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CFD modelling of liquid fluidized beds in slugging mode
Powder Technology 167 (2006) 94 – 103 www.elsevier.com/locate/powtec
CFD modelling of liquid fluidized beds in slugging mode Paola Lettieri
a,⁎
, Renzo Di Felice b , Roberta Pacciani b , Olumuyiwa Owoyemi
a
a
b
Department of Chemical Engineering, University College London, UK Dipartimento di Ingegneria Chimica e di Processo, Universita' di Genova, Italy
Received 9 May 2005; received in revised form 1 May 2006; accepted 16 June 2006 Available online 30 June 2006
Abstract Computational Fluid Dynamics (CFD) modelling has been used to simulate a liquid fluidized bed of lead shot in slugging mode. Simulations have been performed using a commercial code, CFX4.4. The kinetic model for granular flow, which is already available in CFX, has been used during this study. 2D time-dependent simulations have been carried out at different water velocities. Simulated aspects of fluidization such as voidage profiles, slug formation, pressure drop and pressure fluctuations have been analysed. The fluid-bed pressure drop was found to be greater than the theoretical one at all velocities, in agreement with experimental observations reported for fully slugging fluidized beds. Power spectral density analysis of the pressure signal was used to investigate the development of the flow pattern and the structure of the fluid-bed with increasing fluidizing velocity. A comparison between experimental and simulated results is also reported. © 2006 Elsevier B.V. All rights reserved. Keywords: Liquid-fluidized beds; Slugging mode; Pressure drop; CFD; Granular kinetic model
1. Introduction Particles and processes involving particles are of enormous importance in the chemical and allied industries. Fluidized beds are widely employed in industrial operations, ranging from the pharmaceutical and food industry, to processes such as catalytic cracking of petroleum, combustion and biomass gasification. Fluidization is the operation by which particles are transformed into a fluid-like state through suspension in a gas or liquid. Many of the characteristic features of gas-fluidized beds, like the excellent solid mixing, heat and mass transfer properties, can be related to the presence of bubbles and are dominated by their behaviour. It is well known that the gas–solid contacting efficiency is highly dependent on the fluid-bed hydrodynamics and bubble properties. The formation and development of bubbles in the fluid-bed has been studied over a considerable span of time, particularly with the aid of visualization techniques, see Rowe et al. [1].
⁎ Corresponding author. Tel.: +44 20 76797867; fax: +44 20 73832348. E-mail address: [email protected] (P. Lettieri). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.06.012
The recent development of mathematical modelling of particulate solids behaviour together with the increased computing power is now enabling us to simulate the behaviour of fluidized powders and to link fundamental particle properties directly to powder behaviour and predict the interaction between particles and gaseous or liquid fluids. In this regard, Computational Fluid Dynamics (CFD) modelling provides a new tool to support engineering design and research in multiphase systems such as fluidized beds. Most of the fluidization modelling using CFD is applied to gas– solid systems and can be divided into two groups, the Lagrangian– Eulerian models and the Eulerian–Eulerian models, see Crowe et al. [2]. The Lagrangian approach describes the solids phase at a particle level and the gas phase as a continuum. The Eulerian– Eulerian approach, on the other hand, is based on the two-fluid model (TFM) and treats the phases as interpenetrating continua. Using the TFM, numerical simulations of gas–solid fluidbeds have been performed by various research groups: Kuipers et al. [3,4] developed a two-dimensional hydrodynamic model to study the bubble formation process at a single orifice, which was subsequently developed for three-dimensional fluidized bed simulations [5]. Simulations of freely bubbling gas
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fluidized beds using kinetic theory were reported by Ding and Gidaspow [6], van Wachem [7] and Peirano et al. [8]. Using the two-fluid granular temperature model, Pain et al. [9] and Zhang et al. [10] have simulated the behaviour of bubbling and slugging gas fluidized beds. In a recent study, Lettieri et al. [11,12], have used the Eulerian–Eulerian approach to simulate the behaviour of gas fluidized beds under different flow regimes, spanning from bubbling to slugging and turbulent fluidization. Using the CFX4.4 code (Ansys plc, formerly AEA Technology) 2D and 3D simulations of bubbling Geldart Group B [13] fluidized beds were compared using two different modelling approaches, namely, the Eulerian Granular model, which is currently available within CFX-4, and the particle-bed model [14], recently implemented in CFX-4.4 [11]. Quantitative results demonstrated good agreement between the approaches of the two models. The comparison between 2D and 3D simulations showed that 2D models can reproduce reasonable well aspects of the bubbling fluidization. Using the granular kinetic theory, Lettieri et al. [15] subsequently extended the work to quantitatively investigate the transition from bubbling to slugging gas fluidized beds of a Geldart Group B powder. The simulated transition velocity was compared with predictions obtained from the Baeyens and Geldart [16] criterion; simulated values obtained for the maximum slugging bed height were found to be in reasonable agreement with predictions obtained from the Matsen et al. [17] model. In the present study, the CFD modelling work has been extended to investigate liquid fluidized beds. CFD simulations of lead shot particles in the slugging regime have been performed at different fluidizing velocity. The work presented in this paper stems from an experimental study carried out at the University of Genova by Di Felice [18], which investigated the slugging regime of liquid–solid systems and their transition to turbulent fluidization. This work was part of a fundamental study carried out at UCL on the CFD modelling of fluidized beds, sponsored by BP and Tioxide Ltd., aimed at investigating the capability of general multiphase models to simulate and predict aspects of different fluidization regimes and develop and validate an ad hoc fluid-bed model. Slugging fluidization was first identified in 1967 as a type of fluidization regime [19] and since then has been one of the flow regimes which has been the least studied over the years. Slugging normally occurs in beds of high aspect ratio (N1), often with a small diameter with respect to the particle size. A slug is simply a bubble whose diameter is nearly equal to that of the bed itself and can be generally categorized into two types: axisymmetric slugs (round-nose and squared-nose slugs) and wall slugs [19]. This paper presents the formation and development of axisymmetric slugs and wall slugs in the simulated liquidfluidized bed. Simulated aspects of fluidization such as pressure drop, pressure fluctuations and voidage profiles are also presented. The analysis of the power spectral density of the pressure signal is included to describe the development of the flow pattern within the simulated bed. A comparison between experimental and simulated results is also reported.
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Table 1 Governing equations used for the granular kinetic model Granular kinetic model governing equations Continuity Gas phase: Aðeg qg Þ ¯d eg qg u¯g ¼ 0 þj At Solid phase: Aðes qs Þ ¯dðes qs u¯sÞ ¼ 0 þj At Momentum balance Gas phase: A ¯P þ eg qg g¯−b u¯g−u¯s ¯d eg qg u¯g u¯g ¼ j ¯d ¯sg−eg j eg qg u¯g þ j At Solid phase: A ¯d ðes qs u¯s u¯sÞ ¼ j ¯d ¯ss es j ¯P j ¯Ps þ es qs g¯ þ b u¯g u¯s ðes qs u¯sÞ þ j At Granular temperature balance Solid phase:
¯ u¯s ¼ gs −Ps I¯ þ ¯ss : j
Dissipation of granular energy: " rffiffiffiffi # 2 4 H 2 ¯ −jd u¯s gs ¼ 3 1−e g0 es qs H dp p
Constitutive equations solid phase Radial distribution function: g0 ¼
" #−1 3 es 1=3 1− max 5 es
Solids stress tensor: ¯¯ss ¼ 2ls S¯s þ ks − 2 ls tr S¯s I¯ with 3
1 ¯ u¯s þ ðj ¯ u¯s ÞT j S¯s ¼ 2
Solids pressure: Ps ¼ es qs Hs ð1 þ 2g0 es ð1 þ es ÞÞ Bulk viscosity: 0:5 4 Hs ks ¼ es qs dp g0 ð1 þ es Þ 3 p Shear viscosity: 0:5
2 0:5 2 5p qs dp Hs0:5 4 Hs 4 þ 96 1 þ g0 es ð1 þ es Þ ls ¼ es qs dp g0 ð1 þ es Þ 5 5 p ð1 þ es Þes g0
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2. The granular kinetic model The granular kinetic theory model was used during this work. It solves mass and momentum equations for each phase. The momentum balance of the gas phase is given by the Navier– Stokes equations modified to include an inter-phase momentum transfer term. The momentum balance of the solid phase includes description of the solids pressure and of the solid-phase stress tensor, both obtained from granular kinetic theory [20]. Granular kinetic theory describes the particle–particle interactions as binary collisions, resembling those between molecules in the gas kinetic theory. By analogy with the thermodynamic temperature of the gases, granular theory introduces the granular temperature, Θs, to describe the kinetic energy associated with the particle fluctuations, e.g., Chapman and Cowling [21], Ocone and Astarita [22], where 1 2 Hs ¼ bvV N 3 s
ð1Þ
According to this model, in addition to mass and momentum, an energy balance is solved to estimate the granular temperature. In addition to the granular temperature, granular kinetic theory introduces the coefficient of restitution, es (where 0 b es b 1), to account for the non-ideal behaviour or inelasticity of the solids; also introduced is the radial distribution function, go, a parameter which gives a statistical measure of the probability of particle contacting. The function of go is to allow a tight control of the solids volume fraction, so that the maximum packing is not exceeded and more accurate flow characteristics can be achieved. There are a number of different forms of the radial distribution function in the literature; the equation given by Ding and Gidaspow [6] has been used in this work. From knowledge of the granular temperature, the coefficient of restitution and the radial distribution function, both the solid phase pressure and viscosity can be described. The solids pressure represents the solids phase normal forces due to particle collisions. The expression given by Lun at al. [23] has been used in this work. The non-isotropic part of the stress tensor is modelled by considering the solid phase as a Newtonian fluid. It is then related to the strain rate tensor by means of the bulk viscosity, which describes the resistance of the particle phase against compression, and the shear viscosity. In this work, the equation given by Lun et al. [23] has been used for the bulk viscosity and the expression proposed by Gidaspow [20] has been adopted for the shear viscosity. The full set of equations used during the simulations has been previously reported in Lettieri et al. [11]. Table 1 summarizes the continuity, momentum balance, granular temperature and constitutive equations of the granular kinetic model employed in this paper. The inter-phase drag term in the gas and solid phase momentum equations is expressed as F¯D ¼ bðu¯f − u¯sÞ
ð2Þ
with the inter-phase drag coefficient β being expressed as follows: b¼
3 Cd es qf ju¯f − u¯sjE 4 dp
ð3Þ
where |ūf − ūs| is the modulus of the slip velocity vector (relative velocity of the solid to the gas phase), E is a correction coefficient introduced in order to take into account the presence of high particle concentration in the bed. The correction factor E is generally modelled as a function of the solids volume fraction (Gidaspow [20]), i.e., E ¼ ð1−es Þ−1:65
ð4Þ
The correlation established by Ihme et al. [24] was used to calculate CD: CD ¼
24 þ 5:48Re−0:573 þ 0:36 Re
ð5Þ
3. Simulations The simulations were carried out by using the “Multi Fluid Model” (MFM) available in CFX-4.4, which is based on an Eulerian–Eulerian description. This model solves the continuity and momentum equations for a generic multi-phase system and therefore allows the determination of separate flow field solutions for each phase simultaneously. For the present case of two-phase flow, there are seven equations and as many unknowns: the volumetric fractions, the four velocity components and the pressure P (shared by both phases). The numerical solutions were obtained by a finite volume method based on a collocated grid approach that used the Rhie– Chow algorithm [25] to prevent chequerboard oscillations. The continuity and momentum balance equations were coupled using the SIMPLEC algorithm (Semi-Implicit Method for Pressure Linked Equations–Consistent Algorithm). Time-dependent simulations were performed with 10− 4 s time steps and 25 SIMPLEC iterations were required to achieve full numerical convergence. The convergence behaviour of the SIMPLEC algorithm was analysed by monitoring the mass flow residual as a function of the number of iterations at each time step. Under-relaxation factors between 0.6 and 0.7 were adopted for all flow quantities. Pressure was not under-relaxed, as required by the SIMPLEC algorithm. The resulting linear equation sets were solved by using suitable single-block or multi-block verõsions of the Strongly Implicit Procedure [26] for the momentum and scalar-transport equations, and a conjugate gradient method with incomplete Cholesky preconditioning [27] for the pressurecorrection equation. The hybrid-upwind discretization scheme [28] was used for the convective terms. Simulations here reported relate to a rectangular geometry, whose dimensions in 2D are 30 mm width, 1250 mm height; the column-to-particle ratio is chosen in order to be into the slugging regime. The lateral walls were modelled using no-slip velocity boundary conditions for both phases. Dirichlet boundary conditions were employed at the bottom of the bed to specify a uniform gas inlet velocity. Pressure boundary conditions were employed at the top of the freeboard. This implies Dirichlet boundary conditions on pressure, which was set to a reference value of 1.015 × 105 Pa, and Neumann boundary conditions to the gas flow, i.e., all flow quantities are given zero normal gradient.
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Fig. 1. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.08 m/s, 2 mm square cells.
The initial conditions specify the concentration of solids in the bed, and the fluid flow through the bed. The settled bed was considered 376 mm deep, and the solids volume fraction was defined as 0.40. Fluid flow was defined only in the vertical direction. A constant pressure was defined in all horizontal planes up through the bed of particles. The upper section of the simulated geometry, or freeboard, was considered to be occupied by water only.
4. Results and discussion 4.1. Voidage profiles and bed expansion Results presented in this section show 2D simulations of the slugging fluidization of lead-shot having dp = 1.7 mm and ρs = 10,900 kg/m3. Simulations were performed by increasing the fluidizing velocity from u = 0.08 m/s, 0.10 m/s, 0.12 m/s,
Fig. 2. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.10 m/s, 2 mm square cells.
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Fig. 3. 2D-slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.12 m/s, 2 mm square cells.
0.20 m/s and 0.30 m/s. The simulation runs were carried out for 6 s for the first three velocities investigated and up to 10 s real time at the highest velocities 0.20 m/s and 0.30 m/s. At each velocity, aspects of fluidization such as slug formation, pressure drop and pressure fluctuations were examined to study the development of the slugging regime, as experimentally done by Di Felice [18]. Snapshots obtained from the simulations are shown in Figs. 1–5, where the voidage profiles of the 2D slugging simulations at the different velocities investigated are reported. At the start of each simulation, waves of voidage are created, which travel
through the bed and subsequently break to form bubbles as the simulation progresses. At low velocities flat slugs form in the top part of the fluid-bed while the bottom part maintains a fairly homogeneous structure. With large slugging velocities and in beds of large and dense powders, slugs are usually observed to become asymmetric with formation of wall slugs, as reported by Baeyens and Geldart [16]. Initial aspects of this process can be observed in the snapshots of the voidage profile obtained at the fluidizing velocity 0.12 m/s, see Fig. 3. The process of wall slug formation becomes more and more evident with increase in the fluidizing velocity, as represented in Figs. 4 and 5 for velocities
Fig. 4. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.20 m/s, 2 mm square cells.
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Fig. 5. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.30 m/s, 2 mm square cells.
equal to 0.2 m/s and 0.3 m/s. At high fluidizing velocities, the slugs become larger and the phenomenon of coalescence also becomes more noticeable. At the same time, the dense phase develops into a uniform structure characterised by an average dense phase voidage of 0.7. As the computed bubbles and slugs become better defined, regions of voidage distribution at the void edges can be identified, as experimentally observed by Yates et al. [29] using the UCL X-ray imaging facility. The snapshots reported in Fig. 4 show the oscillation of the bed surface and the sudden drop of the bed height as the slugs break through. The oscillation of the fluid-bed height was monitored over time for all velocities investigated. Fig. 6 shows a typical example of the results obtained at the fluidizing velocity u = 0.20 m/s. During the analysis of the results, the first few seconds of each simulation run were discarded until a pseudo-oscillatory behaviour of the pressure drop and bed height fluctuations were observed. This was done to ensure that the data extrapolated corresponded to a pseudo-steady-state solution. Figs. 1–5 show also the considerable relative increase in bed expansion as the fluidizing velocity increases, a 6% increase was obtained at 0.08 m/s, a 20% increase at 0.10 m/s, 41% at 0.20 m/s and up to a 100% increase in bed height at the highest fluidized velocity investigated, 0.30 m/s. Fig. 7 shows a comparison between the values of the bed height obtained from the simulations and the experimental ones obtained by Di Felice [18] for the fluidization velocities investigated with the CFD. The comparison shows a reasonably good match for low velocities up to 0.20 m/s. However, the comparison deviates at the highest velocity investigated, 0.3 m/s, at which the experiments show that the system has fully transitioned to the turbulent regime thus giving a much greater bed expansion, while the CFD simulation still shows the presence of slugs
within a progressively more uniform dense phase. The delayed transition from slugging to turbulent fluidization obtained with the CFD shows the limitations of the applicability of the general multiphase models used to simulate the liquid–solid fluidized bed under study. 4.2. Pressure drop The overall behaviour of fluid–particle interaction processes is governed by the forces acting on individual particles. At minimum fluidization conditions a bed of powder is fully supported by an upward fluid flow, and the drag force exerted by the fluid on the particles, which is proportional to the global pressure drop across the bed, is balanced by the buoyant weight
Fig. 6. Averaged bed height variation with time u = 0.20 m/s.
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Fig. 7. Bed height, comparison between experimental and simulated values.
Fig. 9. Simulated and experimental normalised pressure drop as a function of the water superficial velocity.
of the suspension. Thus, the pressure drop across the bed equates to the buoyant weight of the suspension: where L is the height of the bed, εmf is the bed voidage at minimum fluidization, and ρf and ρs are the fluid and particle density, respectively. Eq. (6) has been experimentally verified since the first publication of fluidization appeared; see Wilhelm and Kwauk [30]. However, it does not hold for channelling and slugging fluidized beds. When slugging occurs, measured pressure drops exceed the theoretically predicted value, with the difference becoming more and more significant as the bed moves into the full slugging regime, as reported by Di Felice [18]. The physical origin of the difference between the observed and theoretical pressure drop is not yet fully understood. It has been associated with the energy needed to accelerate the solids;
but also to the potential energy dissipated by the solids in the slugging regime, see Geldart et al. [31] and Chen et al. [32]. The values of the pressure drop obtained over time were recorded for each simulation run. The results were analysed by sampling the data at a frequency of 100 Hz. A considerable increase in the pressure fluctuation with time was observed as the simulated fluidizing velocity was increased. Fig. 8 shows a typical example obtained for the case at u = 0.20 m/s. Fig. 9 shows a comparison between the values of the pressure drop obtained from the simulations and those obtained experimentally by Di Felice [18] with increasing the water fluidizing velocity. These were obtained by averaging the values of the pressure drop obtained over time for each simulation run and normalising them with respect to the effective bed weight calculated using Eq. (6). The experimental results show a steep increase in the pressure drop up to 0.1 m/s followed by a decrease due to the transitioning from slugging to
Fig. 8. Simulated pressure variation as a function of time, u = 0.20 m/s.
Fig. 10. Simulated pressure standard deviation as a function of the water superficial velocity.
DP ¼ ðqs −qf Þð1−emf ÞgL
ð6Þ
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Fig. 11. Power spectral density analysis of the pressure signal, u = 0.10 m/s.
Fig. 13. Power spectral density analysis of the pressure signal, u = 0.3 m/s.
turbulent regime. In agreement with the experimental observations, the simulated pressure drop was found to be greater than the theoretical one at all velocities investigated, as typical of the slugging regime, with values of the pressure drop increasing steeply up to 0.12 m/s and then levelling off at the higher fluidizing velocities investigated. However, the comparison with the experimental values shows a divergence of the trend at the higher velocities investigated as the CFD predicted slugging fluidization whilst the experiments showed a transition to the turbulent regime. Transforming the data plotted into Fig. 8 into the deviation standard of the pressure signal, the diagram shown in Fig. 10 is obtained. The deviation standard increases with the fluidizing velocity, consistently with the development of the flow pattern of slug flow and increase in bed pressure drop. Values of the standard deviation were found to be of the same order of magnitude as the experimental ones. However, similar to the
results reported in Fig. 9, the trend of the CFD and experimental results differ insomuch the transition from slugging to turbulent regime is delayed in the CFD simulations, as reflected in the absence of a maximum in the pressure standard deviation obtained from the simulations.
Fig. 12. Power spectral density analysis of the pressure signal, u = 0.2 m/s.
4.3. The fluid-bed structure The dynamic behaviour and structure of fluidized beds is often characterised in terms of the spectral density analysis of pressure time series, as also reported by J. van der Schaaf et al. [33], where the flow behaviour is quantified looking at the frequency of the largest values of the power spectral density. Figs. 11–13 report the power spectral density for the simulations run at 0.10 m/s, 0.20 m/s and 0.30 m/s, where the maximum in the variance of the pressure signal is obtained. The power spectra were obtained with a data acquisition frequency of 100 Hz. At the velocity u = 0.10 m/s, maxima can be observed in both the region 1–2 Hz, generally typical of slugging beds, and a slightly larger one at frequencies in the region of 4–8 Hz, generally the dominant frequency found for non-slugging beds, either homogeneous or turbulent. These results seem to be consistent with those shown in Fig. 2 where two different regions in the fluid-bed can be observed: a top region which showed preliminary features of slugging fluidization, with the presence of a few flat slugs, and a bottom part clearly characterized by a more homogeneous structure. Similar features can also be observed in Figs. 12 and 13 where the power spectral densities for 0.20 m/s and 0.30 m/s are reported. For these cases, the maxima in the region 1–2 Hz become more pronounced with higher peak values as the fluidizing velocity increases. This is consistent with the features represented in Figs. 3 and 4 which show the development of the slugging regime as the fluidizing velocity is increased, with the formation of both axial and wall slugs in the fluid-bed. Figs. 12 and 13 show also an increase in the peak values of the maxima in the region 4–8 Hz. This is consistent with the observed
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Fig. 14. Experimental power spectral density analysis of the pressure signal, 0.13 m/s, Di Felice [18].
development of a progressively more uniform structure in the dense phase, thus indicating that the system is approaching the transition to the turbulent fluidization regime. Figs. 12 and 13 show the full power spectral density at frequencies up to 50 Hz, highlighting that the simulated intensity at frequencies above 10 Hz is low, as reported by van der Schaaf et al. [33]. Results reported in Fig. 11 are compared with those obtained experimentally by Di Felice [18] using a data acquisition frequency of 10 Hz, which are represented in Fig. 14. The comparison shows similar qualitative results between the CFD and a typical experimental power spectral density analysis obtained for the same system fluidized at 0.13 m/s. Both the experimental and CFD results show that the highest maxima of the power spectral are present in the region below 2 Hz, which supports that the slugging regime is obtained in both the experimental and CFD cases. 5. Conclusions The Eulerian–Eulerian granular kinetic model available within the CFX-4 code was used to simulate the slugging fluidization of a liquid fluidized bed. 2D time dependent simulations were carried out at three fluidizing velocities. Results from simulations were analysed in terms of voidage profiles, bed expansion, pressure drop and pressure fluctuations. The fluid-bed pressure drop was found to be greater than the theoretical one at all velocities, with the difference between the simulated and theoretical pressure drop increasing with increase in the fluidizing velocity. This is in agreement with experimental observations reported for fully slugging fluidized beds. Standard deviation of the pressure drop increased consistently with the development of the slug flow with increasing fluidizing velocity. The analysis of the power spectral density of the pressure signal was used to investigate the development of the flow pattern and the structure of the fluid-bed with increase in the fluidizing velocity.
The results obtained from the simulations were compared with the experimental findings obtained by Di Felice [18]. The simulated excess pressure drop and its standard deviation showed a good qualitative comparison with the experiments at fluid velocities less than that necessary to cause the experimental system to transition to turbulent fluidisation. The CFD results showed a gradual development of the slugging regime with the increase of the fluidizing velocity, in agreement with the experimental results where the slugging was also observed at low velocities up to 0.10 m/s. However, the agreement between the experimental and simulated results was not fully achieved insomuch the transition from slugging to turbulent fluidization was delayed in the CFD simulations thus affecting the overall dynamics shown in the simulated bed. This reflects the limitations of the general multiphase models employed for the simulations of the liquid–solid fluidized bed case investigated; further work is currently underway to investigate alternative modelling approaches. Notation A cross-section area of the bed (m2) CD particle drag coefficient (–) db bubble diameter (m) dp particle diameter (m) D bed diameter (m) ΔP pressure drop (kg m− 1 s− 2) es particle coefficient of restitution (–) E correction factor for inter-phase drag (–) FD drag force (kg m s− 2) 2 2 ¯ F net force vector per unit volume (kg m− s− ) 2 g acceleration gravity (m s− ) go radial distribution function L bed depth (m) M mass of particles (kg) Re Reynolds number 1 u superficial fluid velocity (m s− ) 1 v′ particle velocity fluctuations (m s− ) Greek Letters β inter-phase drag (kg m− 3 s− 1) εmf voidage at minimum fluidization εs solids volume fraction (–) γ dissipation of granular energy (kg m− 1 s− 3) Θs granular temperature (m2 s− 2) ρf fluid density (kg m− 3) ρs particle density (kg m− 3) Subscripts and superscripts f fluid phase p particle s solid phase
Acknowledgements The authors acknowledge the financial support from the University of Genova, Italy, BP and Tioxide Ltd.
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[17] J.M. Matsen, S. Hovmand, J.F. Davidson, Expansion of fluidized beds in slug flow, Chem. Eng. Sci. 24 (1969) 1743–1754. [18] R. Di Felice, Liquid fluidised beds in slugging mode: pressure drop and flow regime transition, Powder Technol. 123 (2002) 254–261. [19] P.S.B. Stewart, J.F. Davidson, Powder Technol. 1 (1967) 61. [20] D. Gidaspow, Multiphase Flow and Fluidization, Academic Press, London, 1994. [21] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition, Cambridge University Press, 1970. [22] R. Ocone, G. Astarita, A pseudo-thermodynamic theory of granular flow rheology, J. Rheol. 37 (1993) 727–742. [23] C.K.K. Lun, S.B. Savage, D.J. Jefferey, N. Chepurniy, Kinetic theories for granular flow: inelastic particles in couette flow and slightly inelastic particles in a general flow-field, J. Fluid Mech. 140 (1984) 223–256. [24] F. Ihme, H. Schmidt-Traub, H. Brauer, Theoretical studies on mass transfer at and flow past spheres, Chemie-Ing.-Tech., 44(5), 1972, pp. 306–313. [25] C.M. Rhie, W.L. Chow, A numerical study of the turbulent flow past an aerofoil with trailing edge separation, AIAA J. 21 (1983) 1525–1532. [26] H.L. Stone, Iterative solution of implicit approximations of multidimensional partial differential equations, SIAM J. Numer. Anal. 5 (1968) 530–558. [27] J.R. Kightley, The conjugate gradient method applied to turbulent flow calculations, UKAEA Report CSS 184 (HL85/1584), Harwell, UK, 1985. [28] D.B. Spalding, A novel finite-difference formulation for differential expressions involving both first and second derivatives, Int. J. Numer. Methods Eng. 4 (1972) 551. [29] J.G. Yates, D. Cheesman, Y.A. Sergeev, Experimental observations of voidage distribution around bubbles in a fluidized bed, Chem. Eng. Sci. 49 (12) (1994) 1885–1895. [30] R.H. Wilhelm, M. Kwauk, Fluidization of solid particles, Chem. Eng. Prog. 44 (1948) 201–217. [31] D. Geldart, J.M. Hurt, P.H. Wadia, Slugging in beds of large particles, AIChE Symp. Ser. 74 (176) (1978) 60–66. [32] Z. Chen, L.G. Gibilaro, P.U. Foscolo, Fluid pressure loss in slugging fluidised beds, Chem. Eng. Sci. 52 (1987) 55–62. [33] J. van der Schaaf, J.R. van Ommen, F. Takens, J.C. Schouten, C.M. van den Bleek, Similarity between chaos analysis and frequency analysis of pressure fluctuations in fluidized beds, Chem. Eng. Sci. 59 (2004) 1829–1840.
CFD modelling of liquid fluidized beds in slugging mode Paola Lettieri
a,⁎
, Renzo Di Felice b , Roberta Pacciani b , Olumuyiwa Owoyemi
a
a
b
Department of Chemical Engineering, University College London, UK Dipartimento di Ingegneria Chimica e di Processo, Universita' di Genova, Italy
Received 9 May 2005; received in revised form 1 May 2006; accepted 16 June 2006 Available online 30 June 2006
Abstract Computational Fluid Dynamics (CFD) modelling has been used to simulate a liquid fluidized bed of lead shot in slugging mode. Simulations have been performed using a commercial code, CFX4.4. The kinetic model for granular flow, which is already available in CFX, has been used during this study. 2D time-dependent simulations have been carried out at different water velocities. Simulated aspects of fluidization such as voidage profiles, slug formation, pressure drop and pressure fluctuations have been analysed. The fluid-bed pressure drop was found to be greater than the theoretical one at all velocities, in agreement with experimental observations reported for fully slugging fluidized beds. Power spectral density analysis of the pressure signal was used to investigate the development of the flow pattern and the structure of the fluid-bed with increasing fluidizing velocity. A comparison between experimental and simulated results is also reported. © 2006 Elsevier B.V. All rights reserved. Keywords: Liquid-fluidized beds; Slugging mode; Pressure drop; CFD; Granular kinetic model
1. Introduction Particles and processes involving particles are of enormous importance in the chemical and allied industries. Fluidized beds are widely employed in industrial operations, ranging from the pharmaceutical and food industry, to processes such as catalytic cracking of petroleum, combustion and biomass gasification. Fluidization is the operation by which particles are transformed into a fluid-like state through suspension in a gas or liquid. Many of the characteristic features of gas-fluidized beds, like the excellent solid mixing, heat and mass transfer properties, can be related to the presence of bubbles and are dominated by their behaviour. It is well known that the gas–solid contacting efficiency is highly dependent on the fluid-bed hydrodynamics and bubble properties. The formation and development of bubbles in the fluid-bed has been studied over a considerable span of time, particularly with the aid of visualization techniques, see Rowe et al. [1].
⁎ Corresponding author. Tel.: +44 20 76797867; fax: +44 20 73832348. E-mail address: [email protected] (P. Lettieri). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.06.012
The recent development of mathematical modelling of particulate solids behaviour together with the increased computing power is now enabling us to simulate the behaviour of fluidized powders and to link fundamental particle properties directly to powder behaviour and predict the interaction between particles and gaseous or liquid fluids. In this regard, Computational Fluid Dynamics (CFD) modelling provides a new tool to support engineering design and research in multiphase systems such as fluidized beds. Most of the fluidization modelling using CFD is applied to gas– solid systems and can be divided into two groups, the Lagrangian– Eulerian models and the Eulerian–Eulerian models, see Crowe et al. [2]. The Lagrangian approach describes the solids phase at a particle level and the gas phase as a continuum. The Eulerian– Eulerian approach, on the other hand, is based on the two-fluid model (TFM) and treats the phases as interpenetrating continua. Using the TFM, numerical simulations of gas–solid fluidbeds have been performed by various research groups: Kuipers et al. [3,4] developed a two-dimensional hydrodynamic model to study the bubble formation process at a single orifice, which was subsequently developed for three-dimensional fluidized bed simulations [5]. Simulations of freely bubbling gas
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fluidized beds using kinetic theory were reported by Ding and Gidaspow [6], van Wachem [7] and Peirano et al. [8]. Using the two-fluid granular temperature model, Pain et al. [9] and Zhang et al. [10] have simulated the behaviour of bubbling and slugging gas fluidized beds. In a recent study, Lettieri et al. [11,12], have used the Eulerian–Eulerian approach to simulate the behaviour of gas fluidized beds under different flow regimes, spanning from bubbling to slugging and turbulent fluidization. Using the CFX4.4 code (Ansys plc, formerly AEA Technology) 2D and 3D simulations of bubbling Geldart Group B [13] fluidized beds were compared using two different modelling approaches, namely, the Eulerian Granular model, which is currently available within CFX-4, and the particle-bed model [14], recently implemented in CFX-4.4 [11]. Quantitative results demonstrated good agreement between the approaches of the two models. The comparison between 2D and 3D simulations showed that 2D models can reproduce reasonable well aspects of the bubbling fluidization. Using the granular kinetic theory, Lettieri et al. [15] subsequently extended the work to quantitatively investigate the transition from bubbling to slugging gas fluidized beds of a Geldart Group B powder. The simulated transition velocity was compared with predictions obtained from the Baeyens and Geldart [16] criterion; simulated values obtained for the maximum slugging bed height were found to be in reasonable agreement with predictions obtained from the Matsen et al. [17] model. In the present study, the CFD modelling work has been extended to investigate liquid fluidized beds. CFD simulations of lead shot particles in the slugging regime have been performed at different fluidizing velocity. The work presented in this paper stems from an experimental study carried out at the University of Genova by Di Felice [18], which investigated the slugging regime of liquid–solid systems and their transition to turbulent fluidization. This work was part of a fundamental study carried out at UCL on the CFD modelling of fluidized beds, sponsored by BP and Tioxide Ltd., aimed at investigating the capability of general multiphase models to simulate and predict aspects of different fluidization regimes and develop and validate an ad hoc fluid-bed model. Slugging fluidization was first identified in 1967 as a type of fluidization regime [19] and since then has been one of the flow regimes which has been the least studied over the years. Slugging normally occurs in beds of high aspect ratio (N1), often with a small diameter with respect to the particle size. A slug is simply a bubble whose diameter is nearly equal to that of the bed itself and can be generally categorized into two types: axisymmetric slugs (round-nose and squared-nose slugs) and wall slugs [19]. This paper presents the formation and development of axisymmetric slugs and wall slugs in the simulated liquidfluidized bed. Simulated aspects of fluidization such as pressure drop, pressure fluctuations and voidage profiles are also presented. The analysis of the power spectral density of the pressure signal is included to describe the development of the flow pattern within the simulated bed. A comparison between experimental and simulated results is also reported.
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Table 1 Governing equations used for the granular kinetic model Granular kinetic model governing equations Continuity Gas phase: Aðeg qg Þ ¯d eg qg u¯g ¼ 0 þj At Solid phase: Aðes qs Þ ¯dðes qs u¯sÞ ¼ 0 þj At Momentum balance Gas phase: A ¯P þ eg qg g¯−b u¯g−u¯s ¯d eg qg u¯g u¯g ¼ j ¯d ¯sg−eg j eg qg u¯g þ j At Solid phase: A ¯d ðes qs u¯s u¯sÞ ¼ j ¯d ¯ss es j ¯P j ¯Ps þ es qs g¯ þ b u¯g u¯s ðes qs u¯sÞ þ j At Granular temperature balance Solid phase:
¯ u¯s ¼ gs −Ps I¯ þ ¯ss : j
Dissipation of granular energy: " rffiffiffiffi # 2 4 H 2 ¯ −jd u¯s gs ¼ 3 1−e g0 es qs H dp p
Constitutive equations solid phase Radial distribution function: g0 ¼
" #−1 3 es 1=3 1− max 5 es
Solids stress tensor: ¯¯ss ¼ 2ls S¯s þ ks − 2 ls tr S¯s I¯ with 3
1 ¯ u¯s þ ðj ¯ u¯s ÞT j S¯s ¼ 2
Solids pressure: Ps ¼ es qs Hs ð1 þ 2g0 es ð1 þ es ÞÞ Bulk viscosity: 0:5 4 Hs ks ¼ es qs dp g0 ð1 þ es Þ 3 p Shear viscosity: 0:5
2 0:5 2 5p qs dp Hs0:5 4 Hs 4 þ 96 1 þ g0 es ð1 þ es Þ ls ¼ es qs dp g0 ð1 þ es Þ 5 5 p ð1 þ es Þes g0
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2. The granular kinetic model The granular kinetic theory model was used during this work. It solves mass and momentum equations for each phase. The momentum balance of the gas phase is given by the Navier– Stokes equations modified to include an inter-phase momentum transfer term. The momentum balance of the solid phase includes description of the solids pressure and of the solid-phase stress tensor, both obtained from granular kinetic theory [20]. Granular kinetic theory describes the particle–particle interactions as binary collisions, resembling those between molecules in the gas kinetic theory. By analogy with the thermodynamic temperature of the gases, granular theory introduces the granular temperature, Θs, to describe the kinetic energy associated with the particle fluctuations, e.g., Chapman and Cowling [21], Ocone and Astarita [22], where 1 2 Hs ¼ bvV N 3 s
ð1Þ
According to this model, in addition to mass and momentum, an energy balance is solved to estimate the granular temperature. In addition to the granular temperature, granular kinetic theory introduces the coefficient of restitution, es (where 0 b es b 1), to account for the non-ideal behaviour or inelasticity of the solids; also introduced is the radial distribution function, go, a parameter which gives a statistical measure of the probability of particle contacting. The function of go is to allow a tight control of the solids volume fraction, so that the maximum packing is not exceeded and more accurate flow characteristics can be achieved. There are a number of different forms of the radial distribution function in the literature; the equation given by Ding and Gidaspow [6] has been used in this work. From knowledge of the granular temperature, the coefficient of restitution and the radial distribution function, both the solid phase pressure and viscosity can be described. The solids pressure represents the solids phase normal forces due to particle collisions. The expression given by Lun at al. [23] has been used in this work. The non-isotropic part of the stress tensor is modelled by considering the solid phase as a Newtonian fluid. It is then related to the strain rate tensor by means of the bulk viscosity, which describes the resistance of the particle phase against compression, and the shear viscosity. In this work, the equation given by Lun et al. [23] has been used for the bulk viscosity and the expression proposed by Gidaspow [20] has been adopted for the shear viscosity. The full set of equations used during the simulations has been previously reported in Lettieri et al. [11]. Table 1 summarizes the continuity, momentum balance, granular temperature and constitutive equations of the granular kinetic model employed in this paper. The inter-phase drag term in the gas and solid phase momentum equations is expressed as F¯D ¼ bðu¯f − u¯sÞ
ð2Þ
with the inter-phase drag coefficient β being expressed as follows: b¼
3 Cd es qf ju¯f − u¯sjE 4 dp
ð3Þ
where |ūf − ūs| is the modulus of the slip velocity vector (relative velocity of the solid to the gas phase), E is a correction coefficient introduced in order to take into account the presence of high particle concentration in the bed. The correction factor E is generally modelled as a function of the solids volume fraction (Gidaspow [20]), i.e., E ¼ ð1−es Þ−1:65
ð4Þ
The correlation established by Ihme et al. [24] was used to calculate CD: CD ¼
24 þ 5:48Re−0:573 þ 0:36 Re
ð5Þ
3. Simulations The simulations were carried out by using the “Multi Fluid Model” (MFM) available in CFX-4.4, which is based on an Eulerian–Eulerian description. This model solves the continuity and momentum equations for a generic multi-phase system and therefore allows the determination of separate flow field solutions for each phase simultaneously. For the present case of two-phase flow, there are seven equations and as many unknowns: the volumetric fractions, the four velocity components and the pressure P (shared by both phases). The numerical solutions were obtained by a finite volume method based on a collocated grid approach that used the Rhie– Chow algorithm [25] to prevent chequerboard oscillations. The continuity and momentum balance equations were coupled using the SIMPLEC algorithm (Semi-Implicit Method for Pressure Linked Equations–Consistent Algorithm). Time-dependent simulations were performed with 10− 4 s time steps and 25 SIMPLEC iterations were required to achieve full numerical convergence. The convergence behaviour of the SIMPLEC algorithm was analysed by monitoring the mass flow residual as a function of the number of iterations at each time step. Under-relaxation factors between 0.6 and 0.7 were adopted for all flow quantities. Pressure was not under-relaxed, as required by the SIMPLEC algorithm. The resulting linear equation sets were solved by using suitable single-block or multi-block verõsions of the Strongly Implicit Procedure [26] for the momentum and scalar-transport equations, and a conjugate gradient method with incomplete Cholesky preconditioning [27] for the pressurecorrection equation. The hybrid-upwind discretization scheme [28] was used for the convective terms. Simulations here reported relate to a rectangular geometry, whose dimensions in 2D are 30 mm width, 1250 mm height; the column-to-particle ratio is chosen in order to be into the slugging regime. The lateral walls were modelled using no-slip velocity boundary conditions for both phases. Dirichlet boundary conditions were employed at the bottom of the bed to specify a uniform gas inlet velocity. Pressure boundary conditions were employed at the top of the freeboard. This implies Dirichlet boundary conditions on pressure, which was set to a reference value of 1.015 × 105 Pa, and Neumann boundary conditions to the gas flow, i.e., all flow quantities are given zero normal gradient.
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Fig. 1. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.08 m/s, 2 mm square cells.
The initial conditions specify the concentration of solids in the bed, and the fluid flow through the bed. The settled bed was considered 376 mm deep, and the solids volume fraction was defined as 0.40. Fluid flow was defined only in the vertical direction. A constant pressure was defined in all horizontal planes up through the bed of particles. The upper section of the simulated geometry, or freeboard, was considered to be occupied by water only.
4. Results and discussion 4.1. Voidage profiles and bed expansion Results presented in this section show 2D simulations of the slugging fluidization of lead-shot having dp = 1.7 mm and ρs = 10,900 kg/m3. Simulations were performed by increasing the fluidizing velocity from u = 0.08 m/s, 0.10 m/s, 0.12 m/s,
Fig. 2. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.10 m/s, 2 mm square cells.
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Fig. 3. 2D-slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.12 m/s, 2 mm square cells.
0.20 m/s and 0.30 m/s. The simulation runs were carried out for 6 s for the first three velocities investigated and up to 10 s real time at the highest velocities 0.20 m/s and 0.30 m/s. At each velocity, aspects of fluidization such as slug formation, pressure drop and pressure fluctuations were examined to study the development of the slugging regime, as experimentally done by Di Felice [18]. Snapshots obtained from the simulations are shown in Figs. 1–5, where the voidage profiles of the 2D slugging simulations at the different velocities investigated are reported. At the start of each simulation, waves of voidage are created, which travel
through the bed and subsequently break to form bubbles as the simulation progresses. At low velocities flat slugs form in the top part of the fluid-bed while the bottom part maintains a fairly homogeneous structure. With large slugging velocities and in beds of large and dense powders, slugs are usually observed to become asymmetric with formation of wall slugs, as reported by Baeyens and Geldart [16]. Initial aspects of this process can be observed in the snapshots of the voidage profile obtained at the fluidizing velocity 0.12 m/s, see Fig. 3. The process of wall slug formation becomes more and more evident with increase in the fluidizing velocity, as represented in Figs. 4 and 5 for velocities
Fig. 4. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.20 m/s, 2 mm square cells.
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Fig. 5. 2D-Slugging bed, liquid voidage profiles, dp = 1.7 mm, u = 0.30 m/s, 2 mm square cells.
equal to 0.2 m/s and 0.3 m/s. At high fluidizing velocities, the slugs become larger and the phenomenon of coalescence also becomes more noticeable. At the same time, the dense phase develops into a uniform structure characterised by an average dense phase voidage of 0.7. As the computed bubbles and slugs become better defined, regions of voidage distribution at the void edges can be identified, as experimentally observed by Yates et al. [29] using the UCL X-ray imaging facility. The snapshots reported in Fig. 4 show the oscillation of the bed surface and the sudden drop of the bed height as the slugs break through. The oscillation of the fluid-bed height was monitored over time for all velocities investigated. Fig. 6 shows a typical example of the results obtained at the fluidizing velocity u = 0.20 m/s. During the analysis of the results, the first few seconds of each simulation run were discarded until a pseudo-oscillatory behaviour of the pressure drop and bed height fluctuations were observed. This was done to ensure that the data extrapolated corresponded to a pseudo-steady-state solution. Figs. 1–5 show also the considerable relative increase in bed expansion as the fluidizing velocity increases, a 6% increase was obtained at 0.08 m/s, a 20% increase at 0.10 m/s, 41% at 0.20 m/s and up to a 100% increase in bed height at the highest fluidized velocity investigated, 0.30 m/s. Fig. 7 shows a comparison between the values of the bed height obtained from the simulations and the experimental ones obtained by Di Felice [18] for the fluidization velocities investigated with the CFD. The comparison shows a reasonably good match for low velocities up to 0.20 m/s. However, the comparison deviates at the highest velocity investigated, 0.3 m/s, at which the experiments show that the system has fully transitioned to the turbulent regime thus giving a much greater bed expansion, while the CFD simulation still shows the presence of slugs
within a progressively more uniform dense phase. The delayed transition from slugging to turbulent fluidization obtained with the CFD shows the limitations of the applicability of the general multiphase models used to simulate the liquid–solid fluidized bed under study. 4.2. Pressure drop The overall behaviour of fluid–particle interaction processes is governed by the forces acting on individual particles. At minimum fluidization conditions a bed of powder is fully supported by an upward fluid flow, and the drag force exerted by the fluid on the particles, which is proportional to the global pressure drop across the bed, is balanced by the buoyant weight
Fig. 6. Averaged bed height variation with time u = 0.20 m/s.
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Fig. 7. Bed height, comparison between experimental and simulated values.
Fig. 9. Simulated and experimental normalised pressure drop as a function of the water superficial velocity.
of the suspension. Thus, the pressure drop across the bed equates to the buoyant weight of the suspension: where L is the height of the bed, εmf is the bed voidage at minimum fluidization, and ρf and ρs are the fluid and particle density, respectively. Eq. (6) has been experimentally verified since the first publication of fluidization appeared; see Wilhelm and Kwauk [30]. However, it does not hold for channelling and slugging fluidized beds. When slugging occurs, measured pressure drops exceed the theoretically predicted value, with the difference becoming more and more significant as the bed moves into the full slugging regime, as reported by Di Felice [18]. The physical origin of the difference between the observed and theoretical pressure drop is not yet fully understood. It has been associated with the energy needed to accelerate the solids;
but also to the potential energy dissipated by the solids in the slugging regime, see Geldart et al. [31] and Chen et al. [32]. The values of the pressure drop obtained over time were recorded for each simulation run. The results were analysed by sampling the data at a frequency of 100 Hz. A considerable increase in the pressure fluctuation with time was observed as the simulated fluidizing velocity was increased. Fig. 8 shows a typical example obtained for the case at u = 0.20 m/s. Fig. 9 shows a comparison between the values of the pressure drop obtained from the simulations and those obtained experimentally by Di Felice [18] with increasing the water fluidizing velocity. These were obtained by averaging the values of the pressure drop obtained over time for each simulation run and normalising them with respect to the effective bed weight calculated using Eq. (6). The experimental results show a steep increase in the pressure drop up to 0.1 m/s followed by a decrease due to the transitioning from slugging to
Fig. 8. Simulated pressure variation as a function of time, u = 0.20 m/s.
Fig. 10. Simulated pressure standard deviation as a function of the water superficial velocity.
DP ¼ ðqs −qf Þð1−emf ÞgL
ð6Þ
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Fig. 11. Power spectral density analysis of the pressure signal, u = 0.10 m/s.
Fig. 13. Power spectral density analysis of the pressure signal, u = 0.3 m/s.
turbulent regime. In agreement with the experimental observations, the simulated pressure drop was found to be greater than the theoretical one at all velocities investigated, as typical of the slugging regime, with values of the pressure drop increasing steeply up to 0.12 m/s and then levelling off at the higher fluidizing velocities investigated. However, the comparison with the experimental values shows a divergence of the trend at the higher velocities investigated as the CFD predicted slugging fluidization whilst the experiments showed a transition to the turbulent regime. Transforming the data plotted into Fig. 8 into the deviation standard of the pressure signal, the diagram shown in Fig. 10 is obtained. The deviation standard increases with the fluidizing velocity, consistently with the development of the flow pattern of slug flow and increase in bed pressure drop. Values of the standard deviation were found to be of the same order of magnitude as the experimental ones. However, similar to the
results reported in Fig. 9, the trend of the CFD and experimental results differ insomuch the transition from slugging to turbulent regime is delayed in the CFD simulations, as reflected in the absence of a maximum in the pressure standard deviation obtained from the simulations.
Fig. 12. Power spectral density analysis of the pressure signal, u = 0.2 m/s.
4.3. The fluid-bed structure The dynamic behaviour and structure of fluidized beds is often characterised in terms of the spectral density analysis of pressure time series, as also reported by J. van der Schaaf et al. [33], where the flow behaviour is quantified looking at the frequency of the largest values of the power spectral density. Figs. 11–13 report the power spectral density for the simulations run at 0.10 m/s, 0.20 m/s and 0.30 m/s, where the maximum in the variance of the pressure signal is obtained. The power spectra were obtained with a data acquisition frequency of 100 Hz. At the velocity u = 0.10 m/s, maxima can be observed in both the region 1–2 Hz, generally typical of slugging beds, and a slightly larger one at frequencies in the region of 4–8 Hz, generally the dominant frequency found for non-slugging beds, either homogeneous or turbulent. These results seem to be consistent with those shown in Fig. 2 where two different regions in the fluid-bed can be observed: a top region which showed preliminary features of slugging fluidization, with the presence of a few flat slugs, and a bottom part clearly characterized by a more homogeneous structure. Similar features can also be observed in Figs. 12 and 13 where the power spectral densities for 0.20 m/s and 0.30 m/s are reported. For these cases, the maxima in the region 1–2 Hz become more pronounced with higher peak values as the fluidizing velocity increases. This is consistent with the features represented in Figs. 3 and 4 which show the development of the slugging regime as the fluidizing velocity is increased, with the formation of both axial and wall slugs in the fluid-bed. Figs. 12 and 13 show also an increase in the peak values of the maxima in the region 4–8 Hz. This is consistent with the observed
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Fig. 14. Experimental power spectral density analysis of the pressure signal, 0.13 m/s, Di Felice [18].
development of a progressively more uniform structure in the dense phase, thus indicating that the system is approaching the transition to the turbulent fluidization regime. Figs. 12 and 13 show the full power spectral density at frequencies up to 50 Hz, highlighting that the simulated intensity at frequencies above 10 Hz is low, as reported by van der Schaaf et al. [33]. Results reported in Fig. 11 are compared with those obtained experimentally by Di Felice [18] using a data acquisition frequency of 10 Hz, which are represented in Fig. 14. The comparison shows similar qualitative results between the CFD and a typical experimental power spectral density analysis obtained for the same system fluidized at 0.13 m/s. Both the experimental and CFD results show that the highest maxima of the power spectral are present in the region below 2 Hz, which supports that the slugging regime is obtained in both the experimental and CFD cases. 5. Conclusions The Eulerian–Eulerian granular kinetic model available within the CFX-4 code was used to simulate the slugging fluidization of a liquid fluidized bed. 2D time dependent simulations were carried out at three fluidizing velocities. Results from simulations were analysed in terms of voidage profiles, bed expansion, pressure drop and pressure fluctuations. The fluid-bed pressure drop was found to be greater than the theoretical one at all velocities, with the difference between the simulated and theoretical pressure drop increasing with increase in the fluidizing velocity. This is in agreement with experimental observations reported for fully slugging fluidized beds. Standard deviation of the pressure drop increased consistently with the development of the slug flow with increasing fluidizing velocity. The analysis of the power spectral density of the pressure signal was used to investigate the development of the flow pattern and the structure of the fluid-bed with increase in the fluidizing velocity.
The results obtained from the simulations were compared with the experimental findings obtained by Di Felice [18]. The simulated excess pressure drop and its standard deviation showed a good qualitative comparison with the experiments at fluid velocities less than that necessary to cause the experimental system to transition to turbulent fluidisation. The CFD results showed a gradual development of the slugging regime with the increase of the fluidizing velocity, in agreement with the experimental results where the slugging was also observed at low velocities up to 0.10 m/s. However, the agreement between the experimental and simulated results was not fully achieved insomuch the transition from slugging to turbulent fluidization was delayed in the CFD simulations thus affecting the overall dynamics shown in the simulated bed. This reflects the limitations of the general multiphase models employed for the simulations of the liquid–solid fluidized bed case investigated; further work is currently underway to investigate alternative modelling approaches. Notation A cross-section area of the bed (m2) CD particle drag coefficient (–) db bubble diameter (m) dp particle diameter (m) D bed diameter (m) ΔP pressure drop (kg m− 1 s− 2) es particle coefficient of restitution (–) E correction factor for inter-phase drag (–) FD drag force (kg m s− 2) 2 2 ¯ F net force vector per unit volume (kg m− s− ) 2 g acceleration gravity (m s− ) go radial distribution function L bed depth (m) M mass of particles (kg) Re Reynolds number 1 u superficial fluid velocity (m s− ) 1 v′ particle velocity fluctuations (m s− ) Greek Letters β inter-phase drag (kg m− 3 s− 1) εmf voidage at minimum fluidization εs solids volume fraction (–) γ dissipation of granular energy (kg m− 1 s− 3) Θs granular temperature (m2 s− 2) ρf fluid density (kg m− 3) ρs particle density (kg m− 3) Subscripts and superscripts f fluid phase p particle s solid phase
Acknowledgements The authors acknowledge the financial support from the University of Genova, Italy, BP and Tioxide Ltd.
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