Effect of turbulence on particle and bubble slip velocity

Chemical Engineering Science 100 (2013) 120–136

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Effect of turbulence on particle and bubble slip velocity Swapnil V. Ghatage a,d, Mayur J. Sathe c, Elham Doroodchi c, Jyeshtharaj B. Joshi a,b,n, Geoffrey M. Evans c,nn a

Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India Homi Bhabha National Institute, Anushaktinagar, Mumbai 400 094, India c Discipline of Chemical Engineering, University of Newcastle, Callaghan, NSW 2308, Australia d Department of Chemical Engineering, Indian Institute of Technology, Gandhinagar 382424, Gujarat, India b

H I G H L I G H T S







Measurements of rise/settling velocity of bubble/particle in SLFB are carried out. Developed a mathematical model to predict slip velocity based on energy balance approach. The predicted values show good agreement with experimental results. PIV measurements show good correspondence with energy balance approach.

art ic l e i nf o

a b s t r a c t

Article history: Received 3 September 2012 Received in revised form 8 March 2013 Accepted 17 March 2013 Available online 25 March 2013

In multiphase systems involving a dispersed phase, such as fluidized beds, the interphase exchange of mass, heat and momentum transfer can be very different from those from a single particle, droplet or bubble system under terminal conditions. Such differences need to be correctly predicted for proper design of multiphase reactors. However, most existing methodologies still rely heavily on empirical relationships for parameters such as slip velocity, especially for systems operating under turbulent conditions where there is wide difference in the reported results. In this study the hindered settling/rising (slip) velocity of single steel particles (dPD ¼ 5–12 mm) and single air bubbles (dB ¼1–4 mm) has been measured in a solid–liquid fluidized bed of uniform size borosilicate glass beads (dP ¼5 and 8 mm) as a function of liquid superficial velocity. The homogeneity and intensity of the turbulence within the fluidized bed has been quantified and directly related to the slip velocity of the foreign (steel or bubble) particle. It was found that the turbulence resulted in an increase in the computed drag coefficient for all of the experimental conditions investigated. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Slip velocity Turbulence Bubbles Drops Particles Hindered settling velocity

1. Introduction Efficient contacting of the phases is crucial for many industrial processes including mineral flotation, Fischer Tropsch synthesis, extraction, gas absorption, hydrogenation, etc. The continuous phase can be either gas or liquid whilst the dispersed phase can be bubbles, drops or particles depending on the unit operation being considered. In order to carry out CFD simulation of such equipment, accurate closure relations for turbulence and interphase exchange terms (momentum, heat or mass) are necessary. The interaction forces play a vital role in defining the motion of both the continuous and

n Corresponding author at: Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India. Tel.: þ91 22 25597625; fax: þ91 22 33611020. nn Corresponding author. Tel.: þ 61 240339068; fax: þ 61 240339095 E-mail addresses: [email protected] (J.B. Joshi), [email protected] (G.M. Evans).

0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.03.031

dispersed phases. Drag force is an important interphase exchange term and a proper understanding of its influence on particle, droplet and bubble motion is an essential requirement for developing rational design and modelling approaches. The drag (or ‘slip velocity’) between the continuous and discontinuous phases is often estimated based on the terminal velocity of a single particle, droplet or bubble moving in a quiescent infinite fluid. In many cases of practical application, however, the drag is affected by both the presence of other dispersed elements as well as the level of turbulence within the moving fluid. There has been relatively few studies (e.g. Brucato et al., 1998; Aliseda et al., 2002; Friedman and Katz, 2002; Poorte and Biesheuvel, 2002; Yang and Shy, 2003, 2005; Lane et al., 2005b; Doroodchi et al., 2008) that quantitatively report slip velocity as a function of the level of turbulence. The turbulence is usually defined in terms of the Stokes number, which is the ratio of particle relaxation time to turbulent time scale. The studies listed above involve both experimental measurement and computational

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

analysis over a range of different turbulence conditions, including homogeneous, non-homogeneous and non-stationary. Not surprisingly, there are reported different views about the effect of turbulence on slip velocity. In order to experimentally investigate the influence of turbulence on slip velocity, the turbulence generating device ideally should be able to produce homogeneous, stationary turbulence, with easy manipulation of RMS fluctuating velocity and turbulent energy dissipation. The system should allow for measurement of spatial and temporal continuous and dispersed phase velocities, with an accuracy that would allow for various length and time scales of turbulence to be computed. Finally, it would be desirable if the system was of simple construction and easy scalability so that it could be implemented into industry. A solid–liquid fluidized bed (SLFB), especially when the continuous and dispersed phases are refractive index matched, satisfies all of these physical conditions. However, because it is a multi-particle system it does raise some questions regarding the modelling approach. For example, there is some debate as to whether the fluid or mixture (effective) density should be used to compute the buoyancy force on the dispersed phase. Some researchers (Foscolo et al., 1983; Foscolo and Gibilaro, 1984; Gibilaro et al., 1987; Di Felice et al., 1989; Khan and Richardson, 1990; Van der Wielen et al., 1996) argue that since the buoyancy is due to hydrostatic pressure then the mixture density should be used. Conversely, other researchers (Joshi et al., 1980; Joshi and Shah, 1981; Epstein, 1984; Clift et al., 1987; Fan et al., 1987; Lali et al., 1989; Joshi et al., 2001) state that buoyancy should be related to the fluid density in accordance with the principles of Archimedes buoyancy. For the case when a dense ‘foreign’ particle with diameter, dPD, is falling through a SLFB of smaller particles with diameter, dP, the consideration of a pseudo-fluid fluidized bed is quite reasonable when the mixture density, ρM, is not much greater than that of the liquid and the dPD/dP ratio is high. However, for larger values of ρM and smaller dPD/dP ratios, this phenomenon becomes more complex. Di Felice et al. (1989), Grbavcic et al. (1992) and Joshi et al. (2001) have discussed the criterion of using fluid density or mixture density depending upon the critical ratio of dPD/dP. The brief discussion above has highlighted the continued uncertainty in the modelling approaches used to describe the influence of turbulence on the slip velocity. For this reason, the first aim of this study was to provide a brief updated review of the literature. There is also a lack of experimental validation of the modelling approaches, especially for multi-particle (droplet or bubble) systems, which is due mainly to the variation in the turbulence conditions generated by the different sources. For many of these turbulent generating devices, such as moving grids, it is also very difficult to generate sufficiently levels of turbulence comparable to industrial applications. The SLFB potentially can overcome all of the existing limitations, and to test this possibility experimental measurement of the slip velocity of individual foreign particles and bubbles passing through a SLFB were undertaken. The results were then compared with modelled predictions based on both estimations of the energy dissipation rate and actual particle image velocimetry (PIV) measurements within the fluidized bed.

2. Literature review 2.1. Effect of turbulence on settling of particles Richardson and Zaki (1954) quantified the effect of presence of walls on the settling velocity of a fluidized particle as logV SW ¼ logV S∞ þ

dP D

ð1Þ

121

They also proposed the following correlation for the hindered settling velocity of a particle within a fluidized bed: V S ¼ V SW ∈nL ,

ð2Þ

where n ¼3.8 for creeping flow region and 1.4 for the turbulent region. Kennedy and Bretton (1966) modified the Richardson–Zaki correlation to include a fluidized bed of a mixture of species. They proposed the classification velocity of particle size di in a mixture of different size particles as V R,i ¼ ðV L −V i Þ=∈L ,

ð3Þ

where Vi is the superficial liquid velocity required to fluidize the particles of size di to the same void fraction of the mixture in which the classification velocity is calculated. They also proposed that the fluidizing velocity, Vi was related to the void fraction by V i ¼ ai ∈nL i

ð4Þ

The constants, ai and ni were evaluated using the data of Richardson and Zaki (1954), and then related to the Galileo number, Ga, by the following expression: ai ¼

μL expð−2:3di =DÞ , di ρL ðð15:35=Gai Þ þ ð1:363=Ga0:567 ÞÞ i

ð5Þ

and þ 28 ni ¼ 5:57Ga−0:0779 i

di Ga−0:1186 i D

ð6Þ

The classification velocity of a dense foreign particle, VR, in a fluidised bed is given by V R ¼ V SD −V S ,

ð7Þ

where VS and VSD are the relative velocities (with respect to the rising liquid) for the fluidised and foreign particle, respectively. Kunii and Levenspiel (1969) gave the following correlation for the particle hindered settling velocity as (ReP 4500)
0:5 3:1gðρPD −ρM ÞdPD V SD ¼ k , ð8Þ ρM where k is the wall correction factor given as k¼

V SW V SDW ¼ V S∞ V SD∞

ð9Þ

Joshi (1983) and Pandit and Joshi (1998) have derived an equation for hindered settling velocity of a heavy (foreign) particle in a solid–liquid fluidized bed based on energy conservation. The resultant expression for the normalized settling velocity is given by !0:5 V SD 1 ¼ V SD∞ 1 þ ð2ð1:5∈S V S Þ2 =ðC D∞ V SD Þ2 Þ !0:5 1 ¼ , ð10Þ 1 þð40:9∈2S V 2S =V 2SD Þ Lee and Durst (1982) and Tsuji et al. (1984) experimentally studied the drag observed by particles in a gas–solid suspension through a vertical pipe. Based on the analysis of these experimental data, Lee (1987) concluded that the drag coefficient for a particle in a two-phase suspension is always significantly lower than that for a particle in a laminar stream, mostly by an order of magnitude. It was observed that a large and heavy particle experiences drag in the longitudinal direction as it mainly responds to mean fluid motion. Conversely, small and light particles are mainly affected by fluid fluctuations and moves in the transverse direction. Di Felice et al. (1991) proposed a pseudo-fluid model for predicting the particle settling velocity in a fluidized bed. The pseudo-fluid model considered a large dense particle settling in a

122

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pseudo-fluid consisting of fluid and small particles. The density of the pseudo-fluid was assumed to be ρ ¼ ∈S ρS þ ð1−∈S ÞρL ,

ð11Þ

whilst the viscosity of the pseudo-fluid was calculated using the correlation by Happel (1957) for a fluid–particle suspension, i.e. " # 7=3 2=3 4∈S þ 10−ð84=11Þ∈S μ ¼ 1 þ5:5∈S ð12Þ 10=3 4=3 μ 10ð1−∈ Þ−25∈S ð1−∈ Þ S

S

Authors observed that the predicted classification velocities by this correlation were in close agreement with the pseudo-fluid model proposed by Di Felice et al. (1991) for the smaller particles (dP ¼1.6 mm and dPD ¼1.3–1.6 mm). Ruzicka (2006) related the effective density with the fluid and mixture density as ρef f ¼ ð1−f t ÞρL þf t ρM where the value of ft was given as f t ¼ 0:1635ðr−1Þ

The velocity of the pseudo-fluid was calculated as the combined velocity of the fluid and fluidized particles:

and r was the dimensionless particle size given by

V ¼ ∈S V P þ ð1−∈S ÞV L :

r ¼ ð1−∈L Þ0:33

ð13Þ

For estimating the drag coefficient of a particle in a fluid, the correlation proposed by Dallavalle (1948) was used for a dense foreign particle: 2 C D ¼ ð0:63 þ 4:8Re−0:5 PS Þ ,

ð14Þ

whilst the classification velocity was defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4gdPD ðρPD −ρÞμ2L V R ¼ V− 3C D ρμ2

ð15Þ

Grbavcic and Vukovic (1991) described the modified drag coefficient for a dense particle due to presence of fluidizing particles as C D0 ¼

4gdPD ðρPD −ρL Þ 3ρL ððV S =∈L Þ þV SD Þ2

,

ð16Þ

where VSD is defined by Kunii and Levenspiel (1969) and given by Eq. (8). Grbavcic et al. (1992) measured the effective buoyancy and drag for foreign large particles of varying density, settling or rising in a fluidized bed of smaller particles. They considered the force balance over a particle as ρP gvP ¼ F B,ef f þ F D

ð17Þ

Following Wen and Yu (1966), they assumed the drag force on a large particle under hindered conditions to be F D ¼ F D∞ f ð∈L Þ

ð18Þ

Hence, the drag force over a large particle can be written as π 2 F D ¼ dP ρL C D∞ V 2S f ð∈L Þ ð19Þ 8 Substituting Eq. (19) in Eq. (17) and dividing both sides by gvP gives ρP ¼ ρef f þ

3 ρL C D∞ V 2S f ð∈L Þ : 4 gdP

ð20Þ

Grbavcic et al. (1992) plotted ρP against VS2 and the obtained intercept ρeff was compared with the values of ρL and ρM. They proposed that the density of mixture is relevant at values of dPD/dP larger than 5. Also, through the experiments of rising single particle through fluidised bed, they proposed that for low values of dPD/dP ratios (around 3.36), ρeff is equivalent to fluid density, whereas, the effective density approaches mixture density when the ratio dPD/dP equals 16.25. They observed that the ratio of drag force to effective buoyancy force increases with decrease in size of foreign particle or increase in diameter of fluidised particle. Van der Wielen et al. (1996) proposed a correlation for predicting the hindered settling velocity based on the average density of fluidized bed, fluid density, foreign particle density, terminal settling velocity and Richardson–Zaki index of the foreign particle as   V SD ρ −ρ n=4:8 0:79n−1 ¼ PD M ∈L ð21Þ V SD∞ ρPD −ρL

ð22Þ

dPD dP

ð23Þ

ð24Þ

Ruzicka (2006) proposed that the effective density is equivalent to fluid density, when r o1; it is mixture density when r 47.12 and defined by Eq. (22) when 1≤r≤7.12. However, the estimation of drag applied on a particle in fluidized bed was unattended. Grbavcic et al. (2009) performed experiments to study the single particle settling velocities in a fluidized bed. Due to collision between a settling and fluidized particle an additional collision force was taken into account. They claim that the collision force can be one order of magnitude higher than the drag force between these particles and significantly affects the classification velocity of a foreign particle. The number of displaced particles was given as  3 dPD N ¼ ð1−∈L Þ ð25Þ dP The collision coefficient, K was defined as !−2:47  −1:48 ρM −5:07 dPD , K ¼ 2:2∈L dP ρef f

ð26Þ

where ρM is the mixture density and ρeff is the effective or relevant density and can be obtained as defined by Ruzicka (2006). Grbavcic et al. (2009) have given a correlation for the slip velocity between settling dense particle and the liquid as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4dPD gðρPD −ρef f Þ V SD ¼ ð27Þ 3ρef f ½C D þ Kð1−f ÞNðρL =ρef f ÞðdP =dPD Þ2  In the previous discussion, the turbulence was created by the fluidized particles. Table 1 shows the summary of these studies. In another set of studies, the turbulence was generated by other means and the effect of such a controlled turbulence was investigated on the particle settling velocity. Magelli et al. (1990) observed reduced settling velocities in a batch stirred tank. They observed that for dP less than ten times λ the particle settling velocity is equivalent to that in a quiescent liquid. However, larger eddies interact strongly with a particle, reducing the settling velocities by up to 40 per cent. Brucato et al. (1998) measured the drag coefficient for glass spheres settling in a Taylor–Couette flow. They found that the particle drag was either unaffected or increased by free stream turbulence based on particle size and intensity. For all the particle types studied the settling velocity was observed to decrease with an increase in the turbulence level. For the largest particle and at the highest turbulent intensities, drag coefficients up to 40 times higher than for those in a stationary liquid were observed. They proposed a correlation for the estimation of particle drag coefficient:  3 C D −C D∞ dP ¼ 8:76  10−4 : ð28Þ C D∞ λ Yang and Shy (2003) performed experiments of settling of heavy particles in grid generated stationary, near-isotropic turbulence. Comparison was made between the settling velocity of particle swarm in isotropic turbulence generated by a vibrating

Table 1 Summary of previous work on the effect of turbulence on particle settling velocity in SLFB. D (mm)

H (mm) ρL (kg/m3) lL (kg/ms)

dp (mm)

ρP (kg/m3)

Turbulence quantification

Conclusion

1

Richardson and Zaki (1954)

62

183

ReP ¼ 0.4–7150

2

Kennedy and Bretton (1966)

25.4, 50.8

3

Kunii and Levenspiel (1969) Joshi (1983), Pandit and Joshi (1998) Di Felice et al. (1991)

Analysed the published literature

Authors studied the slip velocity of fluidized bed. However, no comment on the classification velocity of foreign particle in uniform fluidized bed was made They modified the Richardson–Zaki correlation and proposed a correlation for the classification velocity of particle size di in a mixture of different size particles Authors gave a correlation for the drag coefficient for a particle settling in hindered condition A correlation is proposed for estimation of the classification velocity depending on the turbulent intensity and regime of operation Proposed a pseudo-fluid model based on fluid density and density of each phase in binary mixture. However, considered published empirical correlation for drag coefficient on particle Proposed a correlation for estimation of drag coefficient for foreign particle

4

5

818–1135

0.001– 0.015

0.25–6.35

2745–10600

1000, 1120

0.001, 0.035

0.9–2.0

2470–10900

Analysed the experimental data published so far in literature and applied energy balance approach Analysed published experimental data

6

Grbavcic and Vukovic (1991)

40

1200

1000

0.001

dPD ¼ 2.98, 5, 10

7

Grbavcic et al. (1992)

40

1800

1000

0.001

dP ¼0.645, 1.2, 1.94, 2.98, 5 ρP ¼ 2507–2680 dPD ¼ 10–19.5 324–8320

8

Van der Wielen et al. (1996)

40.3

1000

1000

0.001

dP ¼1.2, 1.94, 2.98 dPD ¼ 1.3–1.6

9

Grbavcic et al. (2009)

40

1200

1000

0.001

dP ¼1.6 dPD ¼ 10–19.5) dP ¼0.645, 1.2, 1.94, 2.98 and 5.

ρPD ¼ 2509–2540

ρPD ¼ 1100–1594

ρP ¼ 1024 1237–8320

ReP ¼ 700–5300

They observed that the ratio of drag force to effective buoyancy force increases with decrease in size of foreign particle and/or increase in diameter of fluidised particle Proposed a correlation to predict the hindered settling velocity of foreign particle. Predictions of the correlation were observed to be in close agreement with that of pseudo-fluid model proposed by Di Felice et al. (1991) Authors claim that the collision force between the particles can be significantly higher than the drag force. Also proposed a correlation for calculating the classification velocity of the foreign settling particle

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

S. no. References

123

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grid and that of a single particle in stagnant fluid. It was found that the turbulence generally increased the settling velocity of the particles with the maximum effect occurring at St≈1. Authors claimed that at St≈1, the particle motion has the same velocity as the fluid turbulent motion (minimum relative slip) and consequently the local drag force is minimum enhancing the particle settling rate. It should be noted that these experiments were carried out with fine particles of about 60–500 mm. They observed that even at large values of ReP≈17–39 and VS∞/υK≈5–30, the settling velocities were higher than corresponding terminal values. At ReP ¼25 and VS∞/υK≈10, the settling velocity reached its maximum of about 107% of terminal settling velocity when St approaches unity. Some modest decrease in the settling velocity was found at St41.5, where settling velocities were found to be less than terminal settling velocity. The authors have also justified the use of Kolmogorov time scale because the time scale for the vorticity of characteristic eddies is proportional to τk. Yang and Shy (2005) studied two-way interaction between solid particles and stationary near-isotropic air turbulence. The focus on particle settling was somewhat limited, with only 5 experimental data points available on slip velocity modification. The system consisted of a gaseous cruciform apparatus with turbulence supplied by a pair of counter rotating fans and perforated plates at each end. The settling velocity (VS) in turbulent flow was observed to be much higher than the terminal settling velocity (VS∞) in a stationary fluid. Authors claimed that the maximum influence on settling velocity occurs at the St≈1. In order to justify this they showed consistency of their experimental results with published numerical results at constant value of parameter St/(VS∞/u′). They found that the value of (VS−VS∞) reached a maximum of 0.13u′ when Stokes number approached unity and VS∞/u′≈0.5 at ReP o1. Also, the settling velocities were observed to be strongly influenced by Reλ, when St≈1. The characteristic time and velocity scales they proposed are τk and u′. Yang and Shy (2003, 2005) present their data in terms of the Kolmogorov time scale of turbulence rather than the large eddy turnover time scale. We need to remember that only those eddies which are smaller than the particle are capable of influencing momentum transfer between the particle and the fluid hence the drag force. Any eddy larger than the particle will physically carry the particle and therefore is unable to transfer momentum at a level of drag force formulation. Lane et al. (2005b) proposed a correlation to estimate particle drag coefficient based on available literature data. VS ¼ 1−1:4St 0:7 expð−0:6StÞ V S∞

ð29Þ

This correlation shows a continuous decrease in the settling velocity of both solids and bubbles with increase in stokes number till St¼1. The correlation gives minimum settling velocities at St¼ 1and further increase in settling velocity with increase in Stokes number. Doroodchi et al. (2008) examined the effects of dispersed phase density and size on the applied drag force under turbulent conditions created by oscillating grids. The reduction in settling velocity was observed to be a function of both particle size and density and turbulent characteristic. They observed that the significant reduction of settling velocity occurred at low ratios of particle density to liquid density and high turbulence intensities. At u′¼ 19.2 mm/s, the reduction in velocity ratio (VS/VS∞) as high as about 25% was observed. However, when the particle diameter is much greater than the integral length scale of turbulence, the velocity ratio increased to values around unity. To capture the effects of these parameters on the drag coefficient, the Richardson number was employed. They observed that the effect of turbulence on the settling velocity of particle was decreased as the

Richardson number, Ri, increases. Negligible reduction in the settling velocity was observed for Ri greater than 200. Zhou and Cheng (2009) experimentally investigated the role of grid generated turbulence on particle settling. In most of the cases they found reduced settling velocity in the presence of turbulence. However, they observed that the reduced settling velocity could not be correlated with St or Ri. Table 2 summarizes the studies on the effect of turbulence on particle settling velocity in turbulence devices other than a solid– liquid fluidized bed (SLFB). The analysis shows contradictory views about the retardation (Magelli et al., 1990; Brucato et al., 1998; Doroodchi et al., 2008; Zhou and Cheng, 2009) or enhancement (Yang and Shy, 2003, 2005) of settling velocity of particle due to turbulence. However, it should be noted that the turbulence is created in various ways in different studies as well as the range of particle size and density covered is different. Magelli et al. (1990) studied the turbulence effect on fine (140–1000 mm), medium dense (1020–2450 kg/m3) particles in heterogeneous turbulence in stirred tanks. Yang and Shy (2003, 2005) performed experiments in isotropic, stationary turbulence and used fine (60– 505 mm), very dense (2500–19,300 kg/m3) particles. Doroodchi et al. (2008) and Zhou and Cheng (2009) studied the settling of large (2–8 mm) but light (1050–2300 kg/m3) particles in nearisotropic, stationary turbulence. 2.2. Effect of turbulence on bubbles A lot of work has been devoted to the modification of bubble and drop motion due to turbulence. While considering the effect of turbulence on bubble and drop motion, one should note that for bubbles (as well as drops) additional factor of surface changing parameter (tortuosity) needs to be considered. It further increases the complexity of underlying physics as a partial slip (full slip condition for large bubbles having clean interface and no slip for small bubbles and/or contaminated surface) is present over the surface and presence of pressure fluctuations on the surface causes the drag change with its motion. The mean bubble rise velocity is an important indication of the influence of turbulence on bubble rise characteristics. The mean bubble rise velocity in homogeneous regime is generally found to decrease with an increase in the bubble concentration (∈G) (Davidson and Harrison, 1966; Joshi, 1980; Joshi et al., 2002; Lance and Bataille, 1991; Zenit et al., 2001; Panidis and Papailiou, 2002; Poorte and Biesheuvel, 2002; Risso and Ellingsen, 2002; Bunner and Tryggvason, 2002). The hindrance mechanism require a very small volume fraction (∈G o0.1%) to develop. Roig and de Torunemine (2007) showed that potential flow theory is sufficient to predict bubble rise velocity for ∈G o0.02. However, the mean bubble rise velocity has been reported to increase for large bubbles in case of heterogeneous flow regimes (Griffith and Wallis, 1961; Davidson and Harrison, 1966; Zahradnik et al., 1997; Thakre and Joshi, 1999; Joshi et al., 2001; Joshi and Ranade, 2003). The general interpretation for decrease in the rise velocity in homogeneous regime is the enhanced rate of momentum transfer with an increase in the intensity of the liquid turbulence. In heterogeneous regime, the bulk liquid motion modifies the bubble rise velocity with respect to the external observer. Since the bubble concentration is higher in the central core region where the bulk motion is upwards, the net effect of bubble and liquid motion results into an increase in the bubble rise velocity with respect to external observer. It may be emphasized that the increase in actual slip velocity with respect to ∈G is a small fraction of the increase in rise velocity with respect to external observer. Risso (1999) studied motion of quite large bubbles (dB ¼ 18 mm) in a gradient of turbulence under microgravity conditions. Turbulence was created by jet, which was isotropic and decaying exponentially in z direction. By eliminating buoyancy, the average bubble deformation

Table 2 Summary of previous work on the effect of turbulence on particle settling velocity in turbulence devices other than SLFB. Reference

D (mm)

H (mm)

ρL (kg/m3)

μL (kg/ms)

dp (mm)

ρP (kg/m3)

Method for creation of turbulence

1

Magelli et al. (1990)

132 and 240

530 and 980

1000

0.001 and 0.2

1020, 1150, 2450

2

Brucato et al. (1998)

Couette type device External cylinder: ID=56 mm, H=1 m inner cylinder, OD=40 mm

1000

0.001

0.14, 0.23, 0.30, 0.33, 0.98 Silica (0.18– 0.22 and 0.425–0.5), glass beads (0.06–0.07, 0.22–0.25, 0.425–0.5)

Ruston turbine multiple impeller Rotating inner cylinder and stationary outer cylinder

St=0–0.8

3

Yang and Shy (2003)

Square crossection with 75×110 mm2 test section

1.2

0.00002

Tungsten particles (0.06, 0.16) and glass particles (0.36, 0.50)

19,300 and 2500

Oscillating grid generated turbulence

ε=1.24×10−4– 23.59×10−4 and ReP=2 to 39

4

Yang and Shy (2005)

Square with 150×150 mm2

1.2

0.00002

Copper (0.012), glass (0.04), lead (0.024)

8800, 2500, 11,300

A pair of counter rotating fans and perforated plates

St=0.36, 1 and 1.9, ReP=0.03, 0.303, 0.312

5

Lane et al. (2005b)

Simulation results were compared with the experimental measurements of Barigou and Greaves (1992) in stirred tank

Ruston turbine

St=0–2.5

6

Doroodchi et al. (2008)

Rectangular tank 300 mm width

7

Zhou and Cheng (2009)

Square crosssection of 500×500 mm2

150

1000

2500

Turbulence quantification

1000

0.001

Nylon and Teflon (2 to 8)

1140 and 2300

Grid turbulence using a pair of vertically oriented grids

St=0.35–1.6, ReP=183–4108, u′ =7.2–19.2 mm/s

1000

0.001

Grains (2.78– 7.94 mm)

1050 and 1077

Oscillating grids

ReP=66–824

Comments

The settling velocity reduced due to turbulence. Reported settling velocity as low as 15% of the particle terminal velocity in quiescent liquid. The measured settling speed decreases with increasing turbulence (rotational speed of inner cylinder). The turbulence generally increased the settling velocity of the particles with the maximum effect occurring at St≈1. Some modest decrease in the settling velocity was found at St 41.5. The settling velocity (VS) in turbulent flow is much greater than the terminal settling velocity (VS∞) in still fluid. Reported decrease in settling velocity with increase in St, attained minima at St=1 and further increased. The reduction in settling velocity is a function of particle size and density and turbulent characteristics. The maximum interaction between the continuous and dispersed phases was observed at low ratios of particle density to liquid density and high turbulence intensities. The relative settling velocity in presence of turbulence is always smaller than terminal settling velocity.

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Sl. no.

125

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Table 3 Summary of the work on the effect of turbulence on rise velocity of bubbles. H (mm)

ρL (kg/ m3)

μL (kg/ ms)

dB (mm)

Sl. Reference no.

D (mm)

1

Maxey et al. (1996)

2

Risso (1999)

Discussed the interactions microbubbles with turbulence using two-fluid continuum models and direct numerical simulation. 77 600 1000 0.001 13–21 Confined jet

3

Panidis and Papailiou (2002)

4

Poorte and Biesheuvel (2002)

Square column 300  300 mm2 and 1200 mm length Square column 450  450 mm2 and 2000 mm length

Method for creation of turbulence

1000

0.001 3

Grid generated turbulence

1000

0.001 0.34 and 0.57

Grid generated turbulence

Turbulence Turbulence effect on bubble/drop slip velocity quantification

τB/τK ¼0–2

Bubble rise velocities in presence of turbulence were observed to be lower by an extent of 25–50% to that in still liquid.

ReB ¼400– 1100

The large bubbles (dB ¼ 18 mm) move faster than the liquid. They did not comment on the effect of turbulence on the rise velocity of bubbles.

ReB o200, Sto0.04

The reduction in rise velocities up to 35% was noted in presence of turbulence.

Table 4 Summary of the work on DNS simulation on effect of turbulence on particle settling velocity. Sl. Reference no.

Number of grids

Turbulence quantification

Conclusion

1

Wang and Maxey (1993)

323, 483, 643 and 963

Reλ ¼ 21 and 62

2

Maxey et al. (1996)

483 and 963

Reλ ¼ 31 and 62

3

Bagchi and Balachandar (2003) Reddy et al. (2010a, 2010b) Reddy et al. (2013)

141  160  128 ReP ¼ 60–600, turbulence intensity¼10–25%

The slip velocity (VS) of heavy particles is higher than terminal settling velocity by as much as 50%. The effect is higher when particle size and response time are comparative with Kolmogorov scale. Also, the larger particles are subjected to non-linear drag which reduces the net increase in settling velocity. They found that the settling particle travels faster when particle response time was comparable with the Kolmogorov time scale. When the particle response time greatly exceeds the Kolmogorov time, they are controlled by the larger scale eddies and the enhancement effect is lowered. Also, non-linear drag is expected to oppose the settling (rise) enhancement if ReP 41. Within the studied range, it was found that turbulence had little influence on the mean drag of particles. Authors state that the effect of turbulence on non-linear drag of particles is significant only at large Re and when the turbulent fluctuation is quite strong. It was observed that the settling velocity of particle decreased as the number of particles surrounding was increased. At higher turbulence, the settling velocity was found to decrease up to 27% of terminal settling velocity.

4 5

0.6×106 to 2.5×106 6.5  106

ReP ¼ 0.01–1 ReP ¼ 1–200

and relative (slip) velocity was kept very low. The integral length scale of turbulence was constant at 0.021 m. The bubbles move faster than the liquid because their effective mass is less, the relationship obtained pffiffiffi was V B ¼ 3V L . However, the microgravity results are difficult to compare with the other studies performed in the presence of buoyancy. Panidis and Papailiou (2002) studied the influence of a swarm of bubbles on the structure of a near-isotropic turbulent flow field behind a grid. They found that the local velocities had a close link with void distribution. Further, the magnitude and direction of the lift force acting on bubbles was observed to be very sensitive to bubble deformation and Reynolds number. Also, the presence of bubbles destroys the isotropy of turbulence for their system. However, they did not comment on the effect of turbulence on the rise velocity of bubbles. Poorte and Biesheuvel (2002) performed experiments on the motion of gas bubbles in order to validate the numerical simulations presented by Spelt and Biesheuvel (1997). Isotropic turbulence was generated by means of an active grid consisting of interlaced shafts with flaps at regular intervals. For weak turbulence of ‘intermediate’ length scales, they presented the following semi-analytical solution to the bubble motion: VB u2 ¼ 1−0:75π n V B∞ ℓ

ð30Þ

where non-dimensional turbulent intensity is, u ¼ u′=V B∞ and non-dimensional lengthscale, ℓn ¼ ℓ=τB V B∞ . The ratio u=ℓn gives the Stokes number, which is a measure of how quickly the bubble attains its rise velocity to an interacting eddy. The

bubble size was estimated with an assumption on equisized bubbles. The diameter of bubble was calculated from the volumetric gas flow rate and the frequency of bubble generation. They studied bubble motion of sizes 0.34 mm and 0.57 mm in diameter. The range of the Stokes number was small with the maximum value of ∼0.04. The authors observed that bubbles were slowed by the turbulence (up to 35%). They attributed this reduction to the lift force which redirects the bubbles laterally to the regions of relative downflow and viscous forces then make the bubbles adapt their speed to the fluid velocity fluctuations. They concluded that most of the experimentally noted velocities were within 9% of the theoretical predictions of Spelt and Biesheuvel (1997). Table 3 summarizes the studies on the effect of turbulence on bubble rise velocity. Having reviewed the literature and given some critical thought to the problem of particle and bubble slip in turbulence, it is clear there exist great complexity in physical interactions between turbulent fluid–particle or fluid–bubble systems. A single universal theory governing particles of both varying density ratio and length scales is far from being realized. The literature survey brings out the various drag models reported by various researchers in order to quantify the effect of turbulence on motion and settling/rise velocity of a particle/bubble in SLFB. Some of these models are completely empirical, some are purely from theory and the rest are the combination of both. The DNS simulations (Wang and Maxey, 1993; Maxey et al., 1996; Bagchi and Balachandar, 2003; Reddy et al., 2010a, 2010b, 2013) have given fair understanding (Table 4), however, detailed understanding is still constrained by the computational power.

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

127

Foreign particle insertion 8 arrangement Outlet

50

Synchronizer

50

Glass column

1

1500

Outer square column 2

Camera Laser

Distributor 200

3

Solid-liquid fluidized bed

Calming section 4 Vent Inlet

Pump

Rotameter 5

Liquid

Refractive index matched NaI Solution

Fig. 2. PIV experimental setup.

7

PUMP

6 Fig. 1. Characterization velocity experimental setup (all dimensions are in mm): (1) glass column; (2) outer square column; (3) distributor; (4) calming section; (5) rotameter; (6) pump; (7) storage tank; and (8) foreign particle insertion arrangement.

Further, in the published work the influence of free stream turbulence on the drag coefficient of particle or bubble has been experimentally investigated extensively. There is a need to understand the physics of the effect of the turbulence on the momentum transfer in the vicinity of fluid–particle and fluid–bubble interfaces. Therefore, it was thought desirable to investigate the turbulence effect on slip velocity of particles and bubbles in SLFB where the quality of turbulence has been characterized. Secondly, the measured values of slip velocities for bubbles and particles has successfully demonstrated the study of turbulence in fluidized bed resulting in quantitative measurement of normalized slip velocity at higher Stokes numbers than usually obtained in other turbulence-generating devices. It may be pointed out that at high bed voidages the contact forces between particles of fluidized bed and the transiting particle are relatively small but their influence increases at low bed voidages, especially for solid particles. More research is required to quantify these contact forces. Further, analysis of the effect of bed voidage on particle–particle or particle–bubble interaction is being carried out by Discrete Element Modelling (DEM) simulation.

3. Experimental 3.1. Measurement of dense particle classification velocity A schematic of the experimental setup of the SLFB is shown in Fig. 1. It consisted of a glass circular column (1), with an inner diameter of 50 mm and a height of 1500 mm. The circular test

section of the fluidized bed was encased in a square column (2) which was filled with the same liquid to ensure proper photographic images. The distributor was a perforated plate (3) containing 128 holes of 2 mm diameter on a triangular pitch of 3.1 mm. A calming section (4) packed with either 6 or 8 mm glass beads of 0.2 m height was provided to homogenize the liquid flow before it reached the liquid distributor. The required liquid flow rate through the column was maintained using a rotameter (5) and globe valve at the bed inlet. A 2 hp centrifugal pump (6) was used for pumping the liquid from storage tank (7). Borosilicate glass beads with mean diameter of 5 70.03 and 8 70.03 mm have been used as fluidized particles. The glass beads were purchased from Sigmund Lindner (Germany) under the trade name Silibeads type P having refractive index of 1.464. Proper arrangements (8) were made for the insertion of the foreign particle from the top of the fluidized bed. The liquid used for the classification velocity measurements was water. Precision-diameter dense (steel) particles with mean diameters (72 mm) of 5, 6, 7, 8, 9, 10, 11 and 12 mm were used for the classification velocity measurements. The slip velocities of single foreign particle (dense solid particle or gas bubble) were measured along a vertical distance of approximately 300 mm using a high speed video camera and a strong LED lighting behind the bed. Whilst the refractive indices of the water (1.33) and fluidized borosilicate glass beads (1.464) were not the same, it was still possible to observe the motion of the foreign particles with a high speed video camera (model: Photron Fastcam Super-10K). The classification velocity was calculated by tracking the centremotion of the dense particle from images (512  420 pixels) captured at 500 frames per second giving a time resolution of 0.002 s. Due care was taken to ensure that the particles had reached terminal velocity prior to entering the test section and not influenced by the mean flow. The superficial liquid velocity was varied between 0.06 and 0.3 m/s, whilst the fractional liquid hold-up was varied from 0.47 to 0.80. Each settling velocity reported is the average of five measurements, where the reproducibility was observed to be within 72 per cent.

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3.2. Turbulence measurement using Particle Image Velocimetry (PIV) The water–borosilicate glass system described above was satisfactory for determining the classification velocity because there was sufficient clarity in the high speed video images to track the centre-motion of the dense particles. However, quantitative measurement of the liquid velocity field by optical methods (e.g. particle image velocimetry), is limited to only a few per cent for the water–borosilicate glass system. For the high solid concentrations used in this study (more than 50 vol%) the refractive index of the particles needed to be matched to that of the liquid. An excellent review on the subject of RI matching is given by Wiederseiner et al. (2011). Aqueous sodium iodide solution was used to match the refractive index with borosilicate spheres. To prepare the solution, lab reagent grade sodium iodide powder was weighed in proportion to required amount

CLASSIFICATION VELOCITY, VR (mm/s)

1800 1600 1400 1200 1000 800 600 400 200 0

0

2

4

6

8

10

12

of water. Powder was gradually added to water. The best RI match was obtained for RI ¼ 1.464 corresponding to 58.5% w/w sodium iodide in water. To maintain the clarity of solution, sodium thiosulphate was added to the solution. Addition of approximately 0.1 g of sodium thiosulphate was sufficient to clarify 1 l of sodium iodide solution. The PIV measurements were undertaken using the experimental setup shown in Fig. 2. The RI matched sodium iodide solution was filled in fluidized bed and surrounding square column, with the fluidized bed particles being 5 mm or 8 mm diameter high-precision borosilicate glass beads. The sodium iodide solution in fluidised bed was seeded with fluorescent seeding particles. Dantec PIV system was used consisting of Litron LDY 300 laser capable of generating 30 mJ/pulse energy at 1000 Hz. Phantom v640 camera capturing image pairs with resolution of 1600  1600 pixels at 900 Hz and BNC 575 synchronizer were employed. The high speed system provided time resolved data, giving a closer look at the dynamics liquid velocity within fluidised bed. The field of view was 50  50 mm2 covering the complete column width. Post-processing of the captured raw PIV images was undertaken to determine the velocity vectors. The raw PIV images were processed using the image processing routines programmed in MATLAB R2011a. Out-of-plane motion of the seeding particles and strong local velocity gradients caused some spurious velocity vectors. Median filtering, with a threshold value 1.5 times the median of surrounding vectors, was applied to filter the high spurious vectors. A signal-to-noise ratio of 4 was applied to filter the low spurious vectors. Parameters like time difference between laser pulses, light sheet thickness and seeding density were optimized so that spurious vectors remained below 2 per cent. Further details pertaining to PIV measurements are given by Deshpande et al. (2009, 2010) and Sathe et al. (2010). Other excellent work in the field include: Tokuhiro et al. (1998), Deen et al. (1999), Lindken and Merzkirch (1999, 2002), Bröder and Sommerfeld (2003).

14

PARTICLE SIZE, dP (mm) PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

1.0

CLASSIFICATION VELOCITY, VR (mm/s)

1800 1600 1400 1200 1000 800 600 400 200 0

0

2

4

6

8

10

12

14

PARTICLE SIZE, dP (mm) Fig. 3. The measured classification velocity: (A) fitted line, terminal settling velocity of steel balls in quiescent liquid, steel ball in 5 mm glass particles: (Δ) ∈L ¼0.47, (○) ∈L ¼0.52, (☐) ∈L ¼ 0.60, (◇) ∈L ¼0.66, (  ) ∈L ¼ 0.70, ( þ ) ∈L ¼ 0.76, ( ) ∈L ¼ 0.80. (B) Fitted line, terminal settling velocity of steel balls in quiescent liquid, steel ball in 8 mm glass particles: (Δ) ∈L ¼ 0.47, (○) ∈L ¼ 0.52, (☐) ∈L ¼ 0.60, (◇) ∈L ¼0.66, (  ) ∈L ¼ 0.70, ( þ ) ∈L ¼ 0.76, and ( ) ∈L ¼ 0.80.

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 4. Comparison of experimental classification velocity with that predicted using Kennedy and Bretton (1966) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (◯) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, ( ) 12 mm and in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) 11 mm, and ( ) 12 mm.

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

129

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 5. Comparison of experimental classification velocity with that predicted using Kunii and Levenspiel (1969) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (○) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, ( ) 12 mm and in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) 11 mm, and ( ) 12 mm.

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 7. Comparison of experimental classification velocity with that predicted using Di Felice (1991) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (○) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, ( ) 12 mm and in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) 11 mm, and ( ) 12 mm.

2.0

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 6. Comparison of experimental classification velocity with that predicted using Joshi (1983) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (○) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, and ( ) 12 mm and in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) dP ¼ 11 mm, and ( ) 12 mm.

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

0.8

-0.2

1.5

1.0

0.5

0.0 0.0

0.5

1.0

1.5

2.0

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 8. Comparison of experimental classification velocity with that predicted using Pandit and Joshi (1998) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (○) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, and ( ) 12 mm and in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) 11 mm, and ( ) 12 mm.

4. Results and discussion 4.1. Richardson–Zaki exponent The liquid volume fraction of the fluidized bed, without the presence of introduced dense particles, was obtained

experimentally by measuring the expansion of the bed as a function of superficial liquid velocity. The measurements were compared with predictions using the Richardson–Zaki correlation given by Eq. (2). From all experimental measurements the exponent, n, was found to be between 1.33 and 1.40, which is in

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S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

0.8

0.6

0.4

0.2

0.0 0.0

-0.2

0.2

0.4

0.6

0.8

the predicted values were 1.2–5 times the measured values. The comparison with the correlation by Kunii and Levenspiel (1969) is shown in Fig. 5, where it can be seen that the predictions were relatively constant across the measurement range. In Fig. 6, the predictions of Joshi (1983) were generally in good agreement with the experimental observations, especially for the fluidised bed of 5 mm particles where the standard deviation was 720%. The correlation of Di Felice et al. (1991) was found to under-predict the classification velocity for both the 5 and 8 mm fluidised particles (Fig. 7). Conversely, the correlation by Pandit and Joshi (1998) overpredicted the classification velocity for both 5 and 8 mm fluidised bed particles as shown by Fig. 8. Finally, the predictions using the correlation from Grbavcic et al. (2009) were in excellent agreement for the 5 mm diameter fluidized bed particles (Fig. 9), but there was much larger scatter for the 8 mm diameter particles. The foregoing discussion brings out clearly that the range of parameters covered in the present and the previous works are different and accordingly, therefore, the previous correlations do not hold for the results of the present work. However, a careful analysis of the previous work brings out the necessity for understanding the mechanism by which the values of drag coefficient get related to the terminal settling velocities of particles (VS∞ and VSD∞), voidage, etc. For this purpose, it was thought desirable to propose a model describing the role of turbulence on the drag coefficient. It was also thought desirable to undertake turbulence measurements using particle image velocimetry.

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s)

Fig. 9. Comparison of experimental classification velocity with that predicted using Grbavcic et al. (2009) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (○) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, and ( ) 12 mm and in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) 11 mm, and ( ) 12 mm.

excellent agreement with the value of 1.4 recommended by Richardson and Zaki for turbulent flow.

4.4. Classification velocity of particles: mathematical model and comparison with experimental data Joshi (1983) has established the energy balance for the fluidized beds. Some brief features are reproduced below for the sake of clarity and continuity. The frictional pressure drop in a pipeline under turbulent conditions is given by ΔP ¼

4.2. Classification velocity as a function of liquid velocity Fig. 3 shows the experimentally observed classification velocities for the different diameter dense (steel) particles as they passed through the water/borosilicate glass fluidised bed; where the liquid superficial velocity was varied to give liquid volume fraction, ∈L, values of 0.47, 0.52, 0.6, 0.66, 0.7, 0.76 and 0.8. Figs. 3A and B show the results for 5 mm and 8 mm fluidised particles, respectively. The solid line represents a fitted curve to experimentally measured terminal velocities for the steel particles in an infinite quiescent liquid. Firstly, it can be seen that, the classification velocity decreases with a decrease in the bed voidage. For the conditions investigated, the classification velocity of the steel particles was decreased to below even 10 per cent of the unhindered terminal velocity values. This reduction in settling velocity is mainly because of the enhanced momentum transfer associated with the turbulence generated by the fluidized particles. The classification velocities of all the steel particles in the 5 mm bed were observed to be higher than those in the 8 mm bed. The reason for this is that at the same voidage, the upward (opposing to particle settling) superficial liquid velocity was higher for the 8 mm particles. 4.3. Classification velocity: comparison with literature correlations The measured classification velocities were compared with the correlations proposed by various researchers. The comparison with the correlation by Kennedy and Bretton (1966) for the 5 and 8 mm diameter fluidised bed particles is given in Fig. 4. It can be seen that

2f HV L 2 ρL , D

ð31Þ

where the friction factor, f, in the pipeline is related to the rms fluctuating velocity in the radial direction and is given by Davies (1972) as rffiffiffi 0 ur f ð32Þ ¼ 2 VL Eq. (32) holds for smooth as well as rough pipes. In other words, the intensity of turbulence represents the rate of momentum transfer and hence the value of friction factor. Thus we can write f ¼ f ∞ þ f ′,

ð33Þ

where f∞ is the friction factor for smooth pipe and f′ is the additional friction factor due to roughness. Using the above approach, the authors have proposed that, for a particle under terminal settling conditions, the drag coefficient can be written by the following equation: rffiffiffiffiffiffiffiffiffi ur ′ C D∞ ð34Þ ¼ V S∞ 2 The terminal velocity is hindered when the particle becomes a part of a solid–liquid fluidized bed. In fact, under turbulent conditions, the turbulence in fluidized bed enhances the drag coefficient of every particle. Following the procedure of Eq. (33), the drag coefficient can be written as C D ¼ C D∞ þ C D ′

ð35Þ

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

250

BUBBLE RISE VELOCITY, VB (mm/s)

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

0.8

0.6

0.4

0.2

200

150

100

50

0 0.0 0.0

0.2

0.4

0.6

1

0.8

2

3

4

5

250

BUBBLE RISE VELOCITY, VB (mm/s)

Fig. 10. Comparison of experimental classification velocity of particles with that predicted using correlation based on energy balance (Eq. (51)) in a bed of uniform particles of 5 mm: (Δ) 6 mm, (○) 7 mm, (☐) 8 mm, (◇) 9 mm, (∇) 10 mm, and ( ) 12 mm.

0.6

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

0

BUBBLE SIZE, dB (mm)

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s)

0.4

200

150

100

0.2

50

0 0

1

2

3

4

5

BUBBLE SIZE, dB (mm)

0.0 0.0

0.2

0.4

0.6

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 11. Comparison of experimental classification velocity of particles with that predicted using correlation based on energy balance (Eq. (51)) in a bed of uniform particles of 8 mm: (▲) 6 mm, (●) 7 mm, (■) 8 mm, (◆) 9 mm, (▼) 10 mm, (✖) 11 mm, and ( ) 12 mm.

Following the same approach as Eq. (34) then urA ′ ¼ V S∞

131

CD′ 2

!0:5

Fig. 12. Measured bubble rise velocity: (A) fitted line, terminal rise velocity of bubbles in quiescent liquid, air bubble in a bed of uniform particles of 5 mm: (Δ) ∈L ¼ 0.47, (○) ∈L ¼0.52, (☐) ∈L ¼ 0.60, (◇) ∈L ¼ 0.66, (  ) ∈L ¼0.70, ( þ) ∈L ¼0.76, and ( ) ∈L ¼0.80. (B) Fitted line, terminal rise velocity of bubbles in quiescent liquid, air bubble in a bed of uniform particles of 8 mm: (Δ) ∈L ¼ 0.47, (○) ∈L ¼0.52, (☐) ∈L ¼ 0.60, (◇) ∈L ¼0.66, (  )∈L ¼0.70, ( þ)∈L ¼0.76, and ( ) ∈L ¼ 0.80.

Table 5 Rise velocity of bubbles in 5 and 8 mm fluidized beds: data analysis using Eqs. (51) and (53). Sl. Bubble no. diameter (mm)

5 mm fluidized bed

8 mm fluidized bed

Richardson– Constant C Zaki index (n) in Eq. (51)

Richardson– Constant C Zaki index (n) in Eq. (51)

1 2 3 4 5 6 7

1.43 1.37 1.33 1.21 1.08 0.73 0.35

1.43 1.42 1.40 1.39 1.28 1.02 0.73

ð36Þ

where urA′ is the radial component of the additional liquid phase turbulence generated by the particles in the bed. An estimate of CD′ can also be obtained on the basis of energy balance. At the bottom of the fluidized bed, the liquid is introduced against the static bed height. The power needed (energy

1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.50 1.33 1.24 1.05 0.88 0.57 0.33

1.50 1.45 1.41 1.37 1.15 0.81 0.57

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S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

For a single particle, the force balance is given by

input rate) is given by the following equation: π  D2 V L gHð∈S ρS þ ∈L ρL Þ Ei ¼ 4

ð37Þ

The liquid leaving the bed is at height H and has potential power given by π  D2 V L gHρL : ð38Þ EL ¼ 4

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

The net power dissipated (EB) in SLFB is (Ei-EL), which is given by π  D2 V L gH∈S ðρS −ρL Þ: EB ¼ ð39Þ 4

C D πdp

2

π  1 3 ρL V S 2 ¼ dp ðρS −ρL Þg, 2 6

ð40Þ

where CD is the drag coefficient based on total particle area. Substitution of Eq. (40) in Eq. (39) gives EB ¼

  2 3π D HC D V S 2 V L ρL ∈S : 4 dP

ð41Þ

The energy dissipation rate at the particle–liquid interface is calculated on the basis of energy dissipation rate in the vicinity of 2 single particle, which is given by C D πdp 12ρL V S 2 , and the total

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 13. Comparison of experimental bubble classification velocity with that predicted using correlation based on energy balance (Eq. (51)) in a bed of uniform particles of 5 mm: (Δ) 1 mm, (○) 1.5 mm, (☐) 2 mm, (◇) 2.5 mm, (∇) 3 mm, ( ) 3.5 mm, and ( ) 4 mm.

PREDICTED CLASSIFICATION VELOCITY, VR (m/s)

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

EXPERIMENTAL CLASSIFICATION VELOCITY, VR (m/s) Fig. 14. Comparison of experimental bubble classification velocity with that predicted using correlation based on energy balance (Eq. (51)) in a bed of uniform particles of 8 mm: (▲) 1 mm, (●) 1.5 mm, (■) 2 mm, (◆) 2.5 mm, (▼) 3 mm, (✖) 3.5 mm, and ( ) 4 mm.

Fig. 15. PIV snapshots: (A) raw image and (B) processed image showing liquid velocity vectors.

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

133

Table 6 Turbulence parameters estimated from PIV measurements of solid–liquid fluidized beds: 5 mm glass beads. Sl. no.

Liquid volume fraction, ∈L (dimensionless)

Fluctuating liquid velocity, u′ (m/s)

Turbulent intensity (%)

Energy dissipation rate, ε (W/kg)

Turbulent lengthscale, l (mm)

u′/VS

1 2 3 4 5

0.52 0.66 0.73 0.78 0.81

0.0500 0.0682 0.0761 0.0752 0.0764

74 60 58 53 49

0.098 0.094 0.074 0.061 0.054

1.28 3.38 5.93 6.94 8.21

1.324 1.057 1.036 0.951 0.884

(dimensionless)

Table 7 Turbulence parameters estimated from PIV measurements of solid–liquid fluidized beds: 8 mm glass beads. Sl. no.

Liquid volume fraction, ∈L (dimensionless)

Fluctuating liquid velocity, u′ (m/s)

Turbulent intensity (%)

Energy dissipation rate, ε (W/kg)

Turbulent lengthscale, l (mm)

u′/VS

1 2 3 4 5

0.52 0.66 0.73 0.78 0.81

0.066 0.092 0.100 0.111 0.118

85 65 57 51 48

0.113 0.115 0.100 0.086 0.074

2.58 6.70 10.21 12.14 14.13

1.516 1.158 1.021 0.915 0.861

4

Substitution of Eq. (35) in Eq. (43) gives   3π E¼ D2 HV S 3 ρL ∈L ∈S C D ′ 4dP

NORMALIZED VELOCITY, VSD/VSD (-)

3.5

ð44Þ

The power dissipation per unit liquid mass ((π/4)D2H∈LρL) is given by

3

Pm ¼ 2.5

3V S 3 ∈S C D ′ : dP

ð45Þ

The integral length scale (ℓ) is considered to be dP/2. Therefore P m ℓ ¼ 1:5V S 3 ∈S C D ′ :

2

1.5

u′ ¼ C∈S V S

ð47Þ

where u′ is the bulk turbulent fluctuating velocity, value of C takes into account the non-isotropicity and possible deviation of integral length scale from dP/2. Substitution of Eq. (47) in Eq. (36) gives

1

0 0.0001

ð46Þ

Substitution of Eq. (36) in Eq. (46) and knowing P m ℓ ¼ ðu′Þ3 , gives

0.5

C D ′ ¼ 2ðC∈S Þ2 0.001

0.01

0.1

1

10

STOKES NUMBER, St (-) Fig. 16. Comparison of various studies from literature with present study showing effect of turbulence on settling velocity of foreign particle: (☐) Brucato et al. (1998) [215–250 mm]; ( ) Brucato et al. (1998) [425–450 mm]; (▭) Aliseda et al. (2002); (○) Friedman and Katz (2002); (  ) Poorte and Bieshuvel (2002); (◇) Yang and Shy (2003) [solid particles in air]; ( þ) Yang and Shy (2003) [tungsten particles]; (-) Yang and Shy (2003) [glass particles]; Present study: (Δ) 6 mm and ( ) 12 mm dense particle in 5 mm glass beads; (▲) 6 mm and ( ) 12 mm dense particle in 8 mm glass beads.

number of particles:   2 3π D HC D∞ V S 3 ρL ∈L ∈S : ES ¼ 4 dP

ð42Þ

The net energy dissipation rate in the liquid (E) is (EB-ES) and is given by  E¼

(dimensionless)

 3π D2 HV S 3 ρL ∈L ∈S ðC D −C D∞ Þ 4dP

ð43Þ

ð48Þ

This additional drag coefficient is responsible for the hindrance. Therefore, the relationship between the hindered settling velocity and the terminal settling velocity is  0:5 VS C D∞ ¼ ð49Þ ′ V S∞ C D þ C D∞ Substitution of Eq. (48) in Eq. (49) and knowing that CD∞ is 0.11 in the turbulence regime (on the basis of total surface area), gives !0:5 VS 1 ¼ ð50Þ V S∞ 1 þ ð2ðC∈S Þ2 =0:11Þ Similarly, the settling of a foreign particle gets hindered and on the basis of Eq. (50), can be written as !0:5 V SD 1 ¼ , ð51Þ V SD∞ 1 þ ð2ðC∈S Þ2 =0:11Þ and the classification velocity is given by Eq. (7). All the experimental results of the classification velocity in the present work were analysed on the basis of Eqs. (7) and (51) where the correlation coefficients were found to be 0.92 and 0.90. The parity plots are shown in Figs. 10 and 11 for 5 and 8 mm fluidized

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particles. The values of C were found to be 1.5 and 1.93, respectively. The parity plot supports the form of Eq. (51). It may be pointed out that Eq. (51) is based on Eq. (47). Therefore, it was thought desirable to check the validity of Eq. (47) using PIV and discussed in Section 4.6. 4.5. Classification velocity of bubbles: mathematical model and comparison with experimental data The classification velocity for bubbles is given by VR ¼ VB þ VS,

ð52Þ

where VB and VS are the slip velocities for the bubbles and particles, respectively. The values of VB were estimated from Eq. (52) and are plotted in Fig. 12A and B for 5 and 8 mm diameter particle fluidised beds, respectively. The experimentally measured terminal rise velocities of bubbles, VB∞ are shown by fitted line. These results have been analysed using two methods: Firstly, all the VB data will be subjected to the following Richardson–Zaki correlation: VB ¼ ∈nL ð53Þ V B∞ For the bubbles of 1 mm size, the value of n for both the beds was observed to be 1.43 which is in fairly close agreement with the value of 1.4 for solid particles. This means that the interface of a 1 mm diameter bubble can be considered rigid similar to solid particles and no-slip boundary condition prevails for these two cases. However, with an increase in bubble size, the bubble–liquid interface becomes increasingly mobile and accordingly the role of turbulence on the momentum transfer at the interface diminishes. The same effect can be seen in Table 5 where the value of C decreases with an increase in bubble diameter. Figs. 13 and 14 show the comparison of experimentally measured classification velocities of bubble with those predicted using the Eq. (51). Again it may be pointed out that the origin of Eq. (51) is Eq. (47). For checking the validity of Eq. (47), PIV measurements were undertaken and are presented in the following section. 4.6. PIV results PIV measurements were carried out within the same range of liquid voidage where settling velocity measurements of foreign particles have been performed. Fig. 15A shows the raw PIV snapshot of RI matched fluidised bed with 8 mm glass beads. Fig. 15B is the processed image of 8000 images showing time averaged liquid velocity vectors. The turbulence quantities were calculated from the PIV measurements. The values are reported in Tables 6 and 7 for 5 mm and 8 mm fluidized glass particles, respectively. The values of u′ were found to follow the following relationship: 0

u ¼ C∈S V S

ð54Þ

The values of C obtained from Eq. (51) were noted to be 1.5 and 1.93 for 5 and 8 mm glass particles, respectively. The corresponding PIV measurements show good agreement with values of 1.37 and 2.18, respectively. An attempt was made to compare the obtained results of ratio of hindered settling velocity to terminal settling velocity against Stokes number with those noted by other researchers in various turbulence devices. Fig. 16 depicts the comparison. The results are consistent with those other studies shown. The normalized slip velocity was found to monotonically decrease with increasing Stokes number. Interestingly, the results from this study are at higher Stokes number (41) than those reported previously, and exhibit a gradual decrease in slope and possibly a minimum in normalized slip velocity for Stokes number greater than 10. The existence of a minimum value would be consistent with the hypothesis of Lane (2005a).

5. Conclusions The absolute value of the hindered settling velocity, VSD, of a dense foreign particle in a solid–liquid fluidized bed was found to decrease with decreasing bed liquid volume fraction. For the extreme case at very low fluidization level, the hindered settling velocity as low as about 20 per cent that of a single particle settling in a stationary liquid in the absence of fluidized bed particles was observed. For a given fluidized bed volume fraction of glass beads, the classification velocity, defined as VR ¼ VSD−VS, was found to be on average 20–40 per cent higher for 5 mm diameter glass beads as compared to those for the 8 mm glass beads. The reason for the increase in VR with decreased fluidized bed particle diameter was due to a decrease in the superficial liquid velocity, VL, at the same bed liquid volume fraction; as well as an increase in the foreign particle slip velocity due to the changes in isotropicity of turbulence and integral length scales with the bed, as defined by the constant C in Eq. (51). Values of C of 1.5 and 1.93, with correlation coefficient of 90%, were obtained by applying an energy balance analysis to the experimental data for the 5 and 8 mm diameter fluidized glass beads, respectively. Values for the constant, C, was also obtained from Eq. (54) using the measured liquid superficial velocity, volume fraction of glass beads, and PIV measured bulk velocity fluctuations. Values of C of 1.37 and 2.18 were obtained for the 5 and 8 mm fluidized glass beads, respectively, and were in good agreement with those obtained from the energy balance analysis. For bubbles, it has been pointed out that the value of C for mobile interface is expected to be lower than that for rigid interface for which no-slip boundary condition holds. The values of C for different bubble sizes rising in 5 and 8 mm diameter particle fluidized beds are given in Table 5, and found to be varying between 0.33–1.50 and 0.57–1.50 for 5 mm and 8 mm fluidized particles respectively.

Nomenclature ai C CD CD∞ CD′ di dP dPD D E EB Ei EL ES f f∞ f′ ft FB FD FD∞ FG g Ga H k K ℓ

constant in Eq. (4), m/s dimensionless constant in Eq. (47), dimensionless drag coefficient, dimensionless drag coefficient in infinite medium, dimensionless drag coefficient due to presence of other particles, dimensionless diameter of particle of species i with uniform size, m diameter of particle comprising fluidized bed, m diameter of foreign particle, m column diameter, m power dissipation in liquid phase, W power dissipation in SLFB, W power input, W power recovery because of pumping of liquid, W power dissipation at solid–liquid interface, W friction factor, dimensionless friction factor for smooth pipe, dimensionless additional friction factor due to roughness, dimensionless transition function given by Eq. (23), dimensionless buoyancy force, N drag force, N drag force over a particle in quiescent liquid, N gravitational force, N gravitational acceleration, m/s2 3 Galileo number, dPD gρL ðρPD −ρL Þ=μ2 , dimensionless column height, m wall correction factor, dimensionless collision coefficient, dimensionless characteristic turbulence lengthscale, m

S.V. Ghatage et al. / Chemical Engineering Science 100 (2013) 120–136

ℓn n ni N Pm r Re ReP Reλ Ri St u u′ ur′ urA ′ um vP V VB VB∞ Vi VG VL VP VR VS VS∞ VSD VSD∞ VSDW VSW

non-dimensional turbulence lengthscale, ℓ=τB V B∞ , dimensionless Richardson–Zaki index, dimensionless constant in Eq. (4), dimensionless number of displaced particles, dimensionless power dissipation per unit liquid mass, W/kg dimensionless particle size given by Eq. (24), dimensionless liquid Reynolds number, DVLρL/μL, dimensionless particle Reynolds number, dPVPρL/μL, dimensionless Reynolds number based on Taylor microscale, dimensionless    ℓ Richardson number, gðρPρ−ρL Þ 02 , dimensionless L  u 0

ðρP =ρL Þ þ C V Þ Stokes number, ð3=4ÞðC D∞ =dP ÞV S∞ uℓr , dimensionless non-dimensional turbulent intensity, u′/VB∞, dimensionless bulk turbulent fluctuating velocity, m/s rms turbulent velocity in radial direction, m/s radial component of the liquid phase turbulence generated by the particles, m/s mean centerline velocity, m/s volume of particle, m3 superficial velocity of pseudo-fluid, m/s bubble rise velocity, m/s terminal bubble rise velocity, m/s superficial liquid velocity required to fluidize the particles of size di to void fraction of mixture, m/s superficial gas velocity, m/s superficial liquid velocity, m/s particle velocity, m/s classification velocity, m/s interstitial fluidization velocity, VL/∈L, m/s terminal velocity of particle, m/s interstitial fluidization velocity for dense particle, m/s terminal settling velocity of dense particle, m/s bounded settling velocity for dense particle, m/s bounded settling velocity of fluidizing particle, m/s

Greek letters ∈ μ μ ρ ρ τB τK τL τP λ υK ε ΔP

fractional phase hold-up, dimensionless viscosity of fluid, kg/ms viscosity of pseudo-fluid, kg/ms density, kg/m3 density of pseudo-fluid, kg/m3 bubble relaxation time, s Kolmogorov time scale, s integral time scale, s particle relaxation time, s Kolmogorov microscale, m Kolmogorov velocity scale, m/s energy dissipation rate per unit mass, m2/s3 pressure drop, N/m2

Subscripts B D eff F G L M P PS S ∞

bubble dense particle effective fluid gas liquid mixture particle pseudo-fluid solid infinite medium

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