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A conceptual model for analyzing the stability condition and regime transition in bubble columns
Chemical Engineering Science 65 (2010) 517 -- 526
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A conceptual model for analyzing the stability condition and regime transition in bubble columns Ning Yang a, ∗ , Jianhua Chen a,b , Wei Ge a , Jinghai Li a a b
State Key Laboratory of Multi-Phase Complex System, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing, 100190, PR China Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China
A R T I C L E
I N F O
Article history: Received 30 June 2008 Received in revised form 1 June 2009 Accepted 2 June 2009 Available online 10 June 2009 Keywords: Regime transition Bubble column Stability condition Multi-scale Gas–liquid flow
A B S T R A C T
The abrupt change on the curve of gas holdup vs. superficial gas velocity calculated from the dual-bubblesize (DBS) model was physically interpreted as a shift from the homogeneous and transition regimes to the heterogeneous regime for bubble columns in our previous work (Yang et al., 2007. Explorations on the multi-scale flow structure and stability condition in bubble columns. Chem. Eng. Sci. 62, 6978–6991). The fundamentals related to the DBS model and this jump change are further analyzed in this work. A conceptual analysis is performed on the momentum and energy transfer modes between phases and the partition of energy dissipation at different scales, thus the hydrodynamic equations can be closed with a stability condition formulated as a variational criterion, that is, the minimization of micro-scale energy dissipation or the maximization of meso-scale energy dissipation. Model calculation indicates that the stability condition drives the variation and evolution of structure parameters for the two bubble classes and hence causes the jump change of gas holdup which is due to the shift of the location of the global minimum point of the micro-scale energy dissipation from one ellipsoid of iso-surface to another in the 3D space of structure parameters. The stability condition brings about the compromise between small and large bubbles in that these two classes compete with each other to approach a critical diameter at which drag coefficient reaches minimum. For different liquid media, generally only one bubble class could jump to the critical diameter, except the critical state at which the roles of stabilizing and destabilizing flow reach a balance and the two bubble classes jump together to the critical diameter. This may offer a physical explanation on the dual effect of liquid viscosity and surface tension on flow stability and regime transition reported in literature, and the model calculation for this dual effect and the regime map is in reasonable agreement with experimental findings. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Flow structure in gas–liquid systems is complicated not only in micro-scale behavior of single bubbles such as bubble shape, bubble oscillation, bubble wake and path instability (Clift et al., 1978; Fan and Tsuchiya, 1990; Magnaudet and Eames, 2000), but in macroscale phenomena of global systems. For instance, it is reported that gas–liquid flow in bubble columns generally assumes homogeneous (bubbly), transition, and heterogeneous (churn-turbulent) regimes with the increase of gas velocities (Chen et al., 1994; Zahradnik and Fialova, 1996). Liquid movement and bubble properties, e.g., liquid circulation, bubble size distribution and two-phase structures as well as the mode of gas–liquid interaction, are quite different depending on various flow regimes (Tzeng et al., 1993; Chen et al., 1994).
∗ Corresponding author. Tel.: +86 10 8262 7076; fax: +86 10 6255 8065. E-mail address: [email protected] (N. Yang). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.06.014
Comprehension of the mechanisms of regime transition is of crucial importance for modeling and simulation of the hydrodynamics and hence design and optimization of gas–liquid reactors. A large number of experimental methods have been developed to investigate the regime transition in bubble columns (see Shaikh and Al-Dahhan (2007) for a review). Zahradnik and Fialova (1996) and Zahradnik et al. (1997a) reported that regimes can be distinguished from the curve of gas holdup vs. superficial gas velocity: the gas holdup first increases linearly with gas velocity and then the break down of this linearity marks the shift from homogeneous regime to transition regime. In the latter case, the gas holdup curve rises gradually and then forms a plateau, or reaches a maximum and then falls down, exhibiting an S-shaped variation, both of which dampen the tendency of increasing gas holdup. When the gas velocity is greater than a critical value, say 0.125 m/s for the system of Zahradnik et al. (1997a), the gas holdup increases once again and the two-phase flow reaches a so-called fully developed heterogeneous regime. Ruzicka et al. (2001) reported two kinds of
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heterogeneous regimes: for small and closely spaced orifices of sparger or distributor, the homogeneous regime is destabilized with increasing gas velocity and turned into a heterogeneous regime called THeR at high gas velocity. For large orifices, the gas holdup increases monotonously within the whole range of gas velocity and no shoulder or plateau of gas holdup can be observed. The flow regime in the latter case was called PHeR (pure heterogeneous regime) and exhibits similar characteristics with THeR, showing population of large and highly non-uniform bubbles with a strong tendency to coalesce. By analyzing the time series signals of pressure fluctuation, some researchers employed statistical or chaotic methods to identify the various regimes and their transition (Letzel et al., 1997; Ruthiya et al., 2005; among others). Apart from empirical and semi-empirical correlations reported in literature, the linear stability theories have been developed to model regime transition in bubble columns (see Joshi et al., 2001, for a review). Conservation equations were expressed in terms of the perturbation variable of fractional holdup and the magnification of small perturbation with time was considered to signify the destabilization of homogeneous regime. Simonnet et al. (2008) reported that implementation of their drag correlation as a function of local gas holdup into CFD models could obtain more reasonable curves of gas holdup vs. gas velocity than common drag correlations for isolated bubbles, and reproduce some typical characteristics for different regimes. Monahan et al. (2005) and Monahan and Fox (2007) performed a systematic study on the ability of CFD models on predicting regime transition of bubble columns and found that the formulation of closure models, such as those for drag, virtual mass, lift and bubble-induced turbulence, and even the grid resolution play important roles on simulation. With the input of turbulent energy dissipation rate and gas holdup obtained from some empirical correlations or CFD simulation, the population balance equation was solved by Wang et al. (2005) using their kernel models for coalescence and breakup of bubbles, and the sharp decrease in the calculated volume fraction of small bubbles smaller than 10 mm with increasing gas velocity was regarded to indicate a regime transition. The underlying mechanism for regime transition is not yet clearly revealed and there is still no general consensus on the explanation for experimental findings. Lucas et al. (2005) believed that the positive sign of lift coefficient for small bubbles could stabilize the flow; whereas the negative sign for large bubbles may destabilize the flow. Mudde (2005) pointed out that the experiments of Harteveld et al. (2004) and Harteveld (2005) in which needle spargers were employed to generate uniform gas distribution at inlet proved that homogeneous flow could exist at high gas holdup without being destabilized, but CFD models often predict large scale liquid circulation which is the characteristics of heterogeneous regime. Actually the CFD model itself can only provide a framework for simulating two-phase flow and the complicated underlying physics related to the gas–liquid and bubble–bubble interaction is simply attributed to the closure equations which are still empirical and become challenging issues at present. It is not very clear how momentum is transferred between phases and how energy is transferred, or stored temporarily via the existence of some meso-scale structures and dissipated finally on different scales. Therefore investigation on the mode of momentum and energy transfer between the two phases is of significance. The complexity of multi-scale structure has been explored via some variational criteria (Li and Kwauk, 2003; Li et al., 2005). The basic idea is that the multi-scale structure resulting from the correlation between scales can be physically attributed to the compromise between dominant mechanisms which are mathematically expressed as various extremum tendencies (Ge et al., 2007). The socalled stability condition can therefore be formulated by a mutually constrained extremum reflecting the compromise of mechanisms.
This idea was first introduced in the modeling of gas–solid fluidization with the energy-minimization multi-scale (EMMS) model (Li and Kwauk, 1994; Li et al., 1999). A stability condition was proposed to reflect the compromise between two dominant mechanisms: the gas tends to pass through the particle layer with least resistance (Wst →min) and the particles tend to maintain least gravitational potential (→min). The stability condition was formulated as a mutually constrained extremum, that is, minimization of energy consumption for suspending and transporting particles per unit mass Nst = Wst /p (1−). Ge and Li (2002) extended this stability condition to predict choking, the regime transition between dense fluidization and dilute transport, as a jump change between two branches of stable solutions of the model. Since gas–liquid and gas–solid systems bear some analogy, Zhao (2006) and Ge et al. (2007) established a stability condition and a single-bubble-size (SBS) model for gas–liquid systems. Yang et al. (2007) extended the SBS to a dual-bubble-size (DBS) model. The DBS model can generate a jump change on the curve of gas holdup vs. gas velocity, which was consistent with the regime transition point of Zahradnik et al. (1997a) and Camarasa et al. (1999). Our recent work (Chen et al., 2009) revealed that the jump change of gas holdup is due to the shift of the location of minimum point of stability condition between two points of local minima in the 3D space of structure parameters, and the DBS model can reasonably predict the dual effects of liquid viscosity on regime transition. The objective of this article is to further analyze the fundamentals related to the model formulation, especially the mode of momentum and energy exchange in bubble columns, and to clarify the essence of jump change in calculation which can be understood from the perspective of compromise between small bubbles and large bubbles. The effects of liquid viscosity and surface tension on regime transition are investigated with the DBS model and interpreted with the concept of the competition of small and large bubble classes. It should be pointed out that quantitative agreement with specific experiments is not expected as this model does not take account of factors influencing regime transition like column diameter and sparger types, nor consider inertia-induced forces like virtual mass and lift. Under given model assumptions, the DBS model and the stability condition in this study only serves as a conceptual model to establish a framework for understanding the complexity of gas–liquid systems, and the way of scale differentiation and the model formulation on energy dissipation may still require further exploration. 2. Model description 2.1. Partition of energy dissipation and stability condition The exchange of momentum and energy between gas and liquid is more complicated than that in gas–solid systems (Magnaudet and Eames, 2000). Here we use Fig. 1 to illustrate the mode of the momentum transfer and energy dissipation in bubble columns. Very small bubbles resemble rigid particles in the way of surface interaction, i.e., non-slip boundary condition can be imposed in direct numerical simulation, and drag coefficient of small bubbles approximates that of particles with same diameter (case A). With the increase of bubble size, the drag coefficient of bubbles begins to deviate from that of particles due to the slip of liquids along bubble surfaces (case B). Larger bubbles may change its shape or oscillate when undergoing strong interaction with bubble wake and liquid turbulence (case C). With further increase of bubble size and intensity of gas–liquid interaction, arriving eddies with characteristic length scale smaller than these bubbles whilst containing sufficient kinetic energy could induce the breakage of target bubbles (case D). Considering the force balance between drag and buoyancy, the total energy consumption per unit mass of liquid NT can be obtained
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Fig. 1. Mode of momentum transfer and energy dissipation between gas and liquid.
from the rate of work done by the drag force on unit mass of liquid [(1 − fb )l − g ]g i ni Vi uslip,i f U NT = ≈ Ug − b l g (1) (1 − fb )l 1 − fb where ni denotes the number of bubbles of ith class per unit volume, Vi the volume of a bubble of ith class and uslip,i the interstitial slip velocity. For bubble columns with zero superficial liquid velocity, we have NT = Ug g
(2)
If the gas is fully composed of very small bubbles, NT is transferred from bubbles to liquid through shear stress and non-slip boundary condition, and then transfered and finally dissipated in the process of energy cascade of liquid turbulence, as shown in case A of Fig. 1. But for cases B, C and D encountered in bubble columns, only a part of NT , i.e. (CD,p /CD,b )NT , is directly transferred from bubbles to liquid in this way, and the remaining part denoted by Nsurf accounts for the energy consumption due to the slip of liquid along bubble surfaces and shape oscillation and can be formulated as CD,p (3) NT Nsurf = 1 − CD,b though it is only a rough description of the complicated interphase energy exchange. The first part of NT is also not completely dissipated via energy cascade and a portion of this may store temporarily as surface energy generated from bubble breakage Nbreak and finally dissipated in the process of bubble coalescence. This implies that no net surface is generated when the dynamic balance between breakup and coalescence is well established. Nbreak can be formulated as Nbreak =
db
min 0
0.5
(db , ) · Pb (db , , fBV ) · cf d2b · dfBV d (1−fb )l +fb g
(4)
where the arrival frequency (db , ) and the breakage probability Pb (db , , fBV ) can be obtained from the classical statistical theory of isotropic turbulence (Luo and Svendsen, 1996; Wang et al., 2003; Kostoglou and Karabelas, 2005; Zhao and Ge, 2007). It was supposed that the breakup of a bubble in turbulent flow results mainly from the collision with eddies whose characteristic sizes are equal to or smaller than that of the bubble. To break up the target bubbles, the energy content of the colliding eddy with size and kinetic energy e() should be greater than the increase of surface energy in bubble breakage, and the dynamic pressure of eddies should be greater than the capillary pressure of the smaller daughter bubbles.
As a result, the total energy NT was decomposed into three parts NT = Nsurf + Nturb + Nbreak
(5)
by Zhao (2006) and Ge et al. (2007). They supposed that bubbles generally break up at one location and then coalesce at another after traveling a relatively long distance of time and space, so Nbreak should represent meso-scale dissipation compared to Nturb and Nsurf which dissipate directly at micro-scale. Then the stability condition was proposed as the minimization of micro-scale energy dissipation Nsurf + Nturb = min
(6)
or the maximization of meso-scale energy dissipation Nbreak = max
(7)
According to Ge et al. (2007), Eq. (6) physically reflects the compromise of two dominant mechanisms, that is, Nsurf = min and Nturb = min. The former tends to minimize the bubble diameter whereas the latter tends to maximize the bubble diameter. The following assumptions is adopted in this model: (1) viscous dissipation Nturb approximately equals turbulent energy dissipate rate since bubble-induced and shear-induced turbulence is generally pronounced in bubble columns and hence the rate of energy dissipation for mean flow kinetic energy may be negligible. (2) No net surface is generated when the dynamic balance between breakup and coalescence is well established so that Nbreak generated from breakup is completely dissipated for bubbles coalescence. (3) The effect of bubble oscillation due to the interaction with wake and liquid turbulence is simply attributed to a part of Nsurf . 2.2. Structure resolution It has been reported that gas heterogeneity in bubble columns is featured by some bi-modal bubble size distribution and the coexistence of small and large bubbles (De Swart et al., 1996). Yang et al. (2007) extended the single-bubble-size model of Zhao (2006) and Ge et al. (2007) to a dual-bubble-size model so that the state of heterogeneity of the system can be simply specified by the following structure variables of the two bubble classes: bubble diameters (dS , dL ), volume fraction (fS , fL ) and superficial gas velocities (Ug,S , Ug,L ). The liquid structure is assumed to be shared by the two bubble classes and therefore not resolved. The way of partition of energy dissipation can thus be extended to the system with two bubble classes. With the common circumstances of viscous dissipation Nturb ,
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each bubble class dissipates energy via the slip of liquid along bubble surfaces and shape oscillation of bubbles (Nsurf,S , Nsurf,L ). Each bubble class breaks up and generates new surfaces by extracting energy from liquid turbulence (Nbreak,S , Nbreak,L ) and finally dissipates in the course of bubbles coalescence which may occur between the bubbles belonging to same class or between different classes. Therefore the stability condition can be rewritten as the minimization of all of the micro-scale dissipation Nsurf,S + Nsurf,L + Nturb = min
(8)
or the maximization of all of the meso-scale energy dissipation Nbreak,S + Nbreak,L = max
(9)
Note that each energy dissipation term in Eqs. (8) and (9) is a function of structure parameters under a given superficial gas velocity, so the stability condition determines the evolution of system structure with gas velocity. Fig. 2. Model calculation of Yang et al. (2007) and experiments of Camarasa et al. (1999) with multiple orifice nozzle.
2.3. Model equations The mass and force balance equations for the two bubble classes was formulated as fi l g =
2 Ug,i 2 1 Ul · C · − d D,i l 4 i 2 fi 1 − fb /6 · d3i fi
Ug,i = Ug
(10)
(11)
i
The subscript i refers to S and L which simply represent two different bubble classes because the same drag correlations are employed for the two bubble classes and we do not distinguish small and large bubble classes artificially in the model equations. The correlations of Grace et al. (1976) and Clift et al. (1978) were used to calculate the drag coefficient. See Yang et al. (2007) and Chen et al. (2009) for the detail of the model equations. With specified Ug , Ul and physical properties of gas and liquid for the system, Eqs. (10) and (11) can be solved to obtain the structure variables fS , fL and Ug,L by giving the trial value for dS , dL and Ug,S . Then the stability condition, i.e., Eq. (8), is used to determine the set of the six structure variables which can lead to the minimization of the micro-scale energy dissipation. 3. Results and discussion 3.1. Jump change and regime transition The two points marking the regime transition according to Zahradnik et al. (1997a) can be reasonably predicted with the DBS model for air–water system, as illustrated in Fig. 2. The calculation is well consistent with the experiments of Camarasa et al. (1999) for the case of multiple orifice nozzle for Ug less than 0.07 m/s covering the first point at which Ug equals 0.04 m/s, and the dampening of the tendency of increasing gas holdup can be reflected in the calculation. Though the model fails to predict the S-shaped gradual variation in medium gas velocities, a jump change between 0.128 and 0.129 m/s of gas velocities can be captured, which is very close to the second regime transition point of Zahradnik et al. (1997a) and Camarasa et al. (1999). It should be noticed that since the influence of sparger types and column diameters is not considered in this model, the extent of consistency of the calculation with specific experiments should not be emphasized even though the calculation is in good agreement with experiments at lower gas velocity. Zahradnik et al. (1997b) and Camarasa et al. (1999) reported the influence of sparger types and
column geometry on gas holdup curves. With more uniform gas aeration generated from different spargers, the gas holdup curve generally moved up. The occurrence of the first transition point was delayed to a higher gas velocity and the maximum was more pronounced, but the second transition point almost kept invariable and therefore the curve between the maximum point and the second transition point turned to be steeper in their experiments. We may surmise from this variation tendency that for the extreme situation, say, an ideal sparger which could generate absolutely uniform gas injection, the gradual variation between the maximum point and the second transition point may evolve into a jump change as the DBS model predicted. But in practice the abrupt change may be dampened to a gradual and smooth process by the disturbance of nonuniform boundary conditions and hence the strong bubble breakup and coalescence occurs in advance. In this sense, the DBS model may reflect some inherent characteristics of the system evolution. Experiment of Harteveld (2005) and Mudde et al. (2009) suggested that the maximum point was more pronounced to form a summit and then a sharp drop of gas holdup can be observed when using needle spargers and contaminated tap water. 3.2. Comparison between SBS and DBS model Fig. 2 also compares the calculation of the DBS model in this work and the SBS model proposed by Zhao (2006) and Ge et al. (2007). The latter predicts a monotonous increase of gas holdup. Apparently, the difference of the calculation arises from the resolution of gas phase into the small and large bubbles in the DBS model, while only one bubble class with average properties is considered in the SBS model. In fact, Nsurf +Nturb shown in Eq. (8) is linked with more structure variables in the DBS model, and we will show in the following sections that this stability condition leads to the competition between small bubbles and large bubbles. It is interesting to notice that there are also some literature reports claiming that a single gas phase CFD model cannot predict the gas holdup curve which indicates a regime transition, whereas the two gas phases CFD model with some selected correction factors for drag coefficient may do it (Olmos et al., 2003). 3.3. Physical essence of the jump change Although jump change exists for structure parameters and total gas holdup, the energy dissipation terms exhibit relatively
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continuous variation, as shown in Fig. 3. The decrease of the proportion of Nturb to NT indicates that more energy is dissipated for the process of reconstruction of bubble surfaces via breakage, coalescence and oscillation. The micro-scale energy dissipation Nsurf +Nturb decreases and the meso-scale dissipation Nbreak increases with the increase of gas velocity. Fig. 4 illustrates the iso-surfaces of Nsurf +Nturb in the 3D space of structure parameters for the two gas velocities near the jump change. The minimum point jumps from one ellipsoid for 0.128 m/s of gas velocity to another for 0.129 m/s, causing the jump change of the diameter of small bubble from 1.42 to 2.86 mm. Thus it is the stability condition that drives the variation of structure parameters and hence the jump change.
521
We find that dcrit corresponds to the lowest point on the curve of drag coefficient as a function of bubble diameter, as illustrated in Fig. 6. Typically the minimum exists for almost all of drag coefficient correlations, marking the distinction of viscous- and surface tensiondominant regimes (Fan and Tsuchiya, 1990). But the relation of this
3.4. The critical bubble diameter The gas holdup for the corresponding bubble class at different gas velocities is illustrated in Fig. 5 for air–water systems. dS increases and dL decreases with increasing Ug . When jump change of total gas holdup occurs from 0.128 to 0.129 m/s of Ug , the structure parameters for large bubbles (dL , fL ) change little, but dS increases abruptly to a large value denoted by dcrit and the significant drop of fS causes the larger decrease of total gas holdup. Beyond the Ug of jump change, dS is invariable and dL continues to decrease. Here dcrit is defined as the critical bubble diameter of the bubble class for which jump change occurs.
Fig. 3. Energy dissipation terms as a function of superficial gas velocity.
Fig. 5. Gas holdup of the two bubble classes at different Ug (m/s) (adapted from Yang et al., 2007).
Fig. 6. Drag coefficients calculated from Grace et al. (1976), Fan and Tsuchiya (1990) and Tomiyama (1998).
Fig. 4. The iso-surface of micro-scale energy dissipation (Nsurf +Nturb )/NT in the 3D space of Ug,S , dS and dL for two gas velocities.
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Fig. 7. Energy dissipation for bubble oscillation for small and large bubble classes. Note that here Nsurf,L is for small bubbles and Nsurf,S for large bubbles.
Fig. 9. Gas holdup calculated from the DBS model for different liquid viscosities.
Fig. 10. Transition velocity of gas as a function of liquid viscosity. Fig. 8. Influences of liquid viscosity and surface tension on critical bubble diameter.
minimum with jump change is rather complicated due to the nonlinear coupling of the model equations and the stability condition. Fig. 7 shows the jump change of Nsurf,S and Nsurf,L , but the total surface energy dissipation Nsurf does not show significant variation as the jump changes of Nsurf,S and Nsurf,L counteract one another. It can also be seen from Fig. 3 that Nsurf +Nturb show little variation at the Ug where jump change occurs for gas holdup and structure variables. It seems that to hold such a gentle variation while attaining the minimum for stability condition, the structure parameters have to vary in a particular manner with the diameter of at least one bubble class being dcrit . dcrit can be obtained from the Grace drag correlation where the parameter H equals 59.3 (Grace et al., 1976) and is dependent on liquid viscosity and surface tension, as shown in Fig. 8. It should be noticed that the value of dcrit also varies with the form of different drag coefficient correlations. The sensitivity of model calculation to drag coefficient correlations has been investigated in our previous work (Chen et al., 2009). 3.5. Effects of liquid viscosity Liquid viscosity was reported by Ruzicka et al. (2003) to have dual effect on regime transition. Moderate viscosity (3–22 mPa s)
destabilizes the homogeneous regime and advances the transition, whereas low viscosity (1–3 mPa s) stabilizes the homogeneous regime. In addition, gas holdup was reported to increase for lower viscosity and decrease for high viscosity (Eissa and Schugerl, 1975). Low viscosity was supposed to increase drag force which could reduce bubble rise velocity but could not promote coalescence, and thus increase the gas holdup; whereas the tendency of coalescence and polydispersity prevails over the drag reduction for high viscosity. Interestingly, this experimental finding could be reproduced by the calculation with the DBS model, as illustrated in Fig. 9. It can be observed that compared to air–water systems of 1.0 mPa s of liquid viscosity, slight increase in liquid viscosity from 1.0 to 3.0 mPa s delays the jump change to higher gas velocities for air–glycerin solution systems, showing the stabilizing effect on homogeneous regime. Further increase of viscosity from 3.5 to 8.0 mPa s advances the jump change to lower gas velocities, thereby implying the destabilization of the homogeneous regime. Here we define the superficial gas velocity where jump change occurs as a transition velocity Ug,trans . The predicted relationship of Ug,trans with liquid viscosity is plotted in Fig. 10, clearly demonstrating the deferment or advance of regime transition at different viscosities. The dash line represents transition velocity for air–water systems. In contrast to Fig. 5 for air–water systems, Fig. 11 shows three different variation tendencies of structure parameters for three liquid viscosities of air–glycerin solution systems. It can be seen that dcrit
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Fig. 11. Gas holdup of corresponding bubble classes at different Ug (m/s) and the variation of bubble diameter. (a) l = 1.5 mPa s; (b) l = 3.0 mPa s; (c) l = 5.0 mPa s.
increases with the liquid viscosity. The diameter of small bubbles dS increases and then jump to dcrit and the diameter of large bubbles dL first decreases and then keep constant at the jump point for low viscosity (1.5 mPa s, Fig. 11a); whereas this structure variation reverses at high viscosity (5.0 mPa s, Fig. 11c), indicating that dS varies little but dL jumps to dcrit . Both dS and dL jump to dcrit at the medium viscosity (3.0 mPa s, Fig. 11b). The two bubble classes seem to compete with each other to jump to a stable state with bubble diameter being dcrit . But at low viscosity, only the diameter of small bubbles could reach dcrit ; whereas at high viscosity, large bubbles have the priority to reach dcrit . Considering the calculation in Fig. 9, it can be concluded that the homogeneous flow can keep its stability even at higher gas velocity when the small bubble diameter dS jumps to dcrit first, but lose its stability at lower gas velocity if the large bubble diameter dL first achieves dcrit . The jump change of gas holdup for small and large bubbles should also be noticed. For air–water system shown in Fig. 5 and low viscosity shown in Fig. 11(a), only the hold up of one bubble class jumps, but the holdup for both classes jumps together for medium and high viscosities (Fig. 11(b) and (c)). We can also understand the dual effect of liquid viscosity from the comparison of the micro-scale energy dissipation Nsurf +Nturb as a function of gas velocity for different liquid viscosities, as shown in Fig. 12. For higher viscosity, the micro-scale energy dissipation is lower compared to that of lower and medium viscosities, and therefore the meso-scale dissipation Nbreak is higher and more energy is dissipated for bubble breakup and coalescence to destabilize the flow in this case. 3.6. Effects of surface tension Similar to liquid viscosity, surface tension was also reported to have dual effect on regime transition by Ruzicka et al. (2008). Low concentration of surfactant stabilizes and high concentration
Fig. 12. Energy dissipation of (Nsurf +Nturb )/NT at different liquid viscosities.
destabilizes the homogeneous flow. The calculation of the DBS model shown in Fig. 13 could support this finding. Compared to pure water system, the jump change is first delayed with decreasing surface tension from 73 to 40 mN/m and then gradually advanced when further decreasing surface tension. Also note that increase of surface tension from 73 to 90 mN/m advances the jump change. The synergistic effect of various factors makes it difficult to perform a quantitative comparison with literature reports because the liquid used in experiments are usually different from each other not only in surface tension but also in conductivity and viscosity.
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Fig. 13. Gas holdup calculated from the DBS model for different surface tension.
Fig. 14. Transition velocity of gas as a function of surface tension.
The dual effect of surface tension on transition gas velocity can be recognized from Fig. 14, showing a maximum at 40 mN/m of surface tension. But the maximum looks less pronounced compared to that of liquid viscosity shown in Fig. 10. This corresponds to the variation of dcrit in Fig. 8 that the critical bubble diameter varies gently with surface tension but is highly influenced by liquid viscosity. Similar to Fig. 11, there are also three types of jump change of bubble diameter with the decrease of surface tension. dS jumps to dcrit when surface tension is decreased from 90 to 40 mN/m, corresponding to the stabilizing effect on homogeneous regime. dL jumps to dcrit when surface tension is further decreased from 40 to 20 mN/m and the destabilizing effect dominates. 3.7. Regime map Zhang et al. (1997) presented a regime map of superficial gas velocity vs. liquid velocity which distinguished six different regimes for their experimental systems with small column diameter. Taitel et al. (1980) developed several empirical correlations to discriminate bubbly, slug, churn and annular flow patterns for vertical pipe flow and their regime map is illustrated in Fig. 15. The criterion for
Fig. 15. Regime maps of Taitel et al. (1980) and the calculation with the DBS model.
transition from bubbly to slug flow was defined by calculating the maximum allowable packing of bubbles which was 0.25 for the system of Taitel et al. (1980). However, Harteveld (2005) reported that the gas holdup could reach 0.5 for holding homogeneous regime for the system using needle sparger. So another transition curve between bubbly and slug flow is plotted with = 0.5 in Fig. 15. The transition points calculated from the DBS model only separate the bubbly and the churn-turbulent (heterogeneous) flow as column diameter is not taken into consideration in the model and thus slug flow cannot be identified. The calculated points lie between the two curves of calculation of Taitel correlation, showing that the DBS model can reasonably predict the regime map though it is currently only a conceptual model. Analogous to our previous work on employing the EMMS model to investigate the effect of multi-scale structure on momentum and mass transfer in gas–solid fluidization (Yang et al., 2004; Wang et al., 2003, 2005), the conceptual model of this study can also be extended to understand the complicated momentum and mass transfer in gas–liquid systems, and thus establish some drag closure for CFD simulation as explored in our recent work (Chen et al., 2009). 4. Conclusions and prospects Regime transition to fully-developed heterogeneous regime can be reflected in the model calculation through the jump change on the curve of calculated total gas holdup vs. superficial gas velocity. Physically, this jump change of calculation is due to the shift of the minimum point from one ellipsoid of the iso-surfaces of micro-scale energy dissipation to another in the 3D space of structure variables. With the increase of superficial gas velocity, both the two bubble classes tend to approach the critical diameter dcrit , and the stability condition leads to the competition between the small and large bubble classes. If the diameter of small bubbles jumps to dcrit first, the homogeneous flow tends to be stabilized and the regime transition is delayed; otherwise the flow tends to be destabilized and the regime transition is advanced if large bubbles have the priority to approach dcrit . Since only one bubble class is introduced in the SBS model, the competition between the two bubble classes cannot be reflected and the calculation with this model can only generate a monotonous curve of total gas holdup. The finding of competition between the two bubble classes can be used to predict and also interpret the dual effect of liquid viscosity and surface tension on the stability of homogeneous flow and the regime transition. It can be
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concluded that this conceptual model, though still rough for exactly describing the flow structure and momentum and energy exchange between phases, could provide a theoretical framework for analyzing the general trend of structure variation.
PHeR SBS THeR
Notation
Subscripts g l L p S
cf CDb CD0,b CDp CD0,p db dcrit dmin dL dS D DT Eo fb fBV fL fS g lE Mo nb Nbreak Nst Nsurf NT Nturb Pb Ug Ug,L Ug,S Ug,trans Ul Wst
2/3
coefficient of surface area increase, cf =fBV +(1−fBV )2/3 − 1, dimensionless drag coefficient for a bubble in a swarm, dimensionless drag coefficient for a bubble in a quiescent liquid, dimensionless drag coefficient for a particle in multi-particle systems, dimensionless drag coefficient for a particle in a quiescent fluid, dimensionless bubble diameter, m critical bubble diameter, m minimum bubble diameter, mm bubble diameter of large bubbles, m bubble diameter of small bubbles, m tube diameter in Taitel's model, m column diameter, m ¨ Eotvos number, dimensionless volume fraction of gas phase, dimensionless breakup ratio of daughter bubble to its mother bubble, dimensionless volume fraction of large bubbles, dimensionless volume fraction of small bubbles, dimensionless gravitational acceleration, m/s2 the entrance length in Taitel's model, m Morton number, dimensionless number density of bubbles, l/m3 rate of energy consumption due to bubble breakage and coalescence per unit mass, m2 /s3 rate of energy dissipation for suspending and transporting particles per unit mass, m2 /s3 rate of energy dissipation due to bubble oscillation per unit mass, m2 /s3 total rate of energy dissipation rate of energy dissipation in turbulent liquid phase per unit mass, m2 /s3 bubble breakup probability, dimensionless superficial gas velocity, m/s superficial gas velocity for large bubbles, m/s superficial gas velocity for small bubbles, m/s transition gas velocity, m/s superficial liquid velocity, m/s rate of energy dissipation for suspending and transporting particles per unit volume, m2 /s3
Greek letters
void fraction in Taitel's model, dimensionless voidage, dimensionless character size of eddy, m viscosity, Pa s density, kg/m3 surface tension, N/m collision frequency, l/s
Abbreviations DBS EMMS
double-bubble-size energy-minimization multi-scale
525
pure heterogeneous regime single-bubble-size heterogeneous regime resulted from the instability of homogeneous regime Gas Liquid large bubble particle small bubble
Acknowledgement The authors wish to thank the long term support from National Natural Science Foundation of China, in particular, the Grants 20406022, 20221603 and 20490201, and from National Basic Research Program of China under the Grant 2009CB219906. References Camarasa, E., Vial, C., Poncin, S., Wild, G., Midoux, N., Bouillard, J., 1999. Influence of coalescence behaviour of the liquid and of gas sparging on hydrodynamics and bubble characteristics in a bubble column. Chemical Engineering and Processing 38, 329–344. Chen, J., Yang, N., Ge, W., Li, J., 2009. Modeling of regime transition in bubble columns with stability condition. Industrial & Engineering Chemistry Research 48, 290–301. Chen, R.C., Reese, J., Fan, L.-S., 1994. Flow structure in a three-dimensional bubble column and three-phase fluidized bed. A.I.Ch.E. Journal 40, 1093–1104. Chen, J., Yang, N., Ge, W., Li, J., 2009. Computational fluid dynamics simulation of regime transition in bubble columns incorporating the dual-bubblesize model. Industrial and Engineering Chemistry Research 43, in press, doi:10.1021/ie801644d. Clift, R., Grace, J.R., Weber, M.E., 1978. Bubbles, Drops, and Particles. Academic Press, London. De Swart, J.W.A., van Vliet, R.E., Krishna, R., 1996. Size, structure and dynamics of “large” bubbles in a two-dimensional slurry bubble column. Chemical Engineering Science 51, 4619–4629. Eissa, S.H., Schugerl, K., 1975. Holdup and backmixing investigations in cocurrent and countercurrent bubble columns. Chemical Engineering Science 30, 1251–1256. Fan, L.-S., Tsuchiya, K., 1990. Bubble Wake Dynamics in Liquids and Liquid–solid Suspensions. Butterworth-Heinemann, Stoneham. Ge, W., Chen, F., Gao, J., Gao, S., Huang, J., Liu, X., Ren, Y., Sun, Q., Wang, L., Wang, W., Yang, N., Zhang, J., Zhao, H., Zhou, G., Li, J., 2007. Analytical multi-scale method for multi-phase complex systems in process engineering—bridging reductionism and holism. Chemical Engineering Science 62, 3346–3377. Ge, W., Li, J., 2002. Physical mapping of fluidization regimes—the EMMS approach. Chemical Engineering Science 57, 3993–4004. Grace, J.R., Wairegi, T., Nguyen, T.H., 1976. Shapes and velocities of single drops and bubbles moving freely through immiscible liquids. Transactions of IChemE 54, 167–173. Harteveld, W.K., 2005. Bubble columns: structure or stability? Ph.D. thesis, Delft University of Technology, the Netherlands. Harteveld, W.K., Julia, J.E., Mudde, R.F., van den Akker, H.E.A., 2004. Large scale vortical structures in bubble columns for gas fractions in the range 5–25%. In: Proceedings of the 16th International Conference on Chemical and Process Engineering, Prague, Czech Republic. Joshi, J.B., Deshpande, N.S., Dinkar, M., Phanikumar, D.V., 2001. Hydrodynamic stability of multiphase reactors. Advances in Chemical Engineering 26, 1–130. Kostoglou, M., Karabelas, A.J., 2005. Toward a unified framework for the derivation of breakage functions based on the statistical theory of turbulence. Chemical Engineering Science 60, 6584–6595. Letzel, H.M., Schouten, J.C., Krishna, R., van den Bleek, C.M., 1997. Characterization of regimes and regime transitions in bubble columns by chaos analysis of pressure signals. Chemical Engineering Science 52, 4447–4459. Li, J., Cheng, C., Zhang, Z., Yuan, J., Nemet, A., Fett, F.N., 1999. The EMMS model—its application, development and updated concepts. Chemical Engineering Science 54, 5409–5425. Li, J., Ge, W., Zhang, J., Kwauk, M., 2005. Multi-scale compromise and multi-level correlation in complex systems. Chemical Engineering Research and Design 83 (A6), 574–582. Li, J., Kwauk, M., 1994. Particle-fluid Two-phase Flow—The Energy-minimization Multi-scale Method. Metallurgical Industry Press, Beijing. Li, J., Kwauk, M., 2003. Exploring complex systems in chemical engineering—the multi-scale methodology. Chemical Engineering Science 58, 521–535. Lucas, D., Prasser, H.M., Manera, A., 2005. Influence of the lift force on the stability of a bubble column. Chemical Engineering Science 60, 3609–3619.
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Luo, H., Svendsen, H.F., 1996. Theoretical model for drop and bubble breakup in turbulent dispersions. A.I.Ch.E. Journal 42, 1225. Magnaudet, J., Eames, I., 2000. The motion of high-reynolds-number bubbles in inhomogeneous flows. Annual Review of Fluid Mechanics 32, 659–708. Monahan, S.M., Fox, R.O., 2007. Linear stability analysis of a two-fluid model for air–water bubble columns. Chemical Engineering Science 62, 3159–3177. Monahan, S.M., Vitankar, V.S., Fox, R.O., 2005. CFD predictions for flow-regime transitions in bubble columns. A.I.Ch.E. Journal 51, 1897–1923. Mudde, R.F., 2005. Gravity-driven bubbly flows. Annual Review of Fluid Mechanics 37, 393–423. Mudde, R.F., Harteveld, W.K., Van den Akker, H.E.A., 2009. Uniform flow in bubble columns. Industrial and Engineering Chemistry Research 48, 148–158. Olmos, E., Gentric, C., Midoux, N., 2003. Numerical description of flow regime transitions in bubble column reactors by a multiple gas phase model. Chemical Engineering Science 58, 2113–2121. Ruthiya, K.C., Chilekar, V.P., Warnier, M.J.F., van der Schaaf, J., Kuster, B.F.M., Schouten, J.C., van Ommen, J.R., 2005. Detecting regime transitions in slurry bubble columns using pressure time series. A.I.Ch.E. Journal 51, 1951–1965. Ruzicka, M.C., Drahos, J., Mena, P.C., Teixeira, J.A., 2003. Effect of viscosity on homogeneous–heterogeneous flow regime transition in bubble columns. Chemical Engineering Journal 96, 15–22. Ruzicka, M.C., Vecer, M.M., Orvalho, S., Drahos, J., 2008. Effect of surfactant on homogeneous regime stability in bubble column. Chemical Engineering Science 63, 951–967. Ruzicka, M.C., Zahradnik, J., Drahos, J., Thomas, N.H., 2001. Homogeneous– heterogeneous regime transition in bubble columns. Chemical Engineering Science 56, 4609–4626. Shaikh, A., Al-Dahhan, M.H., 2007. A review on flow regime transition in bubble columns. International Journal of Chemical Reactor Engineering 57, 1–68. Simonnet, M., Gentric, C., Olmos, E., Midoux, N., 2008. CFD simulation of the flow field in a bubble column reactor: importance of the drag force formulation to describe regime transitions. Chemical Engineering and Processing: Process Intensification 47, 1726–1737. Taitel, Y., Bornea, D., Dukler, A.E., 1980. Modelling flow pattern transitions for steady upward gas–liquid flow in vertical tubes. A.I.Ch.E. Journal 26, 345–354. Tomiyama, A., 1998. Struggle with computational bubble dynamics. Multiphase Science and Technology 10, 369–405.
Tzeng, J.-W., Chen, R.C., Fan, L.-S., 1993. Visualization of flow characteristics in a 2-D bubble column and three-phase fluidized bed. A.I.Ch.E. Journal 39, 733–744. Wang, T., Wang, J., Jin, Y., 2003. A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow. Chemical Engineering Science 58, 4629– 4637. Wang, T., Wang, J., Jin, Y., 2005. Theoretical prediction of flow regime transition in bubble columns by the population balance model. Chemical Engineering Science 60, 6199–6209. Wang, L., Jin, D., Li, J., 2003. Effect of dynamic change of flow structure on mass transfer between gas and particles. Chemical Engineering Science 58, 5373–5377. Wang, L., Yang, N., Li, J., 2005. Multi-scale mass transfer model for gas–solid twophase flow. Chemical Engineering Communications 192, 1636–1654. Yang, N., Chen, J., Zhao, H., Ge, W., Li, J., 2007. Explorations on the multi-scale flow structure and stability condition in bubble columns. Chemical Engineering Science 62, 6978–6991. Yang, N., Wang, W., Ge, W., Wang, L., Li, J., 2004. Simulation of heterogeneous structure in a circulating fluidized-bed riser by combining the two-fluid model with the EMMS approach. Industrial and Engineering Chemistry Research 43, 5548–5561. Zahradnik, J., Fialova, M., 1996. The effect of bubbling regime on gas and liquid phase mixing in bubble column reactors. Chemical Engineering Science 51, 2491–2500. Zahradnik, J., Fialova, M., Linek, V., Sinkule, J., Reznickova, J., Kastanek, F., 1997b. Dispersion efficiency of ejector-type gas distributors in different operating modes. Chemical Engineering Science 52, 4499–4510. Zahradnik, J., Fialova, M., Ruzicka, M., Drahos, J., Kastanek, F., Thomas, N.H., 1997a. Duality of the gas–liquid flow regimes in bubble column reactors. Chemical Engineering Science 52, 3811–3826. Zhang, J.P., Grace, J.R., Epstein, N., Lim, K.S., 1997. Flow regime identification in gas–liquid flow and three-phase fluidized beds. Chemical Engineering Science 52, 3979–3992. Zhao, H., 2006. Multi-scale modeling of gas–liquid (slurry) reactors. Ph.D. Thesis, Institute of Process Engineering, Chinese Academy of Sciences, Beijing. Zhao, H., Ge, W., 2007. A theoretical bubble breakup model for slurry beds or threephase fluidized beds under high pressure. Chemical Engineering Science 62, 109–115.
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A conceptual model for analyzing the stability condition and regime transition in bubble columns Ning Yang a, ∗ , Jianhua Chen a,b , Wei Ge a , Jinghai Li a a b
State Key Laboratory of Multi-Phase Complex System, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing, 100190, PR China Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China
A R T I C L E
I N F O
Article history: Received 30 June 2008 Received in revised form 1 June 2009 Accepted 2 June 2009 Available online 10 June 2009 Keywords: Regime transition Bubble column Stability condition Multi-scale Gas–liquid flow
A B S T R A C T
The abrupt change on the curve of gas holdup vs. superficial gas velocity calculated from the dual-bubblesize (DBS) model was physically interpreted as a shift from the homogeneous and transition regimes to the heterogeneous regime for bubble columns in our previous work (Yang et al., 2007. Explorations on the multi-scale flow structure and stability condition in bubble columns. Chem. Eng. Sci. 62, 6978–6991). The fundamentals related to the DBS model and this jump change are further analyzed in this work. A conceptual analysis is performed on the momentum and energy transfer modes between phases and the partition of energy dissipation at different scales, thus the hydrodynamic equations can be closed with a stability condition formulated as a variational criterion, that is, the minimization of micro-scale energy dissipation or the maximization of meso-scale energy dissipation. Model calculation indicates that the stability condition drives the variation and evolution of structure parameters for the two bubble classes and hence causes the jump change of gas holdup which is due to the shift of the location of the global minimum point of the micro-scale energy dissipation from one ellipsoid of iso-surface to another in the 3D space of structure parameters. The stability condition brings about the compromise between small and large bubbles in that these two classes compete with each other to approach a critical diameter at which drag coefficient reaches minimum. For different liquid media, generally only one bubble class could jump to the critical diameter, except the critical state at which the roles of stabilizing and destabilizing flow reach a balance and the two bubble classes jump together to the critical diameter. This may offer a physical explanation on the dual effect of liquid viscosity and surface tension on flow stability and regime transition reported in literature, and the model calculation for this dual effect and the regime map is in reasonable agreement with experimental findings. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Flow structure in gas–liquid systems is complicated not only in micro-scale behavior of single bubbles such as bubble shape, bubble oscillation, bubble wake and path instability (Clift et al., 1978; Fan and Tsuchiya, 1990; Magnaudet and Eames, 2000), but in macroscale phenomena of global systems. For instance, it is reported that gas–liquid flow in bubble columns generally assumes homogeneous (bubbly), transition, and heterogeneous (churn-turbulent) regimes with the increase of gas velocities (Chen et al., 1994; Zahradnik and Fialova, 1996). Liquid movement and bubble properties, e.g., liquid circulation, bubble size distribution and two-phase structures as well as the mode of gas–liquid interaction, are quite different depending on various flow regimes (Tzeng et al., 1993; Chen et al., 1994).
∗ Corresponding author. Tel.: +86 10 8262 7076; fax: +86 10 6255 8065. E-mail address: [email protected] (N. Yang). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.06.014
Comprehension of the mechanisms of regime transition is of crucial importance for modeling and simulation of the hydrodynamics and hence design and optimization of gas–liquid reactors. A large number of experimental methods have been developed to investigate the regime transition in bubble columns (see Shaikh and Al-Dahhan (2007) for a review). Zahradnik and Fialova (1996) and Zahradnik et al. (1997a) reported that regimes can be distinguished from the curve of gas holdup vs. superficial gas velocity: the gas holdup first increases linearly with gas velocity and then the break down of this linearity marks the shift from homogeneous regime to transition regime. In the latter case, the gas holdup curve rises gradually and then forms a plateau, or reaches a maximum and then falls down, exhibiting an S-shaped variation, both of which dampen the tendency of increasing gas holdup. When the gas velocity is greater than a critical value, say 0.125 m/s for the system of Zahradnik et al. (1997a), the gas holdup increases once again and the two-phase flow reaches a so-called fully developed heterogeneous regime. Ruzicka et al. (2001) reported two kinds of
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heterogeneous regimes: for small and closely spaced orifices of sparger or distributor, the homogeneous regime is destabilized with increasing gas velocity and turned into a heterogeneous regime called THeR at high gas velocity. For large orifices, the gas holdup increases monotonously within the whole range of gas velocity and no shoulder or plateau of gas holdup can be observed. The flow regime in the latter case was called PHeR (pure heterogeneous regime) and exhibits similar characteristics with THeR, showing population of large and highly non-uniform bubbles with a strong tendency to coalesce. By analyzing the time series signals of pressure fluctuation, some researchers employed statistical or chaotic methods to identify the various regimes and their transition (Letzel et al., 1997; Ruthiya et al., 2005; among others). Apart from empirical and semi-empirical correlations reported in literature, the linear stability theories have been developed to model regime transition in bubble columns (see Joshi et al., 2001, for a review). Conservation equations were expressed in terms of the perturbation variable of fractional holdup and the magnification of small perturbation with time was considered to signify the destabilization of homogeneous regime. Simonnet et al. (2008) reported that implementation of their drag correlation as a function of local gas holdup into CFD models could obtain more reasonable curves of gas holdup vs. gas velocity than common drag correlations for isolated bubbles, and reproduce some typical characteristics for different regimes. Monahan et al. (2005) and Monahan and Fox (2007) performed a systematic study on the ability of CFD models on predicting regime transition of bubble columns and found that the formulation of closure models, such as those for drag, virtual mass, lift and bubble-induced turbulence, and even the grid resolution play important roles on simulation. With the input of turbulent energy dissipation rate and gas holdup obtained from some empirical correlations or CFD simulation, the population balance equation was solved by Wang et al. (2005) using their kernel models for coalescence and breakup of bubbles, and the sharp decrease in the calculated volume fraction of small bubbles smaller than 10 mm with increasing gas velocity was regarded to indicate a regime transition. The underlying mechanism for regime transition is not yet clearly revealed and there is still no general consensus on the explanation for experimental findings. Lucas et al. (2005) believed that the positive sign of lift coefficient for small bubbles could stabilize the flow; whereas the negative sign for large bubbles may destabilize the flow. Mudde (2005) pointed out that the experiments of Harteveld et al. (2004) and Harteveld (2005) in which needle spargers were employed to generate uniform gas distribution at inlet proved that homogeneous flow could exist at high gas holdup without being destabilized, but CFD models often predict large scale liquid circulation which is the characteristics of heterogeneous regime. Actually the CFD model itself can only provide a framework for simulating two-phase flow and the complicated underlying physics related to the gas–liquid and bubble–bubble interaction is simply attributed to the closure equations which are still empirical and become challenging issues at present. It is not very clear how momentum is transferred between phases and how energy is transferred, or stored temporarily via the existence of some meso-scale structures and dissipated finally on different scales. Therefore investigation on the mode of momentum and energy transfer between the two phases is of significance. The complexity of multi-scale structure has been explored via some variational criteria (Li and Kwauk, 2003; Li et al., 2005). The basic idea is that the multi-scale structure resulting from the correlation between scales can be physically attributed to the compromise between dominant mechanisms which are mathematically expressed as various extremum tendencies (Ge et al., 2007). The socalled stability condition can therefore be formulated by a mutually constrained extremum reflecting the compromise of mechanisms.
This idea was first introduced in the modeling of gas–solid fluidization with the energy-minimization multi-scale (EMMS) model (Li and Kwauk, 1994; Li et al., 1999). A stability condition was proposed to reflect the compromise between two dominant mechanisms: the gas tends to pass through the particle layer with least resistance (Wst →min) and the particles tend to maintain least gravitational potential (→min). The stability condition was formulated as a mutually constrained extremum, that is, minimization of energy consumption for suspending and transporting particles per unit mass Nst = Wst /p (1−). Ge and Li (2002) extended this stability condition to predict choking, the regime transition between dense fluidization and dilute transport, as a jump change between two branches of stable solutions of the model. Since gas–liquid and gas–solid systems bear some analogy, Zhao (2006) and Ge et al. (2007) established a stability condition and a single-bubble-size (SBS) model for gas–liquid systems. Yang et al. (2007) extended the SBS to a dual-bubble-size (DBS) model. The DBS model can generate a jump change on the curve of gas holdup vs. gas velocity, which was consistent with the regime transition point of Zahradnik et al. (1997a) and Camarasa et al. (1999). Our recent work (Chen et al., 2009) revealed that the jump change of gas holdup is due to the shift of the location of minimum point of stability condition between two points of local minima in the 3D space of structure parameters, and the DBS model can reasonably predict the dual effects of liquid viscosity on regime transition. The objective of this article is to further analyze the fundamentals related to the model formulation, especially the mode of momentum and energy exchange in bubble columns, and to clarify the essence of jump change in calculation which can be understood from the perspective of compromise between small bubbles and large bubbles. The effects of liquid viscosity and surface tension on regime transition are investigated with the DBS model and interpreted with the concept of the competition of small and large bubble classes. It should be pointed out that quantitative agreement with specific experiments is not expected as this model does not take account of factors influencing regime transition like column diameter and sparger types, nor consider inertia-induced forces like virtual mass and lift. Under given model assumptions, the DBS model and the stability condition in this study only serves as a conceptual model to establish a framework for understanding the complexity of gas–liquid systems, and the way of scale differentiation and the model formulation on energy dissipation may still require further exploration. 2. Model description 2.1. Partition of energy dissipation and stability condition The exchange of momentum and energy between gas and liquid is more complicated than that in gas–solid systems (Magnaudet and Eames, 2000). Here we use Fig. 1 to illustrate the mode of the momentum transfer and energy dissipation in bubble columns. Very small bubbles resemble rigid particles in the way of surface interaction, i.e., non-slip boundary condition can be imposed in direct numerical simulation, and drag coefficient of small bubbles approximates that of particles with same diameter (case A). With the increase of bubble size, the drag coefficient of bubbles begins to deviate from that of particles due to the slip of liquids along bubble surfaces (case B). Larger bubbles may change its shape or oscillate when undergoing strong interaction with bubble wake and liquid turbulence (case C). With further increase of bubble size and intensity of gas–liquid interaction, arriving eddies with characteristic length scale smaller than these bubbles whilst containing sufficient kinetic energy could induce the breakage of target bubbles (case D). Considering the force balance between drag and buoyancy, the total energy consumption per unit mass of liquid NT can be obtained
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519
Fig. 1. Mode of momentum transfer and energy dissipation between gas and liquid.
from the rate of work done by the drag force on unit mass of liquid [(1 − fb )l − g ]g i ni Vi uslip,i f U NT = ≈ Ug − b l g (1) (1 − fb )l 1 − fb where ni denotes the number of bubbles of ith class per unit volume, Vi the volume of a bubble of ith class and uslip,i the interstitial slip velocity. For bubble columns with zero superficial liquid velocity, we have NT = Ug g
(2)
If the gas is fully composed of very small bubbles, NT is transferred from bubbles to liquid through shear stress and non-slip boundary condition, and then transfered and finally dissipated in the process of energy cascade of liquid turbulence, as shown in case A of Fig. 1. But for cases B, C and D encountered in bubble columns, only a part of NT , i.e. (CD,p /CD,b )NT , is directly transferred from bubbles to liquid in this way, and the remaining part denoted by Nsurf accounts for the energy consumption due to the slip of liquid along bubble surfaces and shape oscillation and can be formulated as CD,p (3) NT Nsurf = 1 − CD,b though it is only a rough description of the complicated interphase energy exchange. The first part of NT is also not completely dissipated via energy cascade and a portion of this may store temporarily as surface energy generated from bubble breakage Nbreak and finally dissipated in the process of bubble coalescence. This implies that no net surface is generated when the dynamic balance between breakup and coalescence is well established. Nbreak can be formulated as Nbreak =
db
min 0
0.5
(db , ) · Pb (db , , fBV ) · cf d2b · dfBV d (1−fb )l +fb g
(4)
where the arrival frequency (db , ) and the breakage probability Pb (db , , fBV ) can be obtained from the classical statistical theory of isotropic turbulence (Luo and Svendsen, 1996; Wang et al., 2003; Kostoglou and Karabelas, 2005; Zhao and Ge, 2007). It was supposed that the breakup of a bubble in turbulent flow results mainly from the collision with eddies whose characteristic sizes are equal to or smaller than that of the bubble. To break up the target bubbles, the energy content of the colliding eddy with size and kinetic energy e() should be greater than the increase of surface energy in bubble breakage, and the dynamic pressure of eddies should be greater than the capillary pressure of the smaller daughter bubbles.
As a result, the total energy NT was decomposed into three parts NT = Nsurf + Nturb + Nbreak
(5)
by Zhao (2006) and Ge et al. (2007). They supposed that bubbles generally break up at one location and then coalesce at another after traveling a relatively long distance of time and space, so Nbreak should represent meso-scale dissipation compared to Nturb and Nsurf which dissipate directly at micro-scale. Then the stability condition was proposed as the minimization of micro-scale energy dissipation Nsurf + Nturb = min
(6)
or the maximization of meso-scale energy dissipation Nbreak = max
(7)
According to Ge et al. (2007), Eq. (6) physically reflects the compromise of two dominant mechanisms, that is, Nsurf = min and Nturb = min. The former tends to minimize the bubble diameter whereas the latter tends to maximize the bubble diameter. The following assumptions is adopted in this model: (1) viscous dissipation Nturb approximately equals turbulent energy dissipate rate since bubble-induced and shear-induced turbulence is generally pronounced in bubble columns and hence the rate of energy dissipation for mean flow kinetic energy may be negligible. (2) No net surface is generated when the dynamic balance between breakup and coalescence is well established so that Nbreak generated from breakup is completely dissipated for bubbles coalescence. (3) The effect of bubble oscillation due to the interaction with wake and liquid turbulence is simply attributed to a part of Nsurf . 2.2. Structure resolution It has been reported that gas heterogeneity in bubble columns is featured by some bi-modal bubble size distribution and the coexistence of small and large bubbles (De Swart et al., 1996). Yang et al. (2007) extended the single-bubble-size model of Zhao (2006) and Ge et al. (2007) to a dual-bubble-size model so that the state of heterogeneity of the system can be simply specified by the following structure variables of the two bubble classes: bubble diameters (dS , dL ), volume fraction (fS , fL ) and superficial gas velocities (Ug,S , Ug,L ). The liquid structure is assumed to be shared by the two bubble classes and therefore not resolved. The way of partition of energy dissipation can thus be extended to the system with two bubble classes. With the common circumstances of viscous dissipation Nturb ,
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each bubble class dissipates energy via the slip of liquid along bubble surfaces and shape oscillation of bubbles (Nsurf,S , Nsurf,L ). Each bubble class breaks up and generates new surfaces by extracting energy from liquid turbulence (Nbreak,S , Nbreak,L ) and finally dissipates in the course of bubbles coalescence which may occur between the bubbles belonging to same class or between different classes. Therefore the stability condition can be rewritten as the minimization of all of the micro-scale dissipation Nsurf,S + Nsurf,L + Nturb = min
(8)
or the maximization of all of the meso-scale energy dissipation Nbreak,S + Nbreak,L = max
(9)
Note that each energy dissipation term in Eqs. (8) and (9) is a function of structure parameters under a given superficial gas velocity, so the stability condition determines the evolution of system structure with gas velocity. Fig. 2. Model calculation of Yang et al. (2007) and experiments of Camarasa et al. (1999) with multiple orifice nozzle.
2.3. Model equations The mass and force balance equations for the two bubble classes was formulated as fi l g =
2 Ug,i 2 1 Ul · C · − d D,i l 4 i 2 fi 1 − fb /6 · d3i fi
Ug,i = Ug
(10)
(11)
i
The subscript i refers to S and L which simply represent two different bubble classes because the same drag correlations are employed for the two bubble classes and we do not distinguish small and large bubble classes artificially in the model equations. The correlations of Grace et al. (1976) and Clift et al. (1978) were used to calculate the drag coefficient. See Yang et al. (2007) and Chen et al. (2009) for the detail of the model equations. With specified Ug , Ul and physical properties of gas and liquid for the system, Eqs. (10) and (11) can be solved to obtain the structure variables fS , fL and Ug,L by giving the trial value for dS , dL and Ug,S . Then the stability condition, i.e., Eq. (8), is used to determine the set of the six structure variables which can lead to the minimization of the micro-scale energy dissipation. 3. Results and discussion 3.1. Jump change and regime transition The two points marking the regime transition according to Zahradnik et al. (1997a) can be reasonably predicted with the DBS model for air–water system, as illustrated in Fig. 2. The calculation is well consistent with the experiments of Camarasa et al. (1999) for the case of multiple orifice nozzle for Ug less than 0.07 m/s covering the first point at which Ug equals 0.04 m/s, and the dampening of the tendency of increasing gas holdup can be reflected in the calculation. Though the model fails to predict the S-shaped gradual variation in medium gas velocities, a jump change between 0.128 and 0.129 m/s of gas velocities can be captured, which is very close to the second regime transition point of Zahradnik et al. (1997a) and Camarasa et al. (1999). It should be noticed that since the influence of sparger types and column diameters is not considered in this model, the extent of consistency of the calculation with specific experiments should not be emphasized even though the calculation is in good agreement with experiments at lower gas velocity. Zahradnik et al. (1997b) and Camarasa et al. (1999) reported the influence of sparger types and
column geometry on gas holdup curves. With more uniform gas aeration generated from different spargers, the gas holdup curve generally moved up. The occurrence of the first transition point was delayed to a higher gas velocity and the maximum was more pronounced, but the second transition point almost kept invariable and therefore the curve between the maximum point and the second transition point turned to be steeper in their experiments. We may surmise from this variation tendency that for the extreme situation, say, an ideal sparger which could generate absolutely uniform gas injection, the gradual variation between the maximum point and the second transition point may evolve into a jump change as the DBS model predicted. But in practice the abrupt change may be dampened to a gradual and smooth process by the disturbance of nonuniform boundary conditions and hence the strong bubble breakup and coalescence occurs in advance. In this sense, the DBS model may reflect some inherent characteristics of the system evolution. Experiment of Harteveld (2005) and Mudde et al. (2009) suggested that the maximum point was more pronounced to form a summit and then a sharp drop of gas holdup can be observed when using needle spargers and contaminated tap water. 3.2. Comparison between SBS and DBS model Fig. 2 also compares the calculation of the DBS model in this work and the SBS model proposed by Zhao (2006) and Ge et al. (2007). The latter predicts a monotonous increase of gas holdup. Apparently, the difference of the calculation arises from the resolution of gas phase into the small and large bubbles in the DBS model, while only one bubble class with average properties is considered in the SBS model. In fact, Nsurf +Nturb shown in Eq. (8) is linked with more structure variables in the DBS model, and we will show in the following sections that this stability condition leads to the competition between small bubbles and large bubbles. It is interesting to notice that there are also some literature reports claiming that a single gas phase CFD model cannot predict the gas holdup curve which indicates a regime transition, whereas the two gas phases CFD model with some selected correction factors for drag coefficient may do it (Olmos et al., 2003). 3.3. Physical essence of the jump change Although jump change exists for structure parameters and total gas holdup, the energy dissipation terms exhibit relatively
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continuous variation, as shown in Fig. 3. The decrease of the proportion of Nturb to NT indicates that more energy is dissipated for the process of reconstruction of bubble surfaces via breakage, coalescence and oscillation. The micro-scale energy dissipation Nsurf +Nturb decreases and the meso-scale dissipation Nbreak increases with the increase of gas velocity. Fig. 4 illustrates the iso-surfaces of Nsurf +Nturb in the 3D space of structure parameters for the two gas velocities near the jump change. The minimum point jumps from one ellipsoid for 0.128 m/s of gas velocity to another for 0.129 m/s, causing the jump change of the diameter of small bubble from 1.42 to 2.86 mm. Thus it is the stability condition that drives the variation of structure parameters and hence the jump change.
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We find that dcrit corresponds to the lowest point on the curve of drag coefficient as a function of bubble diameter, as illustrated in Fig. 6. Typically the minimum exists for almost all of drag coefficient correlations, marking the distinction of viscous- and surface tensiondominant regimes (Fan and Tsuchiya, 1990). But the relation of this
3.4. The critical bubble diameter The gas holdup for the corresponding bubble class at different gas velocities is illustrated in Fig. 5 for air–water systems. dS increases and dL decreases with increasing Ug . When jump change of total gas holdup occurs from 0.128 to 0.129 m/s of Ug , the structure parameters for large bubbles (dL , fL ) change little, but dS increases abruptly to a large value denoted by dcrit and the significant drop of fS causes the larger decrease of total gas holdup. Beyond the Ug of jump change, dS is invariable and dL continues to decrease. Here dcrit is defined as the critical bubble diameter of the bubble class for which jump change occurs.
Fig. 3. Energy dissipation terms as a function of superficial gas velocity.
Fig. 5. Gas holdup of the two bubble classes at different Ug (m/s) (adapted from Yang et al., 2007).
Fig. 6. Drag coefficients calculated from Grace et al. (1976), Fan and Tsuchiya (1990) and Tomiyama (1998).
Fig. 4. The iso-surface of micro-scale energy dissipation (Nsurf +Nturb )/NT in the 3D space of Ug,S , dS and dL for two gas velocities.
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Fig. 7. Energy dissipation for bubble oscillation for small and large bubble classes. Note that here Nsurf,L is for small bubbles and Nsurf,S for large bubbles.
Fig. 9. Gas holdup calculated from the DBS model for different liquid viscosities.
Fig. 10. Transition velocity of gas as a function of liquid viscosity. Fig. 8. Influences of liquid viscosity and surface tension on critical bubble diameter.
minimum with jump change is rather complicated due to the nonlinear coupling of the model equations and the stability condition. Fig. 7 shows the jump change of Nsurf,S and Nsurf,L , but the total surface energy dissipation Nsurf does not show significant variation as the jump changes of Nsurf,S and Nsurf,L counteract one another. It can also be seen from Fig. 3 that Nsurf +Nturb show little variation at the Ug where jump change occurs for gas holdup and structure variables. It seems that to hold such a gentle variation while attaining the minimum for stability condition, the structure parameters have to vary in a particular manner with the diameter of at least one bubble class being dcrit . dcrit can be obtained from the Grace drag correlation where the parameter H equals 59.3 (Grace et al., 1976) and is dependent on liquid viscosity and surface tension, as shown in Fig. 8. It should be noticed that the value of dcrit also varies with the form of different drag coefficient correlations. The sensitivity of model calculation to drag coefficient correlations has been investigated in our previous work (Chen et al., 2009). 3.5. Effects of liquid viscosity Liquid viscosity was reported by Ruzicka et al. (2003) to have dual effect on regime transition. Moderate viscosity (3–22 mPa s)
destabilizes the homogeneous regime and advances the transition, whereas low viscosity (1–3 mPa s) stabilizes the homogeneous regime. In addition, gas holdup was reported to increase for lower viscosity and decrease for high viscosity (Eissa and Schugerl, 1975). Low viscosity was supposed to increase drag force which could reduce bubble rise velocity but could not promote coalescence, and thus increase the gas holdup; whereas the tendency of coalescence and polydispersity prevails over the drag reduction for high viscosity. Interestingly, this experimental finding could be reproduced by the calculation with the DBS model, as illustrated in Fig. 9. It can be observed that compared to air–water systems of 1.0 mPa s of liquid viscosity, slight increase in liquid viscosity from 1.0 to 3.0 mPa s delays the jump change to higher gas velocities for air–glycerin solution systems, showing the stabilizing effect on homogeneous regime. Further increase of viscosity from 3.5 to 8.0 mPa s advances the jump change to lower gas velocities, thereby implying the destabilization of the homogeneous regime. Here we define the superficial gas velocity where jump change occurs as a transition velocity Ug,trans . The predicted relationship of Ug,trans with liquid viscosity is plotted in Fig. 10, clearly demonstrating the deferment or advance of regime transition at different viscosities. The dash line represents transition velocity for air–water systems. In contrast to Fig. 5 for air–water systems, Fig. 11 shows three different variation tendencies of structure parameters for three liquid viscosities of air–glycerin solution systems. It can be seen that dcrit
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Fig. 11. Gas holdup of corresponding bubble classes at different Ug (m/s) and the variation of bubble diameter. (a) l = 1.5 mPa s; (b) l = 3.0 mPa s; (c) l = 5.0 mPa s.
increases with the liquid viscosity. The diameter of small bubbles dS increases and then jump to dcrit and the diameter of large bubbles dL first decreases and then keep constant at the jump point for low viscosity (1.5 mPa s, Fig. 11a); whereas this structure variation reverses at high viscosity (5.0 mPa s, Fig. 11c), indicating that dS varies little but dL jumps to dcrit . Both dS and dL jump to dcrit at the medium viscosity (3.0 mPa s, Fig. 11b). The two bubble classes seem to compete with each other to jump to a stable state with bubble diameter being dcrit . But at low viscosity, only the diameter of small bubbles could reach dcrit ; whereas at high viscosity, large bubbles have the priority to reach dcrit . Considering the calculation in Fig. 9, it can be concluded that the homogeneous flow can keep its stability even at higher gas velocity when the small bubble diameter dS jumps to dcrit first, but lose its stability at lower gas velocity if the large bubble diameter dL first achieves dcrit . The jump change of gas holdup for small and large bubbles should also be noticed. For air–water system shown in Fig. 5 and low viscosity shown in Fig. 11(a), only the hold up of one bubble class jumps, but the holdup for both classes jumps together for medium and high viscosities (Fig. 11(b) and (c)). We can also understand the dual effect of liquid viscosity from the comparison of the micro-scale energy dissipation Nsurf +Nturb as a function of gas velocity for different liquid viscosities, as shown in Fig. 12. For higher viscosity, the micro-scale energy dissipation is lower compared to that of lower and medium viscosities, and therefore the meso-scale dissipation Nbreak is higher and more energy is dissipated for bubble breakup and coalescence to destabilize the flow in this case. 3.6. Effects of surface tension Similar to liquid viscosity, surface tension was also reported to have dual effect on regime transition by Ruzicka et al. (2008). Low concentration of surfactant stabilizes and high concentration
Fig. 12. Energy dissipation of (Nsurf +Nturb )/NT at different liquid viscosities.
destabilizes the homogeneous flow. The calculation of the DBS model shown in Fig. 13 could support this finding. Compared to pure water system, the jump change is first delayed with decreasing surface tension from 73 to 40 mN/m and then gradually advanced when further decreasing surface tension. Also note that increase of surface tension from 73 to 90 mN/m advances the jump change. The synergistic effect of various factors makes it difficult to perform a quantitative comparison with literature reports because the liquid used in experiments are usually different from each other not only in surface tension but also in conductivity and viscosity.
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Fig. 13. Gas holdup calculated from the DBS model for different surface tension.
Fig. 14. Transition velocity of gas as a function of surface tension.
The dual effect of surface tension on transition gas velocity can be recognized from Fig. 14, showing a maximum at 40 mN/m of surface tension. But the maximum looks less pronounced compared to that of liquid viscosity shown in Fig. 10. This corresponds to the variation of dcrit in Fig. 8 that the critical bubble diameter varies gently with surface tension but is highly influenced by liquid viscosity. Similar to Fig. 11, there are also three types of jump change of bubble diameter with the decrease of surface tension. dS jumps to dcrit when surface tension is decreased from 90 to 40 mN/m, corresponding to the stabilizing effect on homogeneous regime. dL jumps to dcrit when surface tension is further decreased from 40 to 20 mN/m and the destabilizing effect dominates. 3.7. Regime map Zhang et al. (1997) presented a regime map of superficial gas velocity vs. liquid velocity which distinguished six different regimes for their experimental systems with small column diameter. Taitel et al. (1980) developed several empirical correlations to discriminate bubbly, slug, churn and annular flow patterns for vertical pipe flow and their regime map is illustrated in Fig. 15. The criterion for
Fig. 15. Regime maps of Taitel et al. (1980) and the calculation with the DBS model.
transition from bubbly to slug flow was defined by calculating the maximum allowable packing of bubbles which was 0.25 for the system of Taitel et al. (1980). However, Harteveld (2005) reported that the gas holdup could reach 0.5 for holding homogeneous regime for the system using needle sparger. So another transition curve between bubbly and slug flow is plotted with = 0.5 in Fig. 15. The transition points calculated from the DBS model only separate the bubbly and the churn-turbulent (heterogeneous) flow as column diameter is not taken into consideration in the model and thus slug flow cannot be identified. The calculated points lie between the two curves of calculation of Taitel correlation, showing that the DBS model can reasonably predict the regime map though it is currently only a conceptual model. Analogous to our previous work on employing the EMMS model to investigate the effect of multi-scale structure on momentum and mass transfer in gas–solid fluidization (Yang et al., 2004; Wang et al., 2003, 2005), the conceptual model of this study can also be extended to understand the complicated momentum and mass transfer in gas–liquid systems, and thus establish some drag closure for CFD simulation as explored in our recent work (Chen et al., 2009). 4. Conclusions and prospects Regime transition to fully-developed heterogeneous regime can be reflected in the model calculation through the jump change on the curve of calculated total gas holdup vs. superficial gas velocity. Physically, this jump change of calculation is due to the shift of the minimum point from one ellipsoid of the iso-surfaces of micro-scale energy dissipation to another in the 3D space of structure variables. With the increase of superficial gas velocity, both the two bubble classes tend to approach the critical diameter dcrit , and the stability condition leads to the competition between the small and large bubble classes. If the diameter of small bubbles jumps to dcrit first, the homogeneous flow tends to be stabilized and the regime transition is delayed; otherwise the flow tends to be destabilized and the regime transition is advanced if large bubbles have the priority to approach dcrit . Since only one bubble class is introduced in the SBS model, the competition between the two bubble classes cannot be reflected and the calculation with this model can only generate a monotonous curve of total gas holdup. The finding of competition between the two bubble classes can be used to predict and also interpret the dual effect of liquid viscosity and surface tension on the stability of homogeneous flow and the regime transition. It can be
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concluded that this conceptual model, though still rough for exactly describing the flow structure and momentum and energy exchange between phases, could provide a theoretical framework for analyzing the general trend of structure variation.
PHeR SBS THeR
Notation
Subscripts g l L p S
cf CDb CD0,b CDp CD0,p db dcrit dmin dL dS D DT Eo fb fBV fL fS g lE Mo nb Nbreak Nst Nsurf NT Nturb Pb Ug Ug,L Ug,S Ug,trans Ul Wst
2/3
coefficient of surface area increase, cf =fBV +(1−fBV )2/3 − 1, dimensionless drag coefficient for a bubble in a swarm, dimensionless drag coefficient for a bubble in a quiescent liquid, dimensionless drag coefficient for a particle in multi-particle systems, dimensionless drag coefficient for a particle in a quiescent fluid, dimensionless bubble diameter, m critical bubble diameter, m minimum bubble diameter, mm bubble diameter of large bubbles, m bubble diameter of small bubbles, m tube diameter in Taitel's model, m column diameter, m ¨ Eotvos number, dimensionless volume fraction of gas phase, dimensionless breakup ratio of daughter bubble to its mother bubble, dimensionless volume fraction of large bubbles, dimensionless volume fraction of small bubbles, dimensionless gravitational acceleration, m/s2 the entrance length in Taitel's model, m Morton number, dimensionless number density of bubbles, l/m3 rate of energy consumption due to bubble breakage and coalescence per unit mass, m2 /s3 rate of energy dissipation for suspending and transporting particles per unit mass, m2 /s3 rate of energy dissipation due to bubble oscillation per unit mass, m2 /s3 total rate of energy dissipation rate of energy dissipation in turbulent liquid phase per unit mass, m2 /s3 bubble breakup probability, dimensionless superficial gas velocity, m/s superficial gas velocity for large bubbles, m/s superficial gas velocity for small bubbles, m/s transition gas velocity, m/s superficial liquid velocity, m/s rate of energy dissipation for suspending and transporting particles per unit volume, m2 /s3
Greek letters
void fraction in Taitel's model, dimensionless voidage, dimensionless character size of eddy, m viscosity, Pa s density, kg/m3 surface tension, N/m collision frequency, l/s
Abbreviations DBS EMMS
double-bubble-size energy-minimization multi-scale
525
pure heterogeneous regime single-bubble-size heterogeneous regime resulted from the instability of homogeneous regime Gas Liquid large bubble particle small bubble
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