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Experimental verification of the scaling relationships for bubbling gas-fluidised beds using the PEPT technique
Chemical Engineering Science 57 (2002) 3649 – 3658
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Experimental veri#cation of the scaling relationships for bubbling gas-(uidised beds using the PEPT technique M. Stein1 , Y.L. Ding ∗ , J.P.K. Seville School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 27 June 2001; received in revised form 18 April 2002; accepted 17 May 2002
Abstract An experimental veri#cation of the scaling relationships for bubbling gas-(uidised beds has been carried out by measuring particle cycle frequency. The Positron Emission Particle Tracking technique was used, which enabled non-intrusive measurements of solids motion in three dimensions with a spatial resolution of ∼2 mm under the conditions of this work. Three cylindrical beds (70, 141 and 240 mm ID) equipped with multiple ori#ce-type distributors were tested. It was shown that for geometrically similar beds, the pair of Froude numbers based on the minimum (uidisation and excess gas velocities was su
1. Introduction The scaling of bubbling gas-(uidised beds has been the subject of a number of investigations in the last few decades; see for example Romero and Johanson (1962), Broadhurst and Becker (1973), Glicksman (1984, 1988), and Horio, Nonaka, Sawa, and Muchi (1986). These studies can be divided into two categories, namely, the derivation of the scaling relationships, and validation of the derived similarity criteria. An important concept for scaling is the principle of similarity (Johnstone & Thring, 1957). When scaling a gas-(uidised bed, three types of similarity have to be considered, namely, geometric, kinematic and dynamic. Two general methods can be used to derive the similarity rules. When di=erential equations governing the behaviour of the system are unknown, dimensional analysis based on the Buckingham’s theorem is often used. Romero and ∗
Corresponding author. Current address: Department of Chemical Engineering, University of Leeds, Clarendon Roads, Leeds LS2 9JT, UK. Tel.: +044-133-343-2747; fax: +044-113-343-2405. E-mail address: [email protected] (Y.L. Ding). 1 Present address: Unilever Research Colworth, Bedford MK44 1LQ, UK.
Johanson (1962) and Broadhurst and Becker (1973) were among the earliest workers who made use of this approach to obtain the scaling relationships for the characterisation of (uidisation. The dimensionless groups obtained by Romero and Johanson (1962) are Froude and Reynolds numbers, both based on the minimum (uidisation velocity, the ratio of solid-to-(uid density, and the ratio of bed height at minimum (uidisation to bed diameter. The dimensionless groups developed by Broadhurst and Becker (1973) are similar to those of Romero and Johanson (1962) except that the minimum (uidisation velocity was replaced by the super#cial velocity. When the di=erential equations are known, non-dimensionalisation of these equations gives the similarity criteria. Glicksman (1984) is regarded as the #rst to report the ‘full set’ of scaling relationships for gas-(uidised beds (Eq. (1)) using this method though the credit could equally be attributed to other workers such as Fitzgerald, Bushnell, Crane, and Shieh (1984) and Fitzgerald (1985): f dp u0 ;
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 2 6 4 - 6
u02 ; gdp
f ; s
H ; dp
D ; dp
s ; particle size distribution; bed geometry;
(1)
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M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
where s and f are, respectively, the densities of the solid and gas phases, is the gas viscosity, s is the particle shape factor, dp is the average particle size, D is the bed diameter, H is the bed height, u0 is the super#cial gas velocity, and g is the gravitational acceleration. The #rst two groups are the particle Reynolds number (Rep ) and Froude number (Fr), respectively. At the viscous limit (Rep ¡ 4), the dimensionless groups given in Eq. (1) can be simpli#ed to f dp u0 ;
u02 ; gdp
H ; dp
D ; dp
s ; particle size distribution; bed geometry:
(1a)
At the inertial limit (Rep ¿ 400), one has u02 ; gdp
f ; s
H ; dp
D ; dp
s ; particle size distribution; bed geometry:
(1b)
For a geometrically similar scaling, Horio et al. (1986) obtained the following two criteria based on similarity of bubbling behaviour and other phenomena within the bed: u0 − umf √ ; gD
umf √ ; gD
(2)
where umf is the minimum (uidisation velocity. The dimensionless groups given in Eq. (2) are Froude numbers based on umf and (u0 − umf ), respectively, and have been shown by Glicksman (1988) to be equivalent to a set of relationships obtained by Glicksman (1984) at the viscous limit (Rep ¡ 4). For geometrically similar scaling, Zhang and Yang (1987) also derived a set of scaling relationships, which were similar to the set obtained by Glicksman (1984) and were shown to reduce to the following two numbers for approximate similarity (Zhang & Yang, 1987): gdp ; u02
u0 : umf
(3)
The dimensionless groups in Eq. (3) are similar to those given in Eq. (2). Based on their particle bed model, which includes a particle pressure term in the solid phase momentum equation, Foscolo, Felice, Gibilaro, Pistone, and Piccolo (1990) derived a set of dimensionless groups including the Archimedes number, density ratio, (ow number and geometrical ratios, where the Archimedes number can be derived from the Reynolds and Froude numbers and the density ratio. These groups are compatible with those obtained by Glicksman (1984), Horio et al. (1986) and Zhang and Yang (1987) when the (uid-particle interaction is dominant. Glicksman, Hyre, and Farrell (1994) gave a detailed summary of the scaling work on (uidisation. In addition to the deterministic methodology outlined above, a chaotic scale-up methodology has also been proposed by some investigators (Schouten, Vander Stappen, & Van Den Bleek, 1996). Such a methodology makes use of empirical correlations that relate the chaotic dynamics to (uidised bed size
and operating conditions, and this method is therefore more problematic in practical applications. Various experimental techniques have been used to validate the scaling relationships, which can be classi#ed into two categories: direct and indirect measurements. The direct method includes the measurement of bubble properties (e.g. diameter, growth rate, diameter distribution, frequency and rise velocity), minimum (uidisation velocity, overall bed expansion and entrainment rate constant. High-speed #lming, video analysis, capacitance and optical probes are often used in this approach; see for example Newby and Keairns (1986), Horio et al. (1986), Zhang and Yang (1987), and Felice, Rapagna, and Foscolo (1992). However, measurements of the time-resolved di=erential pressure, an indirect method, are more commonly used in the veri#cation of the hydrodynamic scaling of gas-(uidised beds (Fan, Ho, Hiraoka, & Walawender, 1981; Fitzgerald et al., 1984; Nicastro & Glicksman, 1984; Roy & Davidson, 1989). Bubbling gas-(uidised beds are two-phase systems in which solids motion is driven by the gas. Due to gas–solid and solid–solid interactions, both the gas and solid phases determine the hydrodynamic behaviour of a gas-(uidised bed. This suggests that veri#cation of scaling relationships for gas-(uidised beds should be carried out from the viewpoint of both the gas and solid phases. However, previous work was mainly focused on the gas phase, as summarised above. The objective of the work reported here was therefore set to validate the scaling relationships from the viewpoint of the solid phase. Bubbling gas (uidised beds operated within and close to the viscous limit were considered in this work. The non-intrusive Positron Emission Particle Tracking (PEPT) technique was used to follow particle motion. The so-called dimensionless particle cycle frequency was used as a dependent parameter to test the scaling relationships.
2. Theoretical Consider two hydrodynamically similar (uidised beds operated in the bubbling regime so that solids (ow patterns are similar. Particle cycle frequencies and their distributions, when made properly non-dimensional, should be identical. Particle cycle frequency in a bubbling gas-(uidised bed can be given as (Geldart, 1986) f=
(w + 0:38d )Y (u0 − umf ) ; Hmf
(4)
where w and d are the fractions of the bubble volume corresponding to the solids carried in the bubble wake and drift, Y is the ratio of the volumetric (owrate of bubbles to the excess gas (owrate (deviation from the two-phase theory), and Hmf is the bed height at minimum (uidisation. Kunii and Levenspiel (1977) arrived at a similar relation to Eq. (4).
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
Eq. (4) can be made non-dimensional to give u0 − umf fD D Q f= : = (w + 0:38d )Y umf Hmf umf
(4a)
As the parameters w ; d and Y are functions of Archimedes number (Geldart, 1986), Eq. (4a) can be expressed as D u0 − umf fQ = fn Ar; ; (4b) ; Hmf umf where Ar is de#ned as Ar =
f dp3 (s
− f )g : 2
(4c)
Eq. (4c) can be recast into the following form: Ar =
f umf dp
2
√
s −1 f
gD
umf
2
dp D
:
(4d)
The term (u0 − umf )=umf in Eq. (4b) can be modi#ed to give u0 − umf u0 − umf umf √ √ = : (4e) umf gD gD Combination of Eqs. (4b), (4d) and (4e) yields f umf dp umf u0 − umf s dp D Q √ √ f = fn ; : ; ; ; ; f D Hmf gD gD (5) Inspection of Eq. (5) indicates that the dimensionless particle cycle frequency depends on all the dimensionless numbers required for proper scaling of (uidised beds. In other words, the dimensionless cycle frequencies of two (uidised beds will be identical if all of these dimensionless groups are matched in full. However, in practical applications, relaxation of these constraints is often required and Glicksman (1984) has shown the possibility and conditions for such a relaxation. This work considers the scaling at the viscous limit. As mentioned in Section 1, the criterion for the viscous limit is Rep ¡ 4, a theoretical result from Glicksman (1984). The experimental results of Roy and Davidson (1989) suggest that the viscous limit can be relaxed up to an Rep number of about 30 and the two dimensionless groups given in Eq. (2) have been shown to be adequate up to this limit (Horio et al., 1986; Roy & Davidson, 1989). At the viscous limit, the density ratio in Eq. (5) can be neglected (Glicksman, 1984). The #rst term in Eq. (5) is the Reynolds number, which can be expressed by other groups and is therefore removed from the list. For geometrically similar beds operated within the viscous limit, Eq. (5) can be rewritten as umf u0 − umf fQ = fn √ ; √ : (6) gD gD
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Eq. (6) suggests that the dimensionless cycle frequencies are identical if the two Froude numbers based on umf and (u0 − umf ) are equal to each other. The two dimensionless groups shown in Eq. (6) are the same as those obtained by Horio et al. (1986), and are experimentally tested here by using three types of particles and three geometrically similar beds. 3. Experimental The experimental system used in this work is shown schematically in Fig. 1. It consisted of a (uidised bed (including a cylindrical column and a gas distributor), a gas supply and control system, and a measuring system. Three columns (70, 141 and 240 mm ID) were used. The distributors were of the multi-ori#ce type and were geometrically similar. These distributors had 80 ori#ces with constant ratio of ori#ce to column diameter (1=140). The holes in the distributor were countersunk on the upstream side and arranged in a triangular con#guration. Air from a Roots blower was used as the (uidising gas. A Furnace Controls FC014 micro-manometer was used to monitor bed pressure drop. The inlet gas pressure was controlled with a Spirax-Monnier pressure regulator. Gas (owrate was measured with a rotameter array which was frequently calibrated. During experiments, the column was placed between the two -ray detectors of the Birmingham positron camera. The detectors covered a #eld of 600 mm × 300 mm2 , and o=ered a resolution of ∼ 2 mm under the conditions of this study. The details of the PEPT technique and positron camera can be found elsewhere (Parker, Broadbent, Fowles, Hawkesworth, & McNeil, 1993; Parker, Dijkstra, Martin, & Seville, 1997). Resin beads, foamed glass and glass ballotini particles were used. Image analyses indicated that the particles were approximately spherical. A summary of the physical properties of these particles, together with values of some dimensionless numbers are given in Table 1. As the size ranges used in this work were quite narrow, the size distributions within the narrow ranges were not measured. However, the average sizes of the particles were determined by using image analyses, which are listed in Table 1. The minimum (uidisation velocities shown in Table 1 were determined experimentally using the 70 mm diameter column #tted with a polyethylene porous distributor (FILTROPLASTJ ). The Froude number based on the minimum (uidisation velocity was in the range of ∼ 0:15– 0.47, whereas the Froude number based on the excess gas velocity was controlled between 0.05 and 0.32. The ratio of particle-to-bed diameter ranged from 23 to 141. The gas-to-particle density ratio ranged from 4 × 10−4 to 3:5 × 10−3 . In all experiments, the bed was #lled to a static height of 1.5 times bed diameter to ensure geometric similarity. From Eq. (6), it can be seen that the basis of this work is the measurement of particle cycle frequency at di=erent
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M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
Fig. 1. Experimental system.
Table 1 Details of the materials
Material
dp
dp (ave)a
s 3
D
umf
f umf dp b
D=dp
umf √ gD
f =s c
(mm)
(mm)
(kg=m )
(mm)
(m=s)
Resin beads
0.55 – 0.75
0.65
1100
70
0.13
5.7
108
0.156
1:096 × 10−3
Foamed glass
1.0 –1.4 1.0 –1.4 1.4 –1.7 2.0 – 4.0 2.0 – 4.0
1.2 1.2 1.6 3.1 3.1
390 390 360 340 340
141 70 141 240 70
0.18 0.18 0.27 0.39 0.39
14.4 14.4 28.8 80.6 80.6
118 58 88 77 23
0.154 0.218 0.229 0.254 0.471
3:015 × 10−3 3:015 × 10−3 2:350 × 10−3 3:547 × 10−3 3:547 × 10−3
Glass ballotini
0.85 –1.2
1.0
2930
141
0.53
35.3
141
0.451
4:120 × 10−4
a Obtained
from image analyses. viscosity is taken as 1:81 × 10−5 Pa:s. c Air density at 20◦ C is taken as 1:206 kg=m 3 . b Air
Froude numbers. Due to intensive mixing of gas-(uidised beds, particles will take di=erent routes in di=erent cycles. A de#nition of cycle frequency is therefore necessary. In this work, a cycle was de#ned as a motion of a tracer starting from the lower quarter of the expanded bed height, reaching the upper quarter and returning to the lower quarter of the bed again. This criterion is considered to be adequate for bubbling beds without internals. Fig. 2 shows a sensitivity analysis of such a criterion. It can be seen that the cycle frequency was not a=ected signi#cantly by a change in the criterion from 14 to 13 of bed height. However, the criterion may need to be adjusted to take into account the e=ect of bed con#guration, in particular the presence of bed internals that are not placed in the middle part of the bed. The e=ect of bed internals on particle circulation was considered by Wong, Salleh, Ding, Seville, and Horio (2001).
The tracer particles used in this work were taken from the bulk. For glass ballotini and foamed glass tracers, activation was carried out directly in the Birmingham cyclotron. For small resin beads, instead of bombarding directly in the cyclotron, an ion exchange technique was employed. This was achieved by #rst irradiating water in the cyclotron to produce 18 F (half-life ∼ 110 min), and the tracer particles were then activated via ion exchange with the radioactive water. The tracer particles usually had an activity of ∼ 150 Ci of 18 F which emits positrons. The annihilation of the positrons with local electrons resulted in back-to-back 511 keV -rays. Detection of the -rays enable the particle position to be found by triangulation. A typical run of experiments involved: (a) preparation of the tracers; (b) start-up of the (uidised bed system; (c) start-up of the PEPT machine; and (d) introduction of
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
3653
Density function, 1/Hz
3 0.66H
2.5
0.75H 2 1.5 1 0.5
05 1.
94 0.
83 0.
72 0.
61 0.
50 0.
39 0.
28 0.
17 0.
0.
06
0
Frequency, Hz Fig. 2. Cycle frequency as a function of cycle criterion, 70 mm column, ori#ce-type distributor, resin beads, gas excess velocity u0 − umf = 0:35 m=s.
the tracer into the bed and initiation of the tracking program. Each tracking experiment lasted about 1 h.
4. Results and discussion 4.1. Vertical cyclic movement of particles and cycle frequency distribution For geometrically similar beds, the pro#les of vertical position of di=erent particles should be similar when the Froude numbers based on umf and u0 − umf are equal. Fig. 3 shows such a comparison, where the vertical position has been normalised by dividing by the bed diameter. (Although bed height is a more appropriate length scale for the normalisation, this does not a=ect the conclusion due to the constant value of H=D.) If attention is paid to the number of peaks in a given time interval, a qualitatively similar behaviour can be observed in the two cases, although the di=erence between the density ratios for the two particles is about 3-fold (Table 1). A quantitative similarity can be seen from the particle cycle frequency distribution shown in Fig. 4, which compares the dimensionless particle frequency distribution between foamed glass and resin beads in geometrically similar beds operated at approximately equal values of the two Froude numbers (see Eq. (6)). From Fig. 4, one can also see some di=erence between the formed glass and resin beads at high frequencies. The exact reason for the discrepancy is unclear. It is believed to be associated with the di=erence between the two materials in the following aspects: (a) surface property, (b) restitution coe
may also contribute to the discrepancy; see the caption of Fig. 4. 4.2. Average particle circulation frequency Figs. 5a–c show the dimensionless average particle frequencies as a function of the Froude number based on the excess gas velocity. The dashed lines in these #gures are from linear curve-#tting of the data. It can be seen that reasonable scale-up of geometrically similar bubbling beds can be achieved by matching the two Froude numbers shown in Eq. (6), at least from the viewpoint of vertical solids motion. Regression of the data in Figs. 5a–c gives the following correlations: u0 − umf umf fD √ fQ = = 0:6866 at √ = 0:16; umf gD gD (7) fD = 0:6292 fQ = umf
u0 − umf √ gD
umf at √ = 0:24; gD (7a)
fD = 0:0761 fQ = umf
u0 − umf √ gD
umf at √ = 0:46: gD (7b)
It is interesting to compare Eqs. (7) – (7b) with the theoretical results from Eq. (4a). In order to do this, Eq. (4a) is recast into the following form: (w + 0:38d )Y (D=Hmf ) u0 − umf fD Q √ √ f= = : (4f) umf (umf gD) gD
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M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658 Foamed Glass 1.0-1.4mm foamed glass particles in 141mm bed
z-coordinate/D [-]
1.75 1.25 0.75 0.25 -0.25
4
24
44
64
84
104
124
104
124
Time [s]
0.65mmResin resinBeads beads in 70mm bed
z-coordinate/D [-]
1.75 1.25 0.75 0.25 -0.25
4
24
44
64
84
Time [s]
Fig. 3. Normalised vertical movement of tracer particles for foamed glass and resin beads: (u0 − umf )=(gD)0:5 = 0:144; umf =(gD)0:5 = 0:155.
5 1.0-1.4mm foamed glass in 141 mm bed
4.5
0.65mm resin beads in 70 mm bed
Density function, 1/Hz
4 3.5 3 2.5 2 1.5 1 0.5 0 0
0.2
0.4
0.6
0.8
Dimensionless cycle frequency Fig. 4. Density function of the dimensionless cycle frequency distribution for (u0 − umf )=(gD)1=2 = 0:144; umf =(gD)1=2 ≈ 0:155; f u0 dp = = 27:9 for 1.0 –1:4 mm formed glass and 10.8 for 0:65 mm resin beads.
Under the conditions of this work, w ; d and Y can be estimated very approximately from the results of Geldart (1986) and are listed in Table 2. Utilisation of these values gives the theoretical values of the prefactor in square brackets of Eq. (4f): 0.133 at umf =(gD)0:5 = 0:16, 0.038 at umf =(gD)0:5 = 0:24, and 0.021 at umf =(gD)0:5 = 0:46. A comparison of these data with the prefactors in
Eqs. (7) – (7b) suggests that the experimental results follow the same trend as, but are systematically higher than, the theoretical results. The exact reason for the discrepancy is unclear. The approximation inherent in estimation of Y , w and d may be one of the reasons. The available data for Y; w and d are widely scattered, particularly in the range of interest here. Similar discrepancy
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
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Table 2 Estimation of the parameters in Eq. (4f)
Materials
Resin beads (0:65 mm)
Foamed glass (1.0 –1:4 mm)
Foamed glass (1.4 –1:7 mm)
Foamed glass (2–4 mm)
Glass ballotini (0.85 –1:2 mm)
Ar Y d w Y (d + 0:38w )
10,544 0.22 0.12 0.11 0.0342
24,081 0.2 0.11 0.105 0.0294
46,690 0.12 0.1 0.1 0.0166
319,654 0.1 0.08 0.08 0.0162
110,215 0.11 0.1 0.1 0.0152
0.25
0.2
0.65mm resin beads in 70mm bed
0.16
Linear regression (coefficient=0.86)
Dimensionless average cycle frequency
Dimensionless average cycle frequency
0.2
0.15
0.1
0.14 0.12 0.1 0.08 0.06 0.04
0.05
umf/(gD)0.5~0.24
0.02
umf/(gD)0.5~0.16
0
0 0
(a)
1.4-1.7mm foamed glass in 141mm bed 2.0-4.0mm foamed glass in 240mm bed Linear regression (coefficient =0.86)
0.18
1.0-1.4mm foamed glass in 141mm bed
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.045
0
0.05
(b)
Froude number based on (uo-umf)
0.1
0.15
0.2
0.25
0.3
Froude number based on (uo-umf)
0.85-1.2mm glass ballotini in 141mm bed 2.0-4.0mm foamed glass in 70mm bed
0.04 Dimensionless average cycle frequency
Linear regression (coefficient=0.51)
0.035
umf/(gD)0.5~0.46
0.03 0.025 0.02 0.015 0.01 0.005 0 0
(c)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Froude number based on (uo-umf)
Fig. 5. Dimensionless average cycle frequency as a function of the Froude number based on gas excess velocity: (a) umf =(gD)0:5 ≈ 0:16, (b) umf =(gD)0:5 ≈ 0:24, and (c) umf =(gD)0:5 ≈ 0:46. (a) The highest Reynolds number f u0 dp = = 42:4 for 1.0 –1:4 mm foamed glass in a 141 mm bed and 13.0 for 0:65 mm resin beads in a 70 mm bed, (b) highest Reynolds number f u0 dp = = 61:4 for 1.4 –1:7 mm foamed glass in a 141 mm bed and 111.7 for 2.0 –4:0 mm formed glass in a 240 mm bed. (c) The highest Reynolds number f u0 dp = = 58:6 for 0.85 –1:2 mm glass ballotini in a 141 mm bed and 131.9 for 2.0 –4:0 mm foamed glass in a 70 mm bed.
has also been reported by Stein, Ding, Seville, and Parker (2000). A comparison of Figs. 5a–c shows a considerable disagreement between ballotini and foamed glass at umf =(gD)1=2 ≈ 0:46 (the highest Froude number based on umf , Fig. 5c). Three possible factors may be responsible for this: (a) Reynolds numbers considerably higher than the viscous limit (Glicksman, 1984, 1988); (b) large difference in solid-to-gas density ratio; and (c) di=erence in
particle-to-bed diameter ratio. The last two will be discussed in the following section. 4.3. E3ects of particle-to-bed diameter ratio and gas-to-particle density ratio Various arguments have been advanced for neglecting the ratio of dp =D in the set of governing parameters at the viscous limit. Glicksman (1988) pointed out that when the
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M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
0.24
1.4-1.7mm foamed glass in 141mm bed 2-4mm foamed glass in 240mm bed
Dimensionless average cycle frequency
0.2
1-1.4mm foamed glass in 70mm bed Linear regression of all data in bubbling regime (coefficient=0.85)
0.16
0.12
0.08 0.5
u mf/(gD) ~0.23 0.04
0 0
0.05
0.1
0.15
0.2
0.25
0.3
Froude numbe r base d on (u o -u mf) Fig. 6. Scale-up in the slugging regime.
governing equations are made non-dimensional with respect to the bed dimension, dp =D only appears in combination with the Reynolds number. The particle diameter is also absent from the ratio of particle inertia to gravity forces (the Froude number) since both forces are proportional to the particle volume. Moreover, commonly used and successful correlations for bubble velocity and bubble size, for example, do not contain dp =D. The experiments in this work seem to suggest that neglecting the dp =D ratio is acceptable when dp =D does not vary too much (Fig. 5a: D=dp = 108, 118; Fig. 5b: D=dp = 77, 88; see Table 1). The particle-to-bed diameter ratio may exert some in(uence on the bed scaling when a change in D=dp is signi#cant (see Fig. 5c: D=dp = 23, 141, where the data points are more scattered than those shown in Figs. 5a and b). However, given the large di=erence in D=dp (∼6-fold), the scattering seems to be acceptable. From Table 1, it can be seen that the density ratio changes by approximately 9-fold. However, no signi#cant e=ect has been observed (Fig. 5). The weak in(uence of the gas-to-solid density ratio on (ow in the bubbling regime has also been reported by Farrell and Glicksman (1997). 4.4. Criterion for the viscous limit This work was aimed at veri#cation of the scaling rules at the viscous limit. According to Glicksman (1984, 1988), the limiting Reynolds number is 4. As mentioned earlier, Roy and Davidson (1989) found that the limit could be relaxed to a value of 30. From the results of this work (Fig. 5), it seems that the limit suggested by Roy and Davidson (1989) can be relaxed further up to about 100; see the captions of Figs. 5a–c for the highest particle Reynolds numbers used in these runs. This conclusion clearly needs independent corroboration.
4.5. Slugging bed Several studies have been carried out on bed scaling in the slugging regime; see for example Felice et al. (1992) and Horio et al. (1986). From the correlations for the onset of slugging and slug rise velocity by Stewart and Davidson (1967), Horio et al. (1986) concluded that their scaling rules, i.e. Eq. (2), should work in the slugging regime. Pressure (uctuation measurements by Felice et al. (1992), however, indicated that the scaling rules developed for bubbling beds did not apply to slugging beds. In this work, slugging was observed at a Froude number based on the excess gas velocity of ∼ 0:07, where foamed glass particles were (uidised in the smallest column (70 mm diameter, Fig. 6). Fig. 6 suggests that the scaling rules at the viscous limit do not apply to the slugging regime, supporting the observation of Felice et al. (1992). 4.6. Discussion of the methodology used in this work This work uses the particle cycle frequency to verify the scaling relationships. The cycle frequency is a measure of the vertical movement of particles. From the de#nition of the cycle frequency, only large vertical movement is taken into account. The technique used in this work is believed to be complementary to the techniques based on the pressure (uctuations, which take both the small and large frequencies into account. The technique used in this work attacks the scaling issue from the viewpoint of solids motion, while the pressure (uctuation measurements look at the problem from the viewpoint of the gas phase (mainly bubbles). As both techniques consider the vertical movement, it seems they are only appropriate for relatively deep beds where vertical movement is dominant. Recent work on particle circulation
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
in shallow (uidised beds (aspect ratio less than 0.5) has shown poor qualitative and quantitative agreement with Eq. (4) (McCormack & Seville, 2001). 5. Conclusions Scaling relationships for bubbling gas-(uidised beds have been a subject of substantial experimental and theoretical investigations. Most experimental studies attack the scaling issue from the viewpoint of bubble motion. In this work, experimental veri#cation is carried out by direct measurements of vertical solids motion using the Positron Emission Particle Tracking (PEPT) technique. The following conclusions are obtained: • For geometrically similar beds, the pair of Froude numbers based on the minimum (uidisation and excess gas velocities is su
f d3 (s −f )g
≡ p 2 , Archimedes number, dimensionless particle diameter, L bed diameter, L particle frequency, T−1 dimensionless cycle frequency, dimensionless ≡ u02 =(gdp ), Froude number, dimensionless acceleration due to gravitational action, LT−2 bed height, L bed height at minimum (uidisation, L ≡ f u0 dp =, Reynolds number, dimensionless super#cial gas velocity, LT−1 minimum (uidisation velocity, LT−1 deviation from the two-phase theory, dimensionless
Greek letters w d s s f
wake volume as a fraction of bubble volume, dimensionless drift volume as a fraction of bubble volume, dimensionless shape factor of particles, dimensionless (uid viscosity, ML−1 T−1 particle density, ML−3 (uid density, ML−3
3657
Acknowledgements The authors are grateful to the EPSRC and ETSU of the DTI (UK) for supporting this work under Grants GR=J99414, 75128 and 75135. References Broadhurst, T. E., & Becker, H. A. (1973). The application of the theory of dimensions to (uidised beds. In Proc. Int. Symp. Ste. Chimie Indust., Toulouse, France. Fan, L. T., Ho, T. C., Hiraoka, S., & Walawender, W. P. (1981). Pressure (uctuations in a (uidised bed. AIChE Journal, 27, 388–396. Farrell, P. A., & Glicksman, L. R. (1997). Measurement of bubble characteristics in a scale model of a 70 MW pressurised (uidised bed combustor. In Proceedings of the 14th International ASME Conference on Fluidised Bed Combustion, Vancouver, Canada, May 11–14. Felice, R. D., Rapagna, S., & Foscolo, P. U. (1992). Dynamic similarity rules: Validity check for bubbling and slugging (uidised beds. Powder Technology, 71, 281–287. Fitzgerald, T. (1985). Coarse particle systems. In J. F. Davidson, R. Clift, & D. Harrison (Eds.), Fluidisation (2nd ed.). London: Academic Press. Fitzgerald, T., Bushnell, D., Crane, S., & Shieh, Y. C. (1984). Testing of cold scaled bed model for (uidised-bed combustors. Powder Technology, 38, 107–120. Foscolo, P. U., Felice, R. D., Gibilaro, L. G., Pistone, L., & Piccolo (1990). Scaling relationships for (uidisation: The generalised particle bed model. Chemical Engineering Science, 45, 1647–1651. Geldart, D. (1986). Solids mixing. In G. Geldart (Ed.), Gas Fluidisation Technology (pp. (97–123)). Chichester: Wiley. Glicksman, L. R. (1984). Scaling relationships for (uidised beds. Chemical Engineering Science, 39, 1373–1379. Glicksman, L. R. (1988). Scaling relationships for (uidised beds. Chemical Engineering Science, 43, 1419–1421. Glicksman, L. R., Hyre, M. R., & Farrell, P. A. (1994). Dynamic similarity in (uidisation. International Journal of Multiphase Flow, 20S, 331–386. Horio, M., Nonaka, A., Sawa, Y., & Muchi, I. (1986). A new similarity rule for (uidised bed scale-up. AIChE Journal, 32, 1466–1482. Johnstone, R. W., & Thring, M. W. (1957). Pilot Plants, Models and Scale-up Methods in Chemical Engineering. New York: McGraw-Hill. Kunii, D., & Levenspiel, O. (1977). Fluidisation Engineering. New York: Krieger Publishing. McCormack, A., & Seville, J. P. K. (2001). Solids motion in shallow gas-(uidised beds, unpublished work. Newby, R. A., & Keairns, D. L. (1986). Test of the scaling relationships for (uid-bed dynamics. In K. Ostergaard, & A. Sorensen (Eds.), Fluidisation (pp. (31–38)). New York: Engineering Foundation. Nicastro, M. T., & Glicksman, L. R. (1984). Experimental veri#cation of scaling relationships for (uidised beds. Chemical Engineering Science, 39, 1381–1391. Parker, D. J., Broadbent, C. J., Fowles, P., Hawkesworth, M. R., & McNeil, P. (1993). Positron emission particle tracking—a technique for studying (ow within engineering equipment. Nuclear Instruments and Methods A, 326, 592–607. Parker, D. J., Dijkstra, A. E., Martin, T. W., & Seville, J. P. K. (1997). Positron emission particle tracking studies of spherical particle motion in rotating drums. Chemical Engineering Science, 52, 2011–2022. Romero, J. B., & Johanson, L. N. (1962). Factors a=ecting (uidised bed quality. Chemical Engineering Progress, Symposium Series, 58(38), 28–32. Roy, R., & Davidson, J. F. (1989). Similarity between gas-(uidised beds at elevated temperature and pressure. In J. R. Grace, L. W. Shemilt, & M. A. Bergougnou (Eds.), Fluidisation VI (pp. (293–300)). New York: Engineering Foundation.
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Schouten, J. C., Vander Stappen, M. L. M., & Van Den Bleek, C. M. (1996). Scale-up of chaotic (uidised bed hydrodynamics. Chemical Engineering Science, 51, 1991–2000. Stein, M., Ding, Y. L., Seville, J. P. K., & Parker, D. J. (2000). Solids motion in bubbling gas-(uidised beds. Chemical Engineering Science, 55, 5291–5300. Stewart, P. S. B., & Davidson, J. F. (1967). Slug (ow in (uidised beds. Powder Technology, 1, 61–71.
Wong, Y. S., Salleh, A. M., Ding, Y. L., Seville, J. P. K., & Horio, M. (2001). Particle motion in a relatively thin rectangular (uidised beds with immersed tubes. Proceedings of the PARTEC-2001, Paper No. 248. Nuremburg, Germany, May 27–29, 2001. Zhang, M. C., & Yang, R. Y. K. (1987). On the scaling laws for bubbling gas-(uidised bed dynamics. Powder Technology, 51, 159–165.
www.elsevier.com/locate/ces
Experimental veri#cation of the scaling relationships for bubbling gas-(uidised beds using the PEPT technique M. Stein1 , Y.L. Ding ∗ , J.P.K. Seville School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 27 June 2001; received in revised form 18 April 2002; accepted 17 May 2002
Abstract An experimental veri#cation of the scaling relationships for bubbling gas-(uidised beds has been carried out by measuring particle cycle frequency. The Positron Emission Particle Tracking technique was used, which enabled non-intrusive measurements of solids motion in three dimensions with a spatial resolution of ∼2 mm under the conditions of this work. Three cylindrical beds (70, 141 and 240 mm ID) equipped with multiple ori#ce-type distributors were tested. It was shown that for geometrically similar beds, the pair of Froude numbers based on the minimum (uidisation and excess gas velocities was su
1. Introduction The scaling of bubbling gas-(uidised beds has been the subject of a number of investigations in the last few decades; see for example Romero and Johanson (1962), Broadhurst and Becker (1973), Glicksman (1984, 1988), and Horio, Nonaka, Sawa, and Muchi (1986). These studies can be divided into two categories, namely, the derivation of the scaling relationships, and validation of the derived similarity criteria. An important concept for scaling is the principle of similarity (Johnstone & Thring, 1957). When scaling a gas-(uidised bed, three types of similarity have to be considered, namely, geometric, kinematic and dynamic. Two general methods can be used to derive the similarity rules. When di=erential equations governing the behaviour of the system are unknown, dimensional analysis based on the Buckingham’s theorem is often used. Romero and ∗
Corresponding author. Current address: Department of Chemical Engineering, University of Leeds, Clarendon Roads, Leeds LS2 9JT, UK. Tel.: +044-133-343-2747; fax: +044-113-343-2405. E-mail address: [email protected] (Y.L. Ding). 1 Present address: Unilever Research Colworth, Bedford MK44 1LQ, UK.
Johanson (1962) and Broadhurst and Becker (1973) were among the earliest workers who made use of this approach to obtain the scaling relationships for the characterisation of (uidisation. The dimensionless groups obtained by Romero and Johanson (1962) are Froude and Reynolds numbers, both based on the minimum (uidisation velocity, the ratio of solid-to-(uid density, and the ratio of bed height at minimum (uidisation to bed diameter. The dimensionless groups developed by Broadhurst and Becker (1973) are similar to those of Romero and Johanson (1962) except that the minimum (uidisation velocity was replaced by the super#cial velocity. When the di=erential equations are known, non-dimensionalisation of these equations gives the similarity criteria. Glicksman (1984) is regarded as the #rst to report the ‘full set’ of scaling relationships for gas-(uidised beds (Eq. (1)) using this method though the credit could equally be attributed to other workers such as Fitzgerald, Bushnell, Crane, and Shieh (1984) and Fitzgerald (1985): f dp u0 ;
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 2 6 4 - 6
u02 ; gdp
f ; s
H ; dp
D ; dp
s ; particle size distribution; bed geometry;
(1)
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M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
where s and f are, respectively, the densities of the solid and gas phases, is the gas viscosity, s is the particle shape factor, dp is the average particle size, D is the bed diameter, H is the bed height, u0 is the super#cial gas velocity, and g is the gravitational acceleration. The #rst two groups are the particle Reynolds number (Rep ) and Froude number (Fr), respectively. At the viscous limit (Rep ¡ 4), the dimensionless groups given in Eq. (1) can be simpli#ed to f dp u0 ;
u02 ; gdp
H ; dp
D ; dp
s ; particle size distribution; bed geometry:
(1a)
At the inertial limit (Rep ¿ 400), one has u02 ; gdp
f ; s
H ; dp
D ; dp
s ; particle size distribution; bed geometry:
(1b)
For a geometrically similar scaling, Horio et al. (1986) obtained the following two criteria based on similarity of bubbling behaviour and other phenomena within the bed: u0 − umf √ ; gD
umf √ ; gD
(2)
where umf is the minimum (uidisation velocity. The dimensionless groups given in Eq. (2) are Froude numbers based on umf and (u0 − umf ), respectively, and have been shown by Glicksman (1988) to be equivalent to a set of relationships obtained by Glicksman (1984) at the viscous limit (Rep ¡ 4). For geometrically similar scaling, Zhang and Yang (1987) also derived a set of scaling relationships, which were similar to the set obtained by Glicksman (1984) and were shown to reduce to the following two numbers for approximate similarity (Zhang & Yang, 1987): gdp ; u02
u0 : umf
(3)
The dimensionless groups in Eq. (3) are similar to those given in Eq. (2). Based on their particle bed model, which includes a particle pressure term in the solid phase momentum equation, Foscolo, Felice, Gibilaro, Pistone, and Piccolo (1990) derived a set of dimensionless groups including the Archimedes number, density ratio, (ow number and geometrical ratios, where the Archimedes number can be derived from the Reynolds and Froude numbers and the density ratio. These groups are compatible with those obtained by Glicksman (1984), Horio et al. (1986) and Zhang and Yang (1987) when the (uid-particle interaction is dominant. Glicksman, Hyre, and Farrell (1994) gave a detailed summary of the scaling work on (uidisation. In addition to the deterministic methodology outlined above, a chaotic scale-up methodology has also been proposed by some investigators (Schouten, Vander Stappen, & Van Den Bleek, 1996). Such a methodology makes use of empirical correlations that relate the chaotic dynamics to (uidised bed size
and operating conditions, and this method is therefore more problematic in practical applications. Various experimental techniques have been used to validate the scaling relationships, which can be classi#ed into two categories: direct and indirect measurements. The direct method includes the measurement of bubble properties (e.g. diameter, growth rate, diameter distribution, frequency and rise velocity), minimum (uidisation velocity, overall bed expansion and entrainment rate constant. High-speed #lming, video analysis, capacitance and optical probes are often used in this approach; see for example Newby and Keairns (1986), Horio et al. (1986), Zhang and Yang (1987), and Felice, Rapagna, and Foscolo (1992). However, measurements of the time-resolved di=erential pressure, an indirect method, are more commonly used in the veri#cation of the hydrodynamic scaling of gas-(uidised beds (Fan, Ho, Hiraoka, & Walawender, 1981; Fitzgerald et al., 1984; Nicastro & Glicksman, 1984; Roy & Davidson, 1989). Bubbling gas-(uidised beds are two-phase systems in which solids motion is driven by the gas. Due to gas–solid and solid–solid interactions, both the gas and solid phases determine the hydrodynamic behaviour of a gas-(uidised bed. This suggests that veri#cation of scaling relationships for gas-(uidised beds should be carried out from the viewpoint of both the gas and solid phases. However, previous work was mainly focused on the gas phase, as summarised above. The objective of the work reported here was therefore set to validate the scaling relationships from the viewpoint of the solid phase. Bubbling gas (uidised beds operated within and close to the viscous limit were considered in this work. The non-intrusive Positron Emission Particle Tracking (PEPT) technique was used to follow particle motion. The so-called dimensionless particle cycle frequency was used as a dependent parameter to test the scaling relationships.
2. Theoretical Consider two hydrodynamically similar (uidised beds operated in the bubbling regime so that solids (ow patterns are similar. Particle cycle frequencies and their distributions, when made properly non-dimensional, should be identical. Particle cycle frequency in a bubbling gas-(uidised bed can be given as (Geldart, 1986) f=
(w + 0:38d )Y (u0 − umf ) ; Hmf
(4)
where w and d are the fractions of the bubble volume corresponding to the solids carried in the bubble wake and drift, Y is the ratio of the volumetric (owrate of bubbles to the excess gas (owrate (deviation from the two-phase theory), and Hmf is the bed height at minimum (uidisation. Kunii and Levenspiel (1977) arrived at a similar relation to Eq. (4).
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
Eq. (4) can be made non-dimensional to give u0 − umf fD D Q f= : = (w + 0:38d )Y umf Hmf umf
(4a)
As the parameters w ; d and Y are functions of Archimedes number (Geldart, 1986), Eq. (4a) can be expressed as D u0 − umf fQ = fn Ar; ; (4b) ; Hmf umf where Ar is de#ned as Ar =
f dp3 (s
− f )g : 2
(4c)
Eq. (4c) can be recast into the following form: Ar =
f umf dp
2
√
s −1 f
gD
umf
2
dp D
:
(4d)
The term (u0 − umf )=umf in Eq. (4b) can be modi#ed to give u0 − umf u0 − umf umf √ √ = : (4e) umf gD gD Combination of Eqs. (4b), (4d) and (4e) yields f umf dp umf u0 − umf s dp D Q √ √ f = fn ; : ; ; ; ; f D Hmf gD gD (5) Inspection of Eq. (5) indicates that the dimensionless particle cycle frequency depends on all the dimensionless numbers required for proper scaling of (uidised beds. In other words, the dimensionless cycle frequencies of two (uidised beds will be identical if all of these dimensionless groups are matched in full. However, in practical applications, relaxation of these constraints is often required and Glicksman (1984) has shown the possibility and conditions for such a relaxation. This work considers the scaling at the viscous limit. As mentioned in Section 1, the criterion for the viscous limit is Rep ¡ 4, a theoretical result from Glicksman (1984). The experimental results of Roy and Davidson (1989) suggest that the viscous limit can be relaxed up to an Rep number of about 30 and the two dimensionless groups given in Eq. (2) have been shown to be adequate up to this limit (Horio et al., 1986; Roy & Davidson, 1989). At the viscous limit, the density ratio in Eq. (5) can be neglected (Glicksman, 1984). The #rst term in Eq. (5) is the Reynolds number, which can be expressed by other groups and is therefore removed from the list. For geometrically similar beds operated within the viscous limit, Eq. (5) can be rewritten as umf u0 − umf fQ = fn √ ; √ : (6) gD gD
3651
Eq. (6) suggests that the dimensionless cycle frequencies are identical if the two Froude numbers based on umf and (u0 − umf ) are equal to each other. The two dimensionless groups shown in Eq. (6) are the same as those obtained by Horio et al. (1986), and are experimentally tested here by using three types of particles and three geometrically similar beds. 3. Experimental The experimental system used in this work is shown schematically in Fig. 1. It consisted of a (uidised bed (including a cylindrical column and a gas distributor), a gas supply and control system, and a measuring system. Three columns (70, 141 and 240 mm ID) were used. The distributors were of the multi-ori#ce type and were geometrically similar. These distributors had 80 ori#ces with constant ratio of ori#ce to column diameter (1=140). The holes in the distributor were countersunk on the upstream side and arranged in a triangular con#guration. Air from a Roots blower was used as the (uidising gas. A Furnace Controls FC014 micro-manometer was used to monitor bed pressure drop. The inlet gas pressure was controlled with a Spirax-Monnier pressure regulator. Gas (owrate was measured with a rotameter array which was frequently calibrated. During experiments, the column was placed between the two -ray detectors of the Birmingham positron camera. The detectors covered a #eld of 600 mm × 300 mm2 , and o=ered a resolution of ∼ 2 mm under the conditions of this study. The details of the PEPT technique and positron camera can be found elsewhere (Parker, Broadbent, Fowles, Hawkesworth, & McNeil, 1993; Parker, Dijkstra, Martin, & Seville, 1997). Resin beads, foamed glass and glass ballotini particles were used. Image analyses indicated that the particles were approximately spherical. A summary of the physical properties of these particles, together with values of some dimensionless numbers are given in Table 1. As the size ranges used in this work were quite narrow, the size distributions within the narrow ranges were not measured. However, the average sizes of the particles were determined by using image analyses, which are listed in Table 1. The minimum (uidisation velocities shown in Table 1 were determined experimentally using the 70 mm diameter column #tted with a polyethylene porous distributor (FILTROPLASTJ ). The Froude number based on the minimum (uidisation velocity was in the range of ∼ 0:15– 0.47, whereas the Froude number based on the excess gas velocity was controlled between 0.05 and 0.32. The ratio of particle-to-bed diameter ranged from 23 to 141. The gas-to-particle density ratio ranged from 4 × 10−4 to 3:5 × 10−3 . In all experiments, the bed was #lled to a static height of 1.5 times bed diameter to ensure geometric similarity. From Eq. (6), it can be seen that the basis of this work is the measurement of particle cycle frequency at di=erent
3652
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
Fig. 1. Experimental system.
Table 1 Details of the materials
Material
dp
dp (ave)a
s 3
D
umf
f umf dp b
D=dp
umf √ gD
f =s c
(mm)
(mm)
(kg=m )
(mm)
(m=s)
Resin beads
0.55 – 0.75
0.65
1100
70
0.13
5.7
108
0.156
1:096 × 10−3
Foamed glass
1.0 –1.4 1.0 –1.4 1.4 –1.7 2.0 – 4.0 2.0 – 4.0
1.2 1.2 1.6 3.1 3.1
390 390 360 340 340
141 70 141 240 70
0.18 0.18 0.27 0.39 0.39
14.4 14.4 28.8 80.6 80.6
118 58 88 77 23
0.154 0.218 0.229 0.254 0.471
3:015 × 10−3 3:015 × 10−3 2:350 × 10−3 3:547 × 10−3 3:547 × 10−3
Glass ballotini
0.85 –1.2
1.0
2930
141
0.53
35.3
141
0.451
4:120 × 10−4
a Obtained
from image analyses. viscosity is taken as 1:81 × 10−5 Pa:s. c Air density at 20◦ C is taken as 1:206 kg=m 3 . b Air
Froude numbers. Due to intensive mixing of gas-(uidised beds, particles will take di=erent routes in di=erent cycles. A de#nition of cycle frequency is therefore necessary. In this work, a cycle was de#ned as a motion of a tracer starting from the lower quarter of the expanded bed height, reaching the upper quarter and returning to the lower quarter of the bed again. This criterion is considered to be adequate for bubbling beds without internals. Fig. 2 shows a sensitivity analysis of such a criterion. It can be seen that the cycle frequency was not a=ected signi#cantly by a change in the criterion from 14 to 13 of bed height. However, the criterion may need to be adjusted to take into account the e=ect of bed con#guration, in particular the presence of bed internals that are not placed in the middle part of the bed. The e=ect of bed internals on particle circulation was considered by Wong, Salleh, Ding, Seville, and Horio (2001).
The tracer particles used in this work were taken from the bulk. For glass ballotini and foamed glass tracers, activation was carried out directly in the Birmingham cyclotron. For small resin beads, instead of bombarding directly in the cyclotron, an ion exchange technique was employed. This was achieved by #rst irradiating water in the cyclotron to produce 18 F (half-life ∼ 110 min), and the tracer particles were then activated via ion exchange with the radioactive water. The tracer particles usually had an activity of ∼ 150 Ci of 18 F which emits positrons. The annihilation of the positrons with local electrons resulted in back-to-back 511 keV -rays. Detection of the -rays enable the particle position to be found by triangulation. A typical run of experiments involved: (a) preparation of the tracers; (b) start-up of the (uidised bed system; (c) start-up of the PEPT machine; and (d) introduction of
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
3653
Density function, 1/Hz
3 0.66H
2.5
0.75H 2 1.5 1 0.5
05 1.
94 0.
83 0.
72 0.
61 0.
50 0.
39 0.
28 0.
17 0.
0.
06
0
Frequency, Hz Fig. 2. Cycle frequency as a function of cycle criterion, 70 mm column, ori#ce-type distributor, resin beads, gas excess velocity u0 − umf = 0:35 m=s.
the tracer into the bed and initiation of the tracking program. Each tracking experiment lasted about 1 h.
4. Results and discussion 4.1. Vertical cyclic movement of particles and cycle frequency distribution For geometrically similar beds, the pro#les of vertical position of di=erent particles should be similar when the Froude numbers based on umf and u0 − umf are equal. Fig. 3 shows such a comparison, where the vertical position has been normalised by dividing by the bed diameter. (Although bed height is a more appropriate length scale for the normalisation, this does not a=ect the conclusion due to the constant value of H=D.) If attention is paid to the number of peaks in a given time interval, a qualitatively similar behaviour can be observed in the two cases, although the di=erence between the density ratios for the two particles is about 3-fold (Table 1). A quantitative similarity can be seen from the particle cycle frequency distribution shown in Fig. 4, which compares the dimensionless particle frequency distribution between foamed glass and resin beads in geometrically similar beds operated at approximately equal values of the two Froude numbers (see Eq. (6)). From Fig. 4, one can also see some di=erence between the formed glass and resin beads at high frequencies. The exact reason for the discrepancy is unclear. It is believed to be associated with the di=erence between the two materials in the following aspects: (a) surface property, (b) restitution coe
may also contribute to the discrepancy; see the caption of Fig. 4. 4.2. Average particle circulation frequency Figs. 5a–c show the dimensionless average particle frequencies as a function of the Froude number based on the excess gas velocity. The dashed lines in these #gures are from linear curve-#tting of the data. It can be seen that reasonable scale-up of geometrically similar bubbling beds can be achieved by matching the two Froude numbers shown in Eq. (6), at least from the viewpoint of vertical solids motion. Regression of the data in Figs. 5a–c gives the following correlations: u0 − umf umf fD √ fQ = = 0:6866 at √ = 0:16; umf gD gD (7) fD = 0:6292 fQ = umf
u0 − umf √ gD
umf at √ = 0:24; gD (7a)
fD = 0:0761 fQ = umf
u0 − umf √ gD
umf at √ = 0:46: gD (7b)
It is interesting to compare Eqs. (7) – (7b) with the theoretical results from Eq. (4a). In order to do this, Eq. (4a) is recast into the following form: (w + 0:38d )Y (D=Hmf ) u0 − umf fD Q √ √ f= = : (4f) umf (umf gD) gD
3654
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658 Foamed Glass 1.0-1.4mm foamed glass particles in 141mm bed
z-coordinate/D [-]
1.75 1.25 0.75 0.25 -0.25
4
24
44
64
84
104
124
104
124
Time [s]
0.65mmResin resinBeads beads in 70mm bed
z-coordinate/D [-]
1.75 1.25 0.75 0.25 -0.25
4
24
44
64
84
Time [s]
Fig. 3. Normalised vertical movement of tracer particles for foamed glass and resin beads: (u0 − umf )=(gD)0:5 = 0:144; umf =(gD)0:5 = 0:155.
5 1.0-1.4mm foamed glass in 141 mm bed
4.5
0.65mm resin beads in 70 mm bed
Density function, 1/Hz
4 3.5 3 2.5 2 1.5 1 0.5 0 0
0.2
0.4
0.6
0.8
Dimensionless cycle frequency Fig. 4. Density function of the dimensionless cycle frequency distribution for (u0 − umf )=(gD)1=2 = 0:144; umf =(gD)1=2 ≈ 0:155; f u0 dp = = 27:9 for 1.0 –1:4 mm formed glass and 10.8 for 0:65 mm resin beads.
Under the conditions of this work, w ; d and Y can be estimated very approximately from the results of Geldart (1986) and are listed in Table 2. Utilisation of these values gives the theoretical values of the prefactor in square brackets of Eq. (4f): 0.133 at umf =(gD)0:5 = 0:16, 0.038 at umf =(gD)0:5 = 0:24, and 0.021 at umf =(gD)0:5 = 0:46. A comparison of these data with the prefactors in
Eqs. (7) – (7b) suggests that the experimental results follow the same trend as, but are systematically higher than, the theoretical results. The exact reason for the discrepancy is unclear. The approximation inherent in estimation of Y , w and d may be one of the reasons. The available data for Y; w and d are widely scattered, particularly in the range of interest here. Similar discrepancy
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
3655
Table 2 Estimation of the parameters in Eq. (4f)
Materials
Resin beads (0:65 mm)
Foamed glass (1.0 –1:4 mm)
Foamed glass (1.4 –1:7 mm)
Foamed glass (2–4 mm)
Glass ballotini (0.85 –1:2 mm)
Ar Y d w Y (d + 0:38w )
10,544 0.22 0.12 0.11 0.0342
24,081 0.2 0.11 0.105 0.0294
46,690 0.12 0.1 0.1 0.0166
319,654 0.1 0.08 0.08 0.0162
110,215 0.11 0.1 0.1 0.0152
0.25
0.2
0.65mm resin beads in 70mm bed
0.16
Linear regression (coefficient=0.86)
Dimensionless average cycle frequency
Dimensionless average cycle frequency
0.2
0.15
0.1
0.14 0.12 0.1 0.08 0.06 0.04
0.05
umf/(gD)0.5~0.24
0.02
umf/(gD)0.5~0.16
0
0 0
(a)
1.4-1.7mm foamed glass in 141mm bed 2.0-4.0mm foamed glass in 240mm bed Linear regression (coefficient =0.86)
0.18
1.0-1.4mm foamed glass in 141mm bed
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.045
0
0.05
(b)
Froude number based on (uo-umf)
0.1
0.15
0.2
0.25
0.3
Froude number based on (uo-umf)
0.85-1.2mm glass ballotini in 141mm bed 2.0-4.0mm foamed glass in 70mm bed
0.04 Dimensionless average cycle frequency
Linear regression (coefficient=0.51)
0.035
umf/(gD)0.5~0.46
0.03 0.025 0.02 0.015 0.01 0.005 0 0
(c)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Froude number based on (uo-umf)
Fig. 5. Dimensionless average cycle frequency as a function of the Froude number based on gas excess velocity: (a) umf =(gD)0:5 ≈ 0:16, (b) umf =(gD)0:5 ≈ 0:24, and (c) umf =(gD)0:5 ≈ 0:46. (a) The highest Reynolds number f u0 dp = = 42:4 for 1.0 –1:4 mm foamed glass in a 141 mm bed and 13.0 for 0:65 mm resin beads in a 70 mm bed, (b) highest Reynolds number f u0 dp = = 61:4 for 1.4 –1:7 mm foamed glass in a 141 mm bed and 111.7 for 2.0 –4:0 mm formed glass in a 240 mm bed. (c) The highest Reynolds number f u0 dp = = 58:6 for 0.85 –1:2 mm glass ballotini in a 141 mm bed and 131.9 for 2.0 –4:0 mm foamed glass in a 70 mm bed.
has also been reported by Stein, Ding, Seville, and Parker (2000). A comparison of Figs. 5a–c shows a considerable disagreement between ballotini and foamed glass at umf =(gD)1=2 ≈ 0:46 (the highest Froude number based on umf , Fig. 5c). Three possible factors may be responsible for this: (a) Reynolds numbers considerably higher than the viscous limit (Glicksman, 1984, 1988); (b) large difference in solid-to-gas density ratio; and (c) di=erence in
particle-to-bed diameter ratio. The last two will be discussed in the following section. 4.3. E3ects of particle-to-bed diameter ratio and gas-to-particle density ratio Various arguments have been advanced for neglecting the ratio of dp =D in the set of governing parameters at the viscous limit. Glicksman (1988) pointed out that when the
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M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
0.24
1.4-1.7mm foamed glass in 141mm bed 2-4mm foamed glass in 240mm bed
Dimensionless average cycle frequency
0.2
1-1.4mm foamed glass in 70mm bed Linear regression of all data in bubbling regime (coefficient=0.85)
0.16
0.12
0.08 0.5
u mf/(gD) ~0.23 0.04
0 0
0.05
0.1
0.15
0.2
0.25
0.3
Froude numbe r base d on (u o -u mf) Fig. 6. Scale-up in the slugging regime.
governing equations are made non-dimensional with respect to the bed dimension, dp =D only appears in combination with the Reynolds number. The particle diameter is also absent from the ratio of particle inertia to gravity forces (the Froude number) since both forces are proportional to the particle volume. Moreover, commonly used and successful correlations for bubble velocity and bubble size, for example, do not contain dp =D. The experiments in this work seem to suggest that neglecting the dp =D ratio is acceptable when dp =D does not vary too much (Fig. 5a: D=dp = 108, 118; Fig. 5b: D=dp = 77, 88; see Table 1). The particle-to-bed diameter ratio may exert some in(uence on the bed scaling when a change in D=dp is signi#cant (see Fig. 5c: D=dp = 23, 141, where the data points are more scattered than those shown in Figs. 5a and b). However, given the large di=erence in D=dp (∼6-fold), the scattering seems to be acceptable. From Table 1, it can be seen that the density ratio changes by approximately 9-fold. However, no signi#cant e=ect has been observed (Fig. 5). The weak in(uence of the gas-to-solid density ratio on (ow in the bubbling regime has also been reported by Farrell and Glicksman (1997). 4.4. Criterion for the viscous limit This work was aimed at veri#cation of the scaling rules at the viscous limit. According to Glicksman (1984, 1988), the limiting Reynolds number is 4. As mentioned earlier, Roy and Davidson (1989) found that the limit could be relaxed to a value of 30. From the results of this work (Fig. 5), it seems that the limit suggested by Roy and Davidson (1989) can be relaxed further up to about 100; see the captions of Figs. 5a–c for the highest particle Reynolds numbers used in these runs. This conclusion clearly needs independent corroboration.
4.5. Slugging bed Several studies have been carried out on bed scaling in the slugging regime; see for example Felice et al. (1992) and Horio et al. (1986). From the correlations for the onset of slugging and slug rise velocity by Stewart and Davidson (1967), Horio et al. (1986) concluded that their scaling rules, i.e. Eq. (2), should work in the slugging regime. Pressure (uctuation measurements by Felice et al. (1992), however, indicated that the scaling rules developed for bubbling beds did not apply to slugging beds. In this work, slugging was observed at a Froude number based on the excess gas velocity of ∼ 0:07, where foamed glass particles were (uidised in the smallest column (70 mm diameter, Fig. 6). Fig. 6 suggests that the scaling rules at the viscous limit do not apply to the slugging regime, supporting the observation of Felice et al. (1992). 4.6. Discussion of the methodology used in this work This work uses the particle cycle frequency to verify the scaling relationships. The cycle frequency is a measure of the vertical movement of particles. From the de#nition of the cycle frequency, only large vertical movement is taken into account. The technique used in this work is believed to be complementary to the techniques based on the pressure (uctuations, which take both the small and large frequencies into account. The technique used in this work attacks the scaling issue from the viewpoint of solids motion, while the pressure (uctuation measurements look at the problem from the viewpoint of the gas phase (mainly bubbles). As both techniques consider the vertical movement, it seems they are only appropriate for relatively deep beds where vertical movement is dominant. Recent work on particle circulation
M. Stein et al. / Chemical Engineering Science 57 (2002) 3649–3658
in shallow (uidised beds (aspect ratio less than 0.5) has shown poor qualitative and quantitative agreement with Eq. (4) (McCormack & Seville, 2001). 5. Conclusions Scaling relationships for bubbling gas-(uidised beds have been a subject of substantial experimental and theoretical investigations. Most experimental studies attack the scaling issue from the viewpoint of bubble motion. In this work, experimental veri#cation is carried out by direct measurements of vertical solids motion using the Positron Emission Particle Tracking (PEPT) technique. The following conclusions are obtained: • For geometrically similar beds, the pair of Froude numbers based on the minimum (uidisation and excess gas velocities is su
f d3 (s −f )g
≡ p 2 , Archimedes number, dimensionless particle diameter, L bed diameter, L particle frequency, T−1 dimensionless cycle frequency, dimensionless ≡ u02 =(gdp ), Froude number, dimensionless acceleration due to gravitational action, LT−2 bed height, L bed height at minimum (uidisation, L ≡ f u0 dp =, Reynolds number, dimensionless super#cial gas velocity, LT−1 minimum (uidisation velocity, LT−1 deviation from the two-phase theory, dimensionless
Greek letters w d s s f
wake volume as a fraction of bubble volume, dimensionless drift volume as a fraction of bubble volume, dimensionless shape factor of particles, dimensionless (uid viscosity, ML−1 T−1 particle density, ML−3 (uid density, ML−3
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Acknowledgements The authors are grateful to the EPSRC and ETSU of the DTI (UK) for supporting this work under Grants GR=J99414, 75128 and 75135. References Broadhurst, T. E., & Becker, H. A. (1973). The application of the theory of dimensions to (uidised beds. In Proc. Int. Symp. Ste. Chimie Indust., Toulouse, France. Fan, L. T., Ho, T. C., Hiraoka, S., & Walawender, W. P. (1981). Pressure (uctuations in a (uidised bed. AIChE Journal, 27, 388–396. Farrell, P. A., & Glicksman, L. R. (1997). Measurement of bubble characteristics in a scale model of a 70 MW pressurised (uidised bed combustor. In Proceedings of the 14th International ASME Conference on Fluidised Bed Combustion, Vancouver, Canada, May 11–14. Felice, R. D., Rapagna, S., & Foscolo, P. U. (1992). Dynamic similarity rules: Validity check for bubbling and slugging (uidised beds. Powder Technology, 71, 281–287. Fitzgerald, T. (1985). Coarse particle systems. In J. F. Davidson, R. Clift, & D. Harrison (Eds.), Fluidisation (2nd ed.). London: Academic Press. Fitzgerald, T., Bushnell, D., Crane, S., & Shieh, Y. C. (1984). Testing of cold scaled bed model for (uidised-bed combustors. Powder Technology, 38, 107–120. Foscolo, P. U., Felice, R. D., Gibilaro, L. G., Pistone, L., & Piccolo (1990). Scaling relationships for (uidisation: The generalised particle bed model. Chemical Engineering Science, 45, 1647–1651. Geldart, D. (1986). Solids mixing. In G. Geldart (Ed.), Gas Fluidisation Technology (pp. (97–123)). Chichester: Wiley. Glicksman, L. R. (1984). Scaling relationships for (uidised beds. Chemical Engineering Science, 39, 1373–1379. Glicksman, L. R. (1988). Scaling relationships for (uidised beds. Chemical Engineering Science, 43, 1419–1421. Glicksman, L. R., Hyre, M. R., & Farrell, P. A. (1994). Dynamic similarity in (uidisation. International Journal of Multiphase Flow, 20S, 331–386. Horio, M., Nonaka, A., Sawa, Y., & Muchi, I. (1986). A new similarity rule for (uidised bed scale-up. AIChE Journal, 32, 1466–1482. Johnstone, R. W., & Thring, M. W. (1957). Pilot Plants, Models and Scale-up Methods in Chemical Engineering. New York: McGraw-Hill. Kunii, D., & Levenspiel, O. (1977). Fluidisation Engineering. New York: Krieger Publishing. McCormack, A., & Seville, J. P. K. (2001). Solids motion in shallow gas-(uidised beds, unpublished work. Newby, R. A., & Keairns, D. L. (1986). Test of the scaling relationships for (uid-bed dynamics. In K. Ostergaard, & A. Sorensen (Eds.), Fluidisation (pp. (31–38)). New York: Engineering Foundation. Nicastro, M. T., & Glicksman, L. R. (1984). Experimental veri#cation of scaling relationships for (uidised beds. Chemical Engineering Science, 39, 1381–1391. Parker, D. J., Broadbent, C. J., Fowles, P., Hawkesworth, M. R., & McNeil, P. (1993). Positron emission particle tracking—a technique for studying (ow within engineering equipment. Nuclear Instruments and Methods A, 326, 592–607. Parker, D. J., Dijkstra, A. E., Martin, T. W., & Seville, J. P. K. (1997). Positron emission particle tracking studies of spherical particle motion in rotating drums. Chemical Engineering Science, 52, 2011–2022. Romero, J. B., & Johanson, L. N. (1962). Factors a=ecting (uidised bed quality. Chemical Engineering Progress, Symposium Series, 58(38), 28–32. Roy, R., & Davidson, J. F. (1989). Similarity between gas-(uidised beds at elevated temperature and pressure. In J. R. Grace, L. W. Shemilt, & M. A. Bergougnou (Eds.), Fluidisation VI (pp. (293–300)). New York: Engineering Foundation.
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