensive particle mixing within the bed. Since, in #uidized bed systems, the actual gas velocity distribution and pressure proAle are di;cult to measure, we are led to seek other ways of getting drag force versus velocity
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Fdo =
2 1 2 g ug ACdo :
(5)
Now, if we introduce other particles in to the #ow Aeld, we would expect changes in the hydrodynamics surrounding the particles as a result of a change in the interstitial gas velocity and particle–particle interactions. This change can be incorporated into Eq. (5) by modifying Cdo using the correction function given in Eq. (2) such that Cd = Cdo f(s )m = Cdo [(1 − s )x ]m ;
(6)
where Cdo is the e"ective drag coe;cient when the #ow Aeld is modiAed by the presence of other particles. Now Eq. (5) can be written for a #uidized bed as follows: Fd = d2p f (ug − up )2 [(1 − s )x ]m Cdo (7) 8 or Fd = d2p f (ug − up )2 (1 − s )−n Cdo = Fd 8 = Fdo (1 − s )−n :
(8)
The parameter n is a result of multiplying x by m, and (ug − up ) is the relative velocity between the gas and particle. Before we proceed further, we must make it clear that we decided to evaluate the drag force based on the formulation given in Eq. (8), where the two parameters x and m are incorporated, such that, only one exponent appears with the correction function. Previous experimental investigations have reported a minimum value of x = 2 for spheres in turbulent #ow, and a maximum of x = 10 for rough spheres in creeping #ow, the exponent m varies between 1 and 2 depending on Rep , such that m=1 for laminar interstitial #ow (Rep ¡ 1); m = 2 for turbulent interstitial #ow (Rep ¿ 1000), and an intermediate value for the transition regime (Maude & Whitmore, 1958; Helland et al., 2002). Therefore, according to the ranges given for both x and m, the values of n should practically fall between 2 and 20 for rough or smooth spheres in a wide range of #ow covering the two extremes of creeping and turbulent #ow. The drag coe;cient for an isolated single particle, Cdo , is well documented, and several correlations are available in the literature with small numerical di"erences. In this study, we will employ the Dallavalle equation (Dallavalle, 1948): 2 4:8 Cdo = 0:63 + ∗0:5 : (9) Rep
100 % Cumulative mass fraction (-)
information. This is usually accomplished by using experimental data to construct a correlation function. The force exerted by a #uid on a single suspended particle is well documented in the literature. It is expressed in terms of the characteristic area, A, kinetic energy per unit volume, 12 ug2 , and a dimensionless quantity known as the drag coe;cient Cdo , often referred to as a friction factor. The drag coe;cient is usually expressed in terms of the particle Reynolds number (Rep ) after both A and ug are known. Accordingly, the drag force on a single particle, Fdo , is deAned as follows (Bird, Stewart, & Lightfoot, 1960):
90
2037
mixture 1- dp=0.35 mm mixture 2- dp=0.70 mm
80 70 60 50 40 30 20 10 0 100
1000 Particle size (micron) Fig. 1. Particle size distribution.
This equation is assumed to provide an adequate representation for the full range of practical Rep∗ . 4. Experimental set-up Experiments were conducted in a cold conventional #uidized bed. The #uidization column was constructed from cast acrylic, 1:5 m high and 0:138 m in diameter. A perforated PVC plate with a total free area of 2.5% was used as a gas distributor. Two di"erent spherical glass ballotini mixtures of mean particle diameter 350 and 700 m, both of average density of 2650 kg=m3 were used as the bed material (powder groups B and B/D, respectively). In the next sections, we shall refer to these as mixture 1 and mixture 2, respectively. The cumulative particle distribution for the two mixtures obtained by sieving is shown in Fig. 1. The static bed height was 0:138 m (Hst = Db ). The #uidization air at ambient temperature was introduced to the base of the column from a main compressor after passing through a rotameter and a gas Alter. The sensors were specially designed to slide freely along the column giving the chance for capturing cross-sectional images at di"erent levels. All sensors were connected to a data acquisition module (DAM 2000) and a computer. The detailed description of the experimental arrangement is shown in Fig. 2. 4.1. The ECT sensors and data recording The ECT sensors are used to obtain images of the distribution of two di"erent permittivity materials in a containing vessel (in this case: air and glass ballotini). It does this at a high imaging speed of 100 Hz, with a typical image format of 32 × 32 pixels. The value of each pixel represents the average solid volume fraction (or void) in a grid of 0:43 × 0:43 × 10−2 m2 cross-section and 0:038 m height. The ECT system is compromised of
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Fig. 2. Experimental set-up.
twin adjacent planes each of 0:038 m in axial length, each plane contains eight electrodes and driven axial electrode guards. Before commencing data recording, the system was calibrated for two extreme cases; when the column is empty (Alled with low-permittivity material, air) and when Alled with the particles tightly packed and in a static condition (packed with the high-permittivity material, glass ballotini). The calibration data were stored and loaded for each running test. For online data capturing and control, special software based on the linear back projection algorithm (LBP) was used (ECT32 software). For data image improvement, another piece of oUine software, based on iterative LBP was used (PTL IU2000). Technical details about the system hardware and software are available upon request. Similar applications for the system are also reported in Makkawi and Wright (2001, 2002 a–c), Makkawi et al. (2002), Dyakowski, Edwards, Xie, and Willaims (1997) and Hallow and Fasching (1993). 4.2. Experimental procedure 4.2.1. Pulse gas experiment The edge of the lower ECT sensors was placed exactly above the static bed height. The main inlet valve (upstream valve) was kept closed and the downstream valve before the rotameter was left open at a controlled level such that the inlet gas #ow rate is known, then a one-step sudden opening of the upstream valve introduces the gas pulse and the changes in the particle concentration in the freeboard was
recorded. This process takes approximately less than 5 s and each pulse test was repeated at least 20 times to ensure reasonable level of data representation. In analyzing the data, only the recorded information between the moment of particles’ detection by the lower sensor and the moment the upper sensor shows a decrease in solid fraction is considered. SuperAcial gas velocity in the range of U 0.3– 0:65 m=s was considered for mixture 1, and in the range of U 0.65 –1 m=s for mixture 2. For values of Umf and Umb , please refer to Table 1. During the experiment, visual observation was used to ensure that the maximum bed height does not exceed the sensor limiting area. A schematic diagram for the sensor placement and pixel measuring illustration is shown in Fig. 3. 4.2.2. Fluidization experiments Additional experiments were conducted after placing the ECT sensor at the bottom of the bed. The recorded images during a continuous #uidization process were considered to represent the cross-sectional solid distribution at 0:019 m (lower sensor) and 0:057 m (upper sensor) above the distributor level. Recorded Ales at di"erent gas velocity were stored, a #uidization velocity in the range of U 0.25 –1:0 m=s was considered for mixture 1 and in the range of U 0.6 –1:5 m=s for mixture 2. Each recorded #uidization Ale consisted of 8000 images corresponding to an 80 s experimental span at a data capture rate of 100 Hz. Table 1 summarizes the experimental conditions for both the pulse gas and continuous #uidization experiments.
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Table 1 Summary of experimental conditions
Experimental unit
Operating conditions
Column Distributor Fluidizing gas Static bed height, Hst ECT system
Dia: = 0:138 m; height = 1:5 m, material: cast acrylic Perforated PVC; 150 holes each of 1:5 mm diameter Air at ambient conditions 0:138 m Twin plane, eight electrodes per plane, each of 0:038 m length
Particle size range Particle density, p Particle material Geldart group Min. #uidization velocity, Umf Min. bubbling velocity, Umb Fluidization experiment span Pulse gas velocity, U
Mixture 1 m 300 –400 dp = 1=( xi =dpi ) = 350 m 2650 kg=m3 Glass ballotini B 0:21 m=s 0:21 m=s 80 s, 8000 images 0:3 ⇒ 0:65 m=s
Mixture 2 500 –1000 m dp = 1=( xi =dpi ) = 700 m 2650 kg=m3 Glass ballotini B/D 0:3 m=s 0:63 m=s 80 s, 8000 images 0:65 ⇒ 1:0 m=s
Fig. 3. Illustration on the experimental set-up and the surface bed expansion during pulse gas experiments.
4.3. Method of data analysis The method of data analysis was developed with the main focus on measuring the e"ective drag force; this force was then used in deriving a correlation expression for the effective drag force and the correction function parameter n. The analysis is based on the following methodology and assumptions: • We deAne the particle cluster as a group of particles moving in a conAned volume, the boundary of this volume being deAned from the ECT pixel resolution. As described in the experimental part, each one of the ECT images is composed of 1028 pixels, with each pixel representing the average solid fraction in a volume of 0:7 × 10−6 m3 . This cluster is described schematically in Fig. 4. • Newton’s second law of motion governs the motion of a particle cluster as a result of a forced #uid, such that duc (10) m = Ftotal ; dt
where duc =dt is the average acceleration of a particle cluster of mass m moving with a velocity uc . Ftotal is the sum of the forces acting on a particle cluster. This total force is assumed to be a combination of: (1) gravity force; (2) gas drag force; (3) inter-particle contact forces; (4) electrostatic forces; and (5) bubble lifting force. Now if we assume that the e"ective drag force is the resultant vector quantity of the last three forces and in a hydrodynamic equilibrium with the gravity and acceleration forces, then Eq. (10) can be written for the e"ective drag force as follows: duc + mg: (11) Fd-e" = m dt • The e"ective drag force Fd-e" is the net force required to drag a particle cluster of mass m in the vertical direction only; therefore, the lateral movement of particles during the #uidization process are not discussed in the contents of this study. • The single particle within a cluster up is assumed to have the same velocity as that of the whole cluster (i.e. up =uc ).
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electrodes
pixel cross-section 0.43 x 0.34 x 10-2 m2 solid volume fraction = εs pixel height = 0.038 m pixel
Fig. 4. ECT pixel resolution and particle cluster dimensions.
0.4
solid volume fraction (-)
0.35 0.3
central pixel signal upper sensor lower sensor
0.25 0.2 0.15
τ2
0.1
τ1
0.05 0 38.95
39
39.05
39.1
39.15
39.2
39.25
39.3
39.35
time (s) Fig. 5. Typical tomography measurement illustrating the calculation method for the average acceleration and velocity of a particle cluster during a pulse gas experiment.
• The interstitial gas velocity at a speciAc point inside the bed is assumed to be a function of the measured local solid fraction and the superAcial gas velocity, such that ug =
U : (1 − s )
(12)
The absolute average particle cluster acceleration, duc =dt, is calculated graphically as illustrated in Fig. 5. The quantity duc is simply calculated from the time delay, 1 and the axial length of the sensor, which is 0:038 m ( 1 is the time taken between the Arst detection of particles passing the lower pixel, and the corresponding signal detected by the upper pixel). The quantity dt = 2, is the time taken from the Arst detection of particles in their upward journey as detected by the lower pixel, until the moment the upper pixel shows a
maximum solid concentration before the particles start their downward journey. Accordingly, the approximate particle acceleration is given by duc 0:038= 1 : = dt 2
(13)
The total mass of the particles, m, dragged by the gas was estimated from the tomographic measurements, as illustrated in Fig. 4. In a mathematical form, this is expressed as follows: m = Vc (s1 + s2 )s
(14)
where Vc = 0:7 × 10−6 m3 is the volume of a single pixel, s1 and s2 are the maximum solid volume fraction in the
Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
lower and upper adjacent pixels, before the bed collapses and particles start their downward #ow. Finally, the experimentally determined drag force on a cluster can be written for a single particle by dividing Fd-e" deAned in Eq. (11) by the total number of particles Np dragged upward by the gas force, duc 1 Fd-exp = m (15) − ms g ; Np dt where Np is deAned as follows: m Np = : (=6)d3p p
(16)
Note that, dp applied here is the mean particle diameter of the mixture.
5. Results and discussion 5.1. General observations during pulse experiments In all tests, the pulse-gas velocity was kept within a low range U . 2:5Umf to ensure that particle movements at the bed surface are within the range of the sensor area. From our visual observation, it was conArmed that during the pulse experiment, the lateral movement of particles is negligible; therefore, we assume that the sensor signal #uctuation in a pixel is basically as a result of only the vertical, upward and downward motions of particles within the same pixel. As detected by the tomography signals, as soon as the gas pulse was introduced, the lower sensor detects the solid passage Arst followed by the upper sensor (see Fig. 3). This was always true, even at the walls. It is also worth mentioning that during the pulse experiment, it was observed that a few particles tended to stick to the walls, apparently as a result of electrostatic forces, either particle–particle or particle–wall. For mixture 1, at the bed center, the particles usually move in a relatively dilute phase, so we believe that much of the electrostatic forces, but not all, would be lost as a result of the particle separation during their motion, at low gas velocity (U . 1:5Umf ), we observed particles (in dilute phase) ejecting from the center bed surface, mainly as a result of the bubble eruption (not as a result of gas drag). At the walls, the particles move vertically in the form of dense clusters. When increasing the gas velocity (1:5Umf ¡ U ¡ 2:5Umf ), a relatively uniform #uidization regime was observed, with no distinct variations between the dilute core and the dense annular region. With mixture 2, there were minor local differences in the particle dynamic soon after commencement of bubbling (U &1:25Umf ). At relatively higher gas velocity (1:5Umf ¡U ¡2:5Umf ), the bed started to #uidize uniformly at the surface, somehow similar to a slugging behavior with a noticeable increase in particles’ cluster velocity near the walls. Particles are seen ejecting from the wall region and sinking through the center region.
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From our early understanding of general particle motion in a #uidized bed, and from measurement of the solid fraction #uctuation across the bed, we believe that the motion of particles at the wall possesses di"erent hydrodynamic behavior to those moving at the bed center, and each region falls under the e"ect of di"erent hydrodynamic and static forces. Generally, the particles moving near the walls are uniformly distributed, and move in the form of packed particle clusters, while the particles at the core are subject to heterogeneous behavior as a result of bubble motion. Hence, during the pulse gas experiments, we decided to collect the data from both the wall and the bed center to cover the two extremes of bed behavior. 5.2. Particle cluster velocity at the bed surface Fig. 6a and b shows the particle cluster velocity at the freeboard as a function of the superAcial gas velocity. As expected, the vertical cluster velocity at the walls is lower than that at the center. This is mainly due to the di"erence in particle concentration, bubble lifting and the e"ective friction with the walls. Electrostatic forces and inter-particle contact forces also play some role in this regard. However, Fig. 6b shows some increase in the cluster velocity near the wall when #uidizing mixture 2 at high gas velocity. At the center, particles are projected into the freeboard as a result of bubble eruption. The projected particles are less compact when compared to the dense clusters at the wall; therefore, they are easily carried, and move with higher velocity. Detailed descriptions of particle ejection mechanisms at the bed surface can be found in Chen and Saxena (1978). When increasing the gas velocity, the bed starts to expand, and the particles in the bed center start to move in the form of dense projected clusters. It is also noted that, increasing the gas velocity decreases the di"erences between the wall and center upward particles’ velocity. 5.3. The correction function In order to obtain numerical values for the parameter n, and verify its dependence on Reynolds number, the experimentally determined drag force and the data obtained for the particle velocity and solid fraction were employed as follows: Fd-exp =
2 d f (ug − up )2 (1 − s )−n Cdo 8 p
= Fdo (1 − s )−n :
(17)
Thus, n=
log(Fdo =Fd-exp ) : log(1 − s )
(18)
Note that Fd-exp employed here is estimated from Eq. (15). As mentioned earlier, the particle velocity, up , is assumed
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1.4
wall
1.2
center
uc (m/s)
1 0.8 0.6 0.4 0.2 0 0.25
0.35
0.45
0.55
0.65
U (m/s)
(a)
1.4 1.2
wall center
Fig. 5
uc (m/s)
1 0.8 0.6 0.4 0.2 0 0.55
0.65
0.75
0.85
0.95
1.05
U (m/s)
(b)
Fig. 6. Particle clusters velocity measured at the bed surface: (a) mixture 1; (b) mixture 2.
to be equal to the measured cluster velocity, uc . Fig. 7a and b show the variation of the index n as function of Rep∗ determined from measurements taken at the walls and center region. The experimental values for n are compared with predictions obtained from the correlation reported by Di Felice (1994) as follows: n = 93:7 − 0:65e−# ; where 1:5 − log(Rep∗ ) 2 : #= 2
(19)
(20)
Note that Rep∗ is evaluated at the relative gas velocity as follows: (ug − us )dp ; (21) Rep∗ = $ where ug is deAned as shown in Eq. (12). From Fig. 7a and b, it is evident that n decreases towards unity as Rep∗ increases. We believe that ideally, n should be equal to 1 for the case of fully suspended and uniformly distributed particles without any e"ect of inter-particle forces. At low Rep∗ , a high value of n at the wall is expected in order to correct for the e"ect of inter-particle forces on the e"ective drag force,
while at the bed center, a high value for n is required to correct for the dominant bubble lift force. For the range of Rep∗ shown in Fig. 7, for both mixtures, the value of n at the wall falls within the range of 1– 6. On the other hand, values of n for the center region fall within the range of 1– 45 for mixture 1 and within 1–14 for mixture 2. This obviously shows a clear deviation from most of the reported literature, especially at low Rep∗ , at the same time, proves the dependence of n on the nature of hydrodynamic forces as they vary when moving from the wall towards the center region. It is worth noting that a few studies have reported values of n higher than 45 (Geldart & Wong, 1984). However, for the range of practical Rep∗ applied in this study (10 ¡ Rep∗ ¡ 40 for particle size 350 m, and 40 ¡ Rep∗ ¡ 120 for particle size 700 m), our experimentally determined values of n falls within a limit comparable with the majority of theoretical and experimental studies (Richardson & Zaki, 1954; Godard & Richardson, 1968; Massimilla, Donsi, & Zucchini, 1972; Di Felice, 1994). Given the fact that the center is the region where bubble motion/eruption takes place, it appears that at low Reynolds number, particle motion is mainly as a result of the bubble lifting or eruption at the bed surface. Tomography measurements during pulse experiments with mixture 1 for
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center wall ncenter=49.5Re*p-0.88
40
nwall =6.98Re*p-0.37 Eq. 19 (Di Felice, 1994)
n
30
20
10
0 0
10
20
30
40
Rep* (-)
(a)
14 wall center nwall=12.2exp(-0.021Re*p)
12
ncenter=7.0exp(-0.041Re*p) Eq. 19 (Di Felice, 1994)
10
n
8
6
4
Fig. 6
2
0 0
20
40
(b)
60
80
100
120
Rep* (-)
Fig. 7. Variation of the correction function parameter n with Reynolds number (Re∗ ): (a) mixture 1; (b) mixture 2.
U ¡ 1:5Umf have shown that the particle clusters dragged upward at the bed center of the freeboard are always below 0.2 solid concentrations, and the cluster velocity is higher than the interstitial gas velocity. This thus conArms that at low interstitial velocity, the particle motion at the center of the freeboard is mainly as a result of bubble eruption. As described by Helland et al. (2000a), highly heterogeneous behavior such as that existing at the bed center should increase the value of n. According to their numerical simulation, it was evident that, when increasing the value from
n=1 to 6, a clear increase in the heterogeneous behavior was observed with frequent voids and less packed clusters, exactly mimicking the same behavior we observed at the bed center when operating at low gas velocity. It is also worth noting two additional points. First, the data obtained at low Rep∗ is relatively scattered. Owing to the pulse experiment observation, the cluster velocity and particle concentration varies from one gas pulse to another (most probably due to the variation of bubble characteristics), that is why the pulse experiment was repeated at least 20 times to obtain the best
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possible data representation. Secondly, the values of n for both the wall and center almost coincide at high Rep∗ , most probably due to the uniform distribution of the hydrodynamic forces over the freeboard cross-section as noted from visual observations. However, at low Rep∗ , the values of n depict a non-linear behavior and deviate considerably from the wall data. As mentioned before, this is strong evidence for the sensitivity of the parameter n towards the variation of Rep∗ (i.e. explicitly the variation of solid concentration or implicitly, the variation of acting hydrodynamic forces). This is also in good agreement with a note made by Helland et al. (2000a) when numerically analyzing the e"ect of the variation of n on the bed heterogeneity, it was concluded that increasing the value of n increases the non-linearity in the bed behavior. Finally, based on the experimental measurement, a correlation equation for the parameter n as a function of Rep∗ was obtained via regression as follows: For particle size 350 m; 0 ¡ Rep∗ ¡ 40: wall region:
nwall = 6:98 Rep∗−0:37 ;
bed center:
ncenter = 49:5 Rep∗−0:78 ;
r 2 = 0:55;
(22)
r 2 = 0:66: (23)
For particle size 700 m; 0 ¡ Rep∗ ¡ 120: wall region: r = 0:92; bed center:
(24) ncenter = −3:1 + 9:55 exp(−0:007 Rep∗ );
r 2 = 0:95:
Numerous experimental and theoretical investigations suggested a correlation expression for the drag coe;cient such that, Cd = f(Rep ). The plot of Cd against Rep expressed in terms of the interstitial gas velocity is shown in Fig. 8a and b. Generally, the curve follows the same trend of a falling sphere as reported in many textbooks. For high Rep , the drag coe;cient is approximately constant, while high values with steep variations are noticed at low Rep . Based on our calculation, the empirical correlation for the e"ective drag coe;cient can be expressed as a function of Rep as follows: For particle size 350 m; 5 ¡ Rep ¡ 40: Cd = 3:0 + 303e−0:135 Rep ;
(25)
The values of n obtained from the Di Felice correlation (1994) [Eq. (19)] hardly changes within the range of Rep∗ shown in Fig. 7. However, the predictions are to some extent comparable to the data obtained from the wall measurements and the central bed data at high Rep∗ . This indicates that the accuracy of Eq. (19) of Di Felice decreases with decreasing solid concentration. 5.4. Development of correlation expression for the e7ective drag coe8cient To provide a simple correlation expression for the e"ective drag coe;cient for a single particle in a #uidized bed, it is necessary to relate it to a measurable variable. A correlation expression in terms of the relative gas velocity requires the knowledge of both the interstitial gas velocity (ug = U=(1 − s )), and the average particle velocity. Therefore, instead of relating the drag coe;cient to the relative gas velocity (ug − up ), as in the case of the correction function, we propose here a correlation as a function of the interstitial gas velocity only (i.e. implicitly in terms of Rep = ug d=$), such that only the superAcial gas velocity and solid concentrations are required. The drag force acting on a single
r 2 = 0:89:
(28)
For particle size 700 m; 40 ¡ Re ¡ 140: Cd = 4:5 + 795e−0:07 Rep ;
nwall = 7:0 exp(−0:014 Rep∗ );
2
particle can be expressed theoretically with the particle velocity term omitted as follows: (26) Fd-e" = d2p f (ug )2 Cd : 8 Now, if we equate Eq. (26) with the experimentally measured drag force deAned in Eq. (15), the e"ective drag coe;cient can be expressed as follows: 8 Cd = Fd-exp : (27) (d2p f ug2 )
r 2 = 0:87:
(29)
These two equations reasonably cover the range of Rep applied in our continuous #uidization experiments. The R-square values conArm the good Atting of data. 5.5. Distribution of the e7ective drag force In order to calculate the distribution of the e"ective drag force and verify the applicability of Eqs. (28) and (29), a series of #uidization experiments were conducted, as described earlier, to collect the temporal solid fraction distribution at a selected level above the distributor. The e"ective drag force on a single particle was obtained by using the analytical expression for the drag coe;cient, Cd , given in Eqs. (28) and (29). The interstitial gas velocity was determined using the superAcial gas velocity, and the measured solid fraction as discussed earlier. All #uidized particles used in this study have the same average density; therefore, it is the larger particles that require higher drag force to be fully supported. Considering the particle size range and their corresponding weight for mixture 1, the minimum required drag force to support the smallest particle size (that is ∼ 0:3 mm) should be ∼ 0:395 N and the minimum to support the largest size (that is ∼ 0:4 mm) should be ∼ 0:845 N. For mixture 2, the minimum required drag force to support the smallest particle size (that is ∼ 0:5 mm) should be ∼ 1:7 N and the minimum to support the largest size (that is ∼ 1:0 mm) should be ∼ 13:3 N. This is clearly in the absence of any other restraining force. Figs. 9 and 10 show the temporal e"ective drag force proAle for the two mixtures estimated based on the measured solid distribution
Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
2045
140
wall center Cd=3.0+303exp(-0.135Rep)
120 100
Cd (-)
80 60 40 20 0
10
20
30
40
Rep (-)
(a) 35
wall center
30
C
=4.5+795exp(-0.07Rep)
d-eff
25
Cd
20
15
10
Fig. 7 5
0 40
60
80
(b)
100
120
140
Rep (-)
Fig. 8. E"ective drag coe;cient as function of particle Reynolds number: (a) mixture 1; (b) mixture 2.
at a selected moment of bed expansion (bubble evolution). The bed expansion structure varies from one bubble passage to another; however, the results given here show the general trends, which are assumed reasonable to represent the drag force distribution. Fig. 9a shows the drag force distribution for mixture 1 when the bed expands at a gas velocity U ∼ 2:2Umf . As expected, the particles at the wall are under a low e"ective drag force, just enough to lift the small particles, therefore, during the bed expansion, the heavy particles tend to fall down or sink at the wall region. It is also interesting to note that in Fig. 9a, at some stage during the bed expansion, the e"ective drag force at the bed center (core region) reaches more than 0:98 N. This is high enough to drag upward the whole particle size range, which obviously results in a considerable bed expansion. For mixture 2, when
the bed is operated at excess gas velocity U ∼ 2:6Umf (see Fig. 9b), the drag force is low when compared to the range of particle weights, this indicates a limited bed expansion with a large proportion of the heavy particles remaining stagnant at the bottom during the bubble passage (particle segregation). The annular region dictates a form of packed structure that remains una"ected by the bubble motion. It was also noted that, particle segregation, limited bed expansion and considerable gas by pass are the main characteristics of this powder mixture. Generally, the high e"ective drag force at the core region is a result of contribution of bubble lifting, where the bubbles have the tendency to move in the bed center as they rise towards the top. Fig. 10a and b show the e"ective drag force distribution when #uidizing at higher gas velocity of U ∼ 3:3Umf for
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Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051 1.05 t=10 ms t=40 1.00 t=70 t=100
Fd-eff (µ N)
0.95
0.90
0.85
0.80
0.75 -1
-0.8
-0.6
(a)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
dimensionless coordinate, r/R (-) 1.00 t=10 ms t=30 t=50
0.95 0.90
t=70
Fd-eff (µ N)
0.85 0.80 0.75 0.70
Fig. 8
0.65 0.60 -1
(b)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
dimensionless coordinate, r/R (-)
Fig. 9. E"ective drag force proAle during bed expansion measured at 0:057 m above the distributor level: (a) mixture 1 at U = 0:6 m=s; (b) mixture 2 at U = 0:67 m=s.
mixture 1 and U ∼ 3:6Umf for mixture 2. Generally, it can be concluded that, the dominant bubble lift force at the bed center starts to diminish when increasing the gas velocity and the drag force exhibits cross-sectional uniformity with a considerable improvement in the wall #uidization. Fig. 10a demonstrates a typical turbulent #uidization for mixture 1 where the e"ective drag force is su;cient enough to suspend the complete particle size range with no clear heterogeneity
at the bed center. Fig. 10b, for mixture 2, shows a di"erent drag force proAle. Here, it appears that during the bed expansion the particles in the annular region are #uidized in piston-like slugs (wall slugs) as indicated by the negligible drag force variation during the expansion process, while the force distribution at the core is relatively #at and decreases as the bed expands, most probably, due to the decrease in solid concentration. It is also interesting to note that, as a
Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
2047
5.7 t=10 ms
5.6
t=40 5.5
t=70 t=100
Fd-eff (µ N)
5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 -1
-0.8
-0.6
-0.4
(a)
-0.2
0
0.2
0.4
0.6
0.8
1
dimensionless coordinate, r/R (-) 7.0
6.5
Fd-eff (µ N)
6.0
5.5
5.0
t=10 ms t=40
4.5
t=70 t=100 4.0 -1
(b)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
dimensionless coordinate, r/R (-)
Fig. 10. E"ective drag force proAle during bed expansion measured at 0:057 m above the distributor level: (a) mixture 1 at U = 0:75 m=s; (b) mixture 2 at U = 1:0 m=s.
result of considerable by-pass, the bubble lifting contribution on the e"ective drag force diminishes. When comparing the particle weight with the available e"ective drag force, it is evident that segregation is more likely for mixture 2. In fact, these numerical evaluations are in good agreement with our visual observations, where it was noticed that mixture 2 exhibits some sort of sluggish behavior. In this regime, heavy particles were segregated at the bottom, the particles at the wall region were seen moving in packed clusters, where, once they reached the top they were projected and thrown back to the center where they tended to sink as the bed re-expanded. The observations noted above are conArmed with additional analysis of cross-sectional ECT measurement of
solid distribution. The time-averaged mesh diagrams are shown in Fig. 11a–d for the two cases of high and low gas velocity. Fig. 11a and b indicates that at low gas velocity, the particles are distributed such that the bubble activity region is concentrated at the bed center while the wall region remains almost stagnant as conArmed by the measurement of the standard deviation of solid fraction #uctuation. We are assuming a direct relationship between the standard deviation of solid fraction #uctuation and the e"ective drag force. At higher gas velocity, Fig. 11c and d conArms the dominance of the e"ective drag force over the bubble lifting force as indicated by the relatively uniform particle dynamic movement over the bed cross-section.
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Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
Fig. 11. Tomographic measurements of solid distribution and dynamic particles movement for mixture 1, dimensions 1 and 2 are pixels coordinates: (a) solid fraction at U = 0:47 m=s; (b) standard deviation at U = 0:47 m=s; (c) solid fraction at U = 0:9 m=s; (d) standard deviation at U = 0:9 m=s.
A typical observation is also noticed with mixture 2 as shown in Fig. 12a–d. However, for mixture 2 (Fig. 12d), the bed uniformity at high gas velocity is mainly related to the semi-slugging behavior. The standard deviation at higher gas velocity provides concrete evidence on the conclusion reached here in regard to the uniformity of drag force distribution as the gas velocity increases. These observations also provide additional evidence that the proposed relationship for Cd given in Eqs. (28) and (29) provides a simple yet accurate estimate of the e"ective drag force distribution for the range of Rep considered. 6. Conclusion The dependence of the void function on the Reynolds number (Rep∗ ) has been investigated experimentally by measuring the e"ective drag force on a bed of two di"erent particle mixtures belonging to Geldart groups B and B/D. For the range of Rep∗ considered in this study, the values of n at
the wall have been found to vary within a limited range between 1 and 6 for both particle mixtures considered. At the bed center, the value of n varies in a wide range between 1 and 45 for mixture 1 and between 1 and 14 for mixture 2. However, for a practical range of Rep∗ usually considered in conventional #uidization processes (above simple bubbling up to turbulent #uidization), the range of n over the entire bed cross-section and for both particle mixtures varies approximately between 1 and 10. Many of the recent numerical simulations on #uidized beds adopted a constant value for n, which is assumed to be valid over a wide range of particle Reynolds number, and over the entire bed cross-section. At this point, we must make it clear that such an assumption is far from accurate when applied to a #uidized bed. The void function, which is mainly introduced to take into account the e"ect of the neighboring particles on a single particle moving within in a cluster, must take into consideration the local heterogeneous behavior of the bed. Generally, it is reasonable to believe that the particles near the walls move vertically in uniform
Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
2049
Fig. 12. Tomographic measurements of solid distribution and dynamic particles movement for mixture 2, dimensions 1 and 2 are pixels coordinates: (a) solid fraction at U = 0:7 m=s; (b) standard deviation at U = 0:7 m=s; (c) solid fraction at U = 1:1 m=s; (d) standard deviation at U = 1:1 m=s.
packed clusters as a result of dominant gas drag force. On the other hand, particles at the bed center are under two major forces: dominant gas drag force at high Rep∗ and dominant bubble lift force at low Rep∗ . Analytical expressions for the e"ective drag coe;cient as a function of Rep (implicitly in terms of interstitial gas velocity, ug ) has been proposed for two di"erent particle mixtures. These equations can serve in estimating the e"ective drag force once the superAcial gas velocity and solid fraction are known. The applicability of these equations has been successfully tested in a real #uidization process. Further experimental investigations considering di"erent particle mixtures of di"erent physical properties are recommended to generalize the correlation expressions. Based on the visual observations and the e"ective drag force measurement given here, we made an attempt to provide a simple pictorial description of the particles #ow structure at the bed surface for the two particle mixtures as shown in Fig. 13. It is also concluded from this study that, measurement of the e"ective drag force can be extended to predict the degree of particle mixing/segregation behavior in a
multi-sized or multi-density powder mixture in a #uidized bed. Notation A Cd Cdo dp Db Fd Fdo Fd-exp Fd-e" Ftotal g
particle characteristic area, m2 e"ective drag coe;cient for a particle in a suspended cluster drag coe;cient for a single isolated suspended particle particle diameter, m bed, column diameter, m drag force on a particle in a suspended cluster, N drag force on a single isolated particle, N experimentally determined e"ective drag force calculated as per Eq. (15), N e"ective drag force calculated, N sum of forces acting on a particle in a suspended cluster, N gravitational acceleration, m=s2
2050
Y. T. Makkawi, P. C. Wright / Chemical Engineering Science 58 (2003) 2035 – 2051
solid
(a)
effective gas drag force solid
solid
(b)
effective gas drag force
effective gas drag force
(c)
Fig. 13. Proposed solid distributions: (a) mixtures 1 and 2 at low gas velocity (U ¡ 1:5Umf ); (b) mixture 1 at high gas velocity (U ¿ 2:5Umf ); (c) mixture 2 at high gas velocity (U ¿ 2Umf ).
Hst m n Np r R Rep Rep∗ t U Umf uc ug up ut ; ut -o Vc x
static bed height, m mass of a particles cluster deAned in Eq. (14), kg correction function exponent, deAned in Eqs. (8) and (18) number of particles in a cluster, deAned in Eq. (16) radial coordinate, m column, bed radius, m particle Reynolds number in terms of interstitial velocity (=g ug dp =$g ) particle Reynolds number in terms of relative velocity (=g (ug − uc )dp =$g ) time, s superAcial gas velocity, m/s superAcial gas velocity at minimum #uidization, m/s particle cluster velocity, m/s interstitial gas velocity, =U=(1 − s ), m/s particle velocity, m/s terminal velocity of single isolated particle and in a suspension respectively, m/s volume of a tomography pixel, m3 correction function parameter deAned in Eq. (2)
Greek letters % # mf s $ g s 1 2
parameter deAned in Eq. (1) parameter deAned in Eq. (20) void fraction at minimum #uidization solid volume fraction gas viscosity, kg=m s gas density, kg=m3 particle density, kg=m3 time delay between upper and lower signals, illustrated in Fig. 3, s time required for a cluster to reach a maximum height, illustrated in Fig. 3, s
Acknowledgements The authors thank the UK’s Engineering and Physical Science Research Council (EPSRC) for a research grant (Ref. GR/M66851). P. Wright acknowledges the EPSRC for provision of Advanced Research Fellowship (GR/A11311/01). Y. Makkawi thanks Heriot-Watt University for a Ph.D. scholarship.
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