Multiscale multiphase phenomena in bubble column reactors: A review

Renewable Energy 141 (2019) 613e631

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Renewable Energy journal homepage: www.elsevier.com/locate/renene

Multiscale multiphase phenomena in bubble column reactors: A review Shuli Shu, David Vidal, François Bertrand*, Jamal Chaouki**
Department of Chemical Engineering, Ecole Polytechnique de Montr
eal, P.O. Box 6079, Stn. C.V., Montr
eal, QC, H3C 3A7, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 January 2019 Received in revised form 29 March 2019 Accepted 5 April 2019 Available online 8 April 2019

This article presents a state-of-the-art review focusing on the current understanding of the multiscale multiphase phenomena inside Bubble Column Reactors (BCRs). Although many reviews are available on BCRs, little attention has been devoted to summarizing multiscale multiphase phenomena, which are common fundamental issues encountered in their applications. These issues range from the microscale of single bubble dynamics to the mesoscale of bubble swarms and up to the macroscale of the reactor. Understanding these phenomena in all relevant scales can help the rational design, scale-up and optimization of BCRs. The microscale bubble dynamics including the bubble shape and motion, the relevant forces involved and the single bubble mass transfer, is summarized. At the mesoscale, the hydrodynamics of a bubble swarm is influenced by the bubble-bubble or bubble-liquid interactions and hence the overall transport properties of a bubble swarm are not linearly related to that of a single bubble. The bubble swarm effect and the bubble breakage and coalescence mechanisms are discussed in detail. In the end, the macroscale or reactor scale dynamics is strongly governed by the interplay between microscale and mesoscale phenomena, but more research focusing on mesoscale phenomena will be particularly needed for improving our understanding of BCRs. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Bubble column reactor Gas-liquid flows Multiphase flows Multiscale phenomena Hydrodynamics

1. Introduction Bubble Column Reactors (BCRs) are not only widely used in traditional chemical and petrochemical industries to carry out gasliquid and gas-liquid-solid chemical reactions but also in the renewable bioenergy sector, e.g., to cultivate biomass such as microalgae. Traditional applications of BCRs include oxidation, hydrogenation, chlorination, carboxylation as well as FischerTropsch (F-T) and methanol synthesis, to name but a few [1]. BCRs produce over 30 million tons of intermediate and finished products every year in chemical and petrochemical industries [2]. BCRs have lots of attractive features such as economic large-scale construction and operation, good heat transfer performance and uniform temperature distribution, no mechanic-induced crushing of catalyst particles and easy withdrawal or addition of catalyst particles. A vertically-arranged cylindrical vessel mounted with a gas distributor at the bottom is the basic configuration of a simplest

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (F. Bertrand), jamal.chaouki@ polymtl.ca (J. Chaouki). https://doi.org/10.1016/j.renene.2019.04.020 0960-1481/© 2019 Elsevier Ltd. All rights reserved.

BCR, which makes the large-scale construction of a BCR quite easy and economical. In industrial applications, the capacity of a BCR can reach about 300 m3 [3]. Hence BCRs allow for chemical reactions demanding a high residence time of the liquid phase [4]. Moreover, BCRs do not need shaft-sealing inherent to mechanically moving parts encountered in agitated reactors, which makes the operation of BCRs in high-pressure and high-temperature conditions safer and cheaper. The heat transfer coefficient of a BCR is about two orders of magnitude larger than that of a single-phase flow [5]. Coherent structures, i.e. both large-scale liquid circulation and small-scale bubble wake can enhance the heat transfer rate and lead to the uniform temperature distribution inside traditional BCRs [6]. Good heat transfer performance makes BCRs a good alternative for exothermic reactions. Moreover, for microalgae cultivation, the light utilization efficiency can be enhanced by these coherent structures, as they intensify the mixing along the light gradient [7]. The shear rate in the BCRs is relatively lower compared to that in the stirred tanks. Low shear rate is an attractive characteristic that can reduce the damage to shear-sensitive suspended solids or microorganism cells such as microalgae [8]. As of today, the design, optimization and scale-up of BCRs still rely on an empirical know-how. Following this, the configuration

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and operating conditions of BCRs are always empirically chosen or adapted to meet requirements such as the residence time or backmixing of the various phases involved [1]. Longer gas-phase and shorter liquid-phase residence times can be achieved with a downflow BCR, in which both gas and liquid phases are fed at the top and the liquid phase flows downwards. In this case, the liquid velocity should be larger than the terminal rise velocity of gas bubbles, resulting in a shorter liquid-phase residence time. As a result, gas bubbles can either flow with the liquid phase or remain suspended in the liquid phase, leading to a longer gas-phase residence time. Shorter gas-phase or longer liquid-phase residence times can be achieved with a simple BCR fed with gas at the bottom. The gas phase rises upwards and the liquid phase flows either co- or counter-currently, resulting in shorter gas-phase and longer liquidphase residence times [3]. The schematic diagrams of a simple BCR and a downflow BCR are shown in Fig. 1. In a simple BCR, the backmixing of the liquid phase is strong. The degree of backmixing can be reduced by either the installation of internals such as trays or packing [9] or connecting BCRs in series [10]. In microalgae cultivation processes, internals such as baffles are usually used to enhance the mixing along the light gradient [11,12]. The inherently multiscale behavior of multiphase flows (gasliquid) in a BCR as depicted in Fig. 2, makes their rational design, scale-up and optimization rather challenging. The macroscale or reactor scale flow is strongly influenced by the complex mesoscale hydrodynamics involving bubble swarm dynamics, bubble-bubble or bubble-liquid interactions on scales several orders of magnitude smaller than that of the bulk flow. The mesoscale hydrodynamics is determined by the interfacial dynamics (such as the gasliquid interface) at the microscale or bubble scale much smaller than that of mesoscale phenomena. The multiscale multiphase phenomena govern the hydrodynamics, mass transfer and heat transfer of a BCR, which are core issues in chemical engineering. At

Fig. 2. The multiscale phenomena of multiphase flow in BCRs (adapted from Refs. [13e16]).

the microscale, the relative motion between a bubble and its surrounding liquid enhances mass transfer and the bubble wake promotes the micro-mixing or heat transfer rate. However, the collective effect of a bubble swarm at the mesoscale either hinders or accelerates the rising of bubbles and thus has a large influence on the hydrodynamics or mass transfer. Furthermore, the bubblebubble or bubble-liquid interactions at the mesoscale lead to the coalescence or breakage of bubbles and the change of bubble sizes. These further increase the complexity of the multiphase flow. The change of bubble sizes not only affects the motion or mass transfer behavior of the bubble swarm, but also leads to the flow regime transition at the macroscale or reactor scale. Weak bubble breakage and coalescence results in a narrow bubble size distribution which always leads to a homogeneous flow. Conversely, strong bubble breakage and coalescence results in a wider bubble size distribution which always results in a heterogeneous flow with a strong liquid circulation. At the reactor scale, the complex liquid circulation pattern further makes the description of the hydrodynamics, mass transfer as well as heat transfer more difficult. The objective of this work is to give a state-of-the-art review of the literature on multiscale multiphase phenomena taking place inside BCRs. It is organized as follows. In Section 2, the microscale or single bubble-scale phenomena, including the single bubble motion and shape, interphase interaction forces, single bubble mass transfer, are presented. In Section 3, studies on the mesoscale or bubble swarm scale phenomena, including the bubble swarm effect on the drag coefficient and mass transfer, bubble breakage and coalescence and bubble size distribution, are reviewed. In Section 4, we focus on the macroscale or reactor-scale phenomena such as flow regime, flow structure and bubble size. To our knowledge, it is the first time that such a review is done. 2. Microscale phenomena

Fig. 1. Two classical operating methods used to tune the residence time of the gas and liquid phases: A) bubble column reactor; B) downflow bubble column reactor (adapted from Ref. [3]).

A better understanding of a single bubble behavior at the microscale can reinforce the knowledge of that at the mesoscale. However, the single bubble dynamics is quite intricate with the complex shape change in its surrounding liquid, such that the description of shape and motion of a single bubble is more daunting than that of a particle with rigid boundaries. Bubble deformation comes from the interplay between the fluid stresses (including shear stress and pressure) and surface tension forces. The Laplace pressure tends to maintain the spherical shape of a bubble, while fluid dynamics is the driving force for its deformation. The bubble deformation changes the surface area, internal gas circulation and the probability of wake separation, impacting the drag force coefficient and bubble terminal velocity, and the heat transfer and mass transfer coefficients. In this section, microscale phenomena such as

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bubble deformation, bubble internal circulation, bubble wake and the forces exerted on a bubble, which are illustrated in Fig. 3, are discussed. 2.1. Bubble shape The bubble shape at the terminal velocity can be determined by dimensionless parameters: Reb (Reynolds number), Mo (Morton € (Eo € tvo €s number). Reb is the bubble Reynolds number, number), Eo defined as:







rl

U slip

db Reb ¼ ml

(1)

where db is the bubble equivalent diameter, rl is the liquid density, Uslip is the relative velocity between the bubble and surrounding liquids and ml is the liquid viscosity. € represents the ratio of the buoyancy to the surface tension Eo forces:

€¼ Eo

  g rl  rg d2b

s

(2)

where rg is the density of the gas bubble, s is the surface tension and g is the gravitational acceleration. Mo is a characteristic of the multiphase fluid properties including viscosity, surface tension and density, and is defined as

Mo ¼

  gm4l rl  rg

r2l s3

(3)

€ and Mo numbers are only related to the physical In fact, the Eo properties of the system as well as the bubble size. A classical bubble shape regime map based on the experimental analysis of the terminal bubble shape of a single bubble rising in a quiescent liquid was given by Clift et al. [19] and is shown in Fig. 4. € and As can be seen, the terminal bubble shape is dictated by the Eo Mo numbers, which means the liquid viscosity, surface tension and the bubble size play an important role in determining the bubble shape. A high surface tension tends to drive the bubble towards a spherical shape. For a bubble immersed in a high viscosity liquid, the bubble can maintain a spherical shape at larger bubble sizes. Then the larger bubble transits to a dimpled ellipsoidal or skirted one. If the viscosity of the liquid is very low such as in air-water

Fig. 4. The shape regime diagram of bubbles rising under gravity (taken from Ref. [19] with permission).

systems, the bubble shape can be spherical, ellipsoidal, wobbling or a spherical cap with the increase of bubble size. Moore [20] suggested that a distorted gas bubble in a small viscosity liquid (Mo < 108) is an oblate spheroid and the ratio of the cross-stream axis to the parallel axis, c, of the bubbles is related to We, the Weber number. It represents the ratio of the dynamic pressure seen on the bubble surface to the surface tension stresses:





2

rl

U slip

db We ¼ : s

(4)

and c is then given by:

Fig. 3. Important phenomena in microscale or single bubble scale: a) quasi-steady bubble shape (taken from Ref. [17] with permission); b) flow inside and outside of a bubble (taken from Ref. [18] with permission).

616

c¼1þ

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  9 We þ O We2 64

(5)

Davies et al. [21] asserted the front part of a large spherical cap bubble in a liquid of small viscosity is indeed spherical and that:

.   Vb R3b ¼ p 2=3  cos qm þ 1=3 cos3 qm

(6)

where Vb is the volume of the bubble, Rb is the frontal radius of the bubble curvature and qm is the so-called maximum angle. The shape of a bubble of intermediate diameter in a liquid of very low viscosity such as in air-water systems (Mo ¼ 2.6  1011) is expected to change continuously. In fact, the liquid flow inside a BCR is not quiescent and the flow conditions are more complicated. Takagi et al. [22] numerically investigated the bubble shape under different simple shear flow conditions as shown Fig. 5. The change in local shear experienced by the bubble can lead to a change in bubble shape even if the Reynolds and Weber numbers are kept constant. The decrease of the local shear U of the liquid would result in large deformations of the bubble. 2.2. Bubble internal circulation Experimental investigation of the flow inside a bubble is arduous due to the lack of measurement techniques. Garner et al. [23] studied the motion inside gas bubbles by the addition of ammonium chloride fog during their formation. They observed the circulation inside clean bubbles larger than 3 mm in diameter and concluded that the addition of surfactant can prohibit the circulation in bubbles up to 6 mm in diameter. Direct Numerical Simulation (DNS) of bubbles is an alternative approach to investigate the bubble internal circulation. Hayashi et al. [24] conducted DNS simulations of both clean bubbles and contaminated bubbles, and demonstrated that the internal circulation in gas bubbles is caused by a free-slip interface and prohibited by the immobile surface in the presence of a surfactant. The internal circulation in a gas bubble

can be described by the so-called Hill’s vortex [25]. The presence of internal circulation in gas bubbles decreases their stability in the liquid, especially in high pressure systems. 2.3. Bubble wake The occurrence of bubble wake in the liquid phase has been demonstrated to control the diffusion of gas species from bubbles to the liquid phase as well as the micro-mixing of liquid or solid phases [26]. On the other hand, the manifestation of bubble wake near the wall can also break up the boundary layer therein and enhance the radial mass or heat exchange [27]. The bubble wake can also lead to the lateral motion of bubbles, which can further enhance the mass transfer between gas and liquid [28e31]. The onset of the bubble wake is triggered at a critical Reynolds number value, above which the streamlines of the liquid phase tend to branch off from the outline of a gas bubble and rejoin at its rear to form a closed region. The shape of wake behind a bubble is quite similar to those behind solid bodies as shown in Fig. 6. However, the bubble interface is a zero-shear-stress rather than a no-slip one, which makes the mechanism of vorticity production on the bubble surface different [32]. There is a pair of symmetric standing vortex on the rear of a rectilinear rising bubble of intermediate size [33,34]. The occurrence of vortex shedding or wake instability, which is not well understood for the moment, can lead to the lateral motion of bubbles [35]. In air-water systems, the onset of wake-instability occurs for bubbles of about 1.4 mm [26]. Shu et al. [34] conducted a DNS simulation of air bubble motion in water and found that vortex shedding can cause the lateral motion of a pair of bubbles. The strong coupling between wake instability and shape oscillations is also not well understood. 2.4. Motion of a single bubble The rising path of a bubble is closely related to its size or shape. A small spherical or large spherical cap bubble rises rectilinearly in

Fig. 5. The bubble shape and streamline of a bubble (We ¼ 8, Reb ¼ 20) in different liquid flow conditions with shear rate magnitude, U, such as: a) U ¼ 0, b) U ¼ 0.1 and c) U ¼ 0.2 (adapted from Ref. [22]).

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the total drag. However, it is difficult to identify the viscous and pressure effects separately. In fact, the drag depends on the shape and size of the bubble, the liquid properties as well as the relative motion between the bubble and the surrounding liquid. The drag, Fd, can be expressed as:



1

F d ¼  Cd0 Arl U slip
U slip
2

(7)

where Cd0 is the drag coefficient of a single bubble, A is the projected area of the bubble onto the plane perpendicular to the flow. In the case of a rectilinear rise, bubbles with a stable shape reaches their terminal velocity, the drag force balancing with the buoyancy force. The drag coefficient can then be expressed as:

Cd0

  4 rl  rg db g ¼ 3 rl UT2

(8)

where UT is its terminal velocity. From Eq. (8), the terminal velocity of a bubble can be easily calculated from the drag force coefficient. We should also emphasize that the effect of bubble shape or projected area A is lumped into the drag force coefficient Cd0. However, as the rise velocity of a wobbling bubble always changes, we can use either the square of the average velocity or the average of the velocity squared in such a case. Dijkhuizen et al. [37] pointed out that the difference between these two methods is only 5%. €, Grace et al. [38] found that Cd0 is a function of Reb, Mo and Eo which gives: Fig. 6. Photographs of wakes (circulation regions) behind solid bodies of different shapes at various Reynolds number (taken from Ref. [26] with permission): (a)e(c) a solid sphere, (d)e(f) an oblate spheroid, (g) a flat plate.

Cd0

sffiffiffiffiffiffiffiffi Eo3 ¼ 2 Mo 3Re 4

(9)

b

a quiescent liquid. Hence, drag and buoyancy are the dominant forces when the bubble reaches its terminal velocity, whereas the virtual mass force is very important when the bubble is accelerated. However, an intermediate size bubble might rise in a zigzag or spiral way in a quiescent liquid. The source of lateral motion of the bubble is the lift force caused by the vortex shedding. If a bubble is immersed in a simple shear flow, a lift force called the Saffman force [36] will also be generated. In practical applications, the presence of a wall would tend to push the bubbles away from the wall and this is another type of lateral force. In fact, the liquid is not always pure but contaminated with surfactants or other rigid solids. The presence of such impurities in the liquid can lead to a change in both drag and lift forces. If the bubble interface is fully covered with a surfactant, the bubble interface is no-slip, which might lead to the increase of the drag coefficient. In this condition, if the bubble is rotating, it will also be submitted to a lateral force called the Magnus force. If the bubble is partly covered with a surfactant, the additional Marangoni force, which opposes the surface flow, is created due to the difference in surface tension, and which can further increase the drag coefficient. The key of describing a single bubble’s motion is to understand the relevant governing forces. The drag, lift, virtual mass forces as well as the wall lubrication force in pure liquids are discussed next. 2.4.1. Drag force The drag acts opposite to the relative velocity between the gas bubble and the liquid. Both viscous drag (skin drag) and pressure drag (form drag) contribute to the overall drag. There is no general theory on the skin drag except in the creeping flow regime [32]. With the increase of bubble size and rising velocity, the pressure variations over the bubble surface contributes increasingly more to

Eq. (9) clearly shows that the relationship between the three characteristic dimensionless numbers and the drag force coefficient is non-linear. Many other drag coefficient correlations for a single bubble are available in the literature and four widely used ones are listed in Table 1. Schiller et al. [39] proposed a correlation for rigid spherical particles based on an experimental investigation, which might be applicable for the contaminated spherical bubbles in the correlation of Tomiyama [40]. However, the Schiller-Naumann model is not applicable for non-spherical bubbles with large deformations. Fig. 7 compares the drag coefficient as a function of a single bubble equivalent diameter for all correlations listed in Table 1. The drag coefficient of a large bubble (db > 10 mm) predicted by Tomiyama [40], Fan et al. [26], and Ishii et al. [41], are in close agreement at about 2.7. However, the drag coefficient calculated by Schiller et al. [39] model is significantly lower at 0.44. In fact, a large bubble is a spherical cap, which leads to a larger projected area and hence to a more complex bubble wake. Therefore, the larger drag coefficient predicted by Tomiyama [40] is more suitable for bubbles with large deformations. The drag coefficient correlation of Grace et al. [38] is applicable for deformed or non-spherical bubbles. The drag coefficient of a bubble of diameter from about 2 to 20 mm predicted with this correlation is similar to that of Tomiyama [40], Fan et al. [26], and Ishii et al. [41]. However, the drag coefficient for bubbles larger than 20 mm predicted by Grace et al. [38] increases with the bubble diameter while the other correlations yield constant value. The former drag coefficient model is acceptable for air-water systems as the maximum stable bubble diameter in this case is about 17 mm [38]. The presence of a surfactant or other contaminants can change the motion behavior of a single bubble as well. As shown in Fig. 7, the addition of such impurities can increase the drag

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Table 1 Summary of correlations available in the literature for a single bubble drag coefficient. Authors

System

Schiller et al. [39]

A rigid spherical particle in an infinite stagnant fluid

Grace et al. [38]

Non-spherical bubble

Ishii et al. [41]

Bubble and droplet

Fan et al. [26]

Bubble in different liquids

Tomiyama [40]

€<103, A single bubble at various conditions (102
100

Expressions   24 Cd0 ¼ max ð1 þ 0:15Re0:687 Þ; 0:44 b Reb m 4 gdb ðrl  rg Þ Cd0 ¼ ; UT ¼ l Mo0:149 ðJ  0:857Þ; 3 UT2 rl rl db ( !0:14 ml 4 0:94b0:757 2 < b < 59:3 € Mo0:149 ; mref ¼ 0.0009 kg/m.s ;b ¼ Eo J ¼ 0:441 b > 59:3 mref 3 3:42b  pffiffiffiffiffiffi  24 2 8 €; Eo ð1 þ 0:15Re0:687 Þ; min Cd0 ¼ max b Reb 3 3 4 gdb ðrl  rg Þ ; Cd0 ¼ 3 UT2 rl  0:25 " n0  n0 =2 #1=n0 € € 0:5 sg Eo 2c1 Eo þ þ ;Kb ¼ UT ¼ 0:25 0:5 € rl 2 Eo Kb Mo  Kb < 12 12 ; with n0 ¼ 1.6 and Kb0 ¼ 14.7 (air-water system) Kb0 Mo0:038 Kb > 12 For purified water: 

 € 16 48 8 Eo ð1 þ 0:15Re0:687 Þ; ; Cd0 ¼ max min ; b €þ4 Reb Reb 3 Eo For slightly contaminated water: 

 € 24 72 8 Eo ð1 þ 0:15Re0:687 Þ; ; ; Cd0 ¼ max min b € Reb Reb 3 Eo þ 4 For contaminated water:

€ 24 8 Eo ð1 þ 0:15Re0:687 Þ; ; Cd0 ¼ max b €þ4 Reb 3 Eo

2.4.2. Lift force The lift force is more complicated to describe as its origin is more complex. The origin of the lift force exerted on a bubble is believed to come from three sources: shear-induced lift force or Saffman force [36], spinning-induced lift force or Magnus force and vortex-induced lift force. Early stage investigations of the lift force only considered the shear-induced lift or Saffman force. The shear-induced lift force, FL, acting on a bubble can be formulated as [51e53]:

Schiller et al.[39] Ishii et al.[41] Tomiyama [40] / (pure) Tomiyama [40] / (comtaminated) Fan et al.[28] Grace et al. [38]

Cd0

10

1 Cd0=2.7 db=17 mm

0.1 0.1

1

10

100

db(mm) Fig. 7. Drag coefficient Cd0 as a function of a single bubble diameter predicted by empirical correlations from the literature (air-water system under ambient conditions).

coefficient significantly when the bubble diameter is less than 3 mm, i.e. decrease the terminal rising velocity. In practical modeling applications and underlying Computational Fluid Dynamics (CFD) simulations, there is still no general agreement on how to choose the drag coefficient. The drag coefficient chosen varies from author to author, going from constant values throughout the reactor [42,43] to values obtained from various detailed correlations accounting for local variations [44e48]. Among these drag coefficient correlations, the one proposed by Ishii et al. [41] is the most widely used and has been shown to give relatively good predictions of gas hold-ups [44,49,50]. It must be emphasized that drag coefficient correlations for a single bubble might not be suitable to describe the motion of a group of bubbles in BCRs, where bubble swarms are frequent. In such cases, the drag coefficient needs to be corrected, a topic that will be discussed in Section 3.

p

F L ¼ CL rl d3b ðU b  U l Þ  ðV  U l Þ: 6

(10)

where CL is the lift force coefficient, Ub is the velocity of the bubble and Ul is the velocity of the liquid phase. Different values for the shear-induced lift coefficients CL such as 0.5 [51], 0.1 or 0.25 [54] have been reported. Legendre et al. [29] pointed out that the CL increases with Re and approaches 0.5, when the bubble Reynolds number is small. Spinning-induced lift force or Magnus force only exists when the bubbles are contaminated. In this case, contaminated bubbles behave like a solid particle and can therefore rotate. Experimental results in a rotating flow show that the increase of rotational speed leads to an increase in lift force coefficient [55,56]. The lift force coefficient for rotational contaminated bubbles tends to be 1e1.2, which is at least twice as much as the value originated from the shear-induced lift force [51]. The lift force produced by the shedding vortex can be one order of magnitude larger than the buoyancy force [57]. Tomiyama et al. [31] assumed that the lift force acting on a bubble caused by a trailing vortex is similar to that induced by the shear. The following expressions for the lift force coefficient CL lump the effects of both shear and shedding vortex-induced lift forces:

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8 € d Þ < min½0:288 tanhð0:121Reb Þ; f ðEo €d Þ CL ¼ f ðEo : 0:28

Thomson [69] gave a classical definition of the virtual mass force by assuming it is the acceleration of the sphere volume times onehalf the mass of the displaced fluid:

€d < 4 Eo € d  10:7 4  Eo € d > 10:7 Eo (11)

€d Þ ¼ 0:00105Eo € 3d  0:0159Eo €2d  0:0204Eo €d þ 0:474and where f ðEo €d is calculated based on the hydraulic diameter Eo p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 € 0:757 [58]. Hence, large bubbles tend to dh ¼ db 1 þ 0:163Eo migrate to the center of the reactor and small bubbles tend to move to the wall, which is consistent with observations in BCRs [59]. In CFD simulations, the predicted profile of the radial distribution of gas hold-up is very sensitive to the value of the lift force coefficient. With a positive lift force coefficient, the radial distribution of gas hold-up is flatter and centerline velocities become lower and vice versa for a negative coefficient [60]. However, how to determine the lift force coefficient in CFD simulations is still an open question. In modeling BCRs, CL has been usually chosen as a constant varying from 0.5 [61] to 0.5 [47,62,63]. In recent years, the model of Tomiyama et al. [31] for the lift force coefficient of a single bubble has become more and more popular [64]. For industrial applications, the effect of Magnus lift force might need to be taken into account due to the possible contaminated bubble surface. However, there is still lack of CFD simulations that account for this effect. Moreover, Kulkarni [65] pointed out that the swarm effect should be taken into consideration as they found that the sign of CL changes from negative in the center to positive in the region close to the wall of the BCR. 2.4.3. Wall lubrication force According to experimental observations, bubbles in a vertical pipe are kept away from the wall [66]. A phenomenological force called “wall lubrication force” is assumed to push bubbles away from the wall. Antal et al. [66] proposed a wall lubrication force model formulated for a bubble approaching a wall as

FW

  

2 2 d CW1 þ CW2 b ¼ rl

U slip

n db 2y

(12)

where FW is the wall lubrication force, y is the distance between bubble and wall, n is the unit vector normal to the wall, CW1 ¼ 0.104-0.06|Uslip| and CW2 ¼ 0.147 are model coefficients. However, this model indicates that the wall forces exerted on bubbles located far away from the wall are directed toward it. Tomiyama et al. [67] proposed a new wall lubrication force model based on experiments for which:

FW ¼

 2

2 2 d

CW3 b rl
U slip
n db 2y

(13)

where CW3 is a model parameter which can be determined by:

 CW3 ¼

€ þ 1:79Þ 1 < Eo €<5 expð  0:933Eo € þ 0:04 € < 33 0:007Eo 5 < Eo

619

(14)

Tomiyama et al. [67] also suggested that a reliable correlation for CW3 is required. The inclusion of wall lubrication forces in CFD simulations of BCRs may result in an overestimation of the velocity profiles [49]. In practice, wall lubrication forces are rarely used to simulate the flow in BCRs. 2.4.4. Virtual mass force The virtual mass force is caused by the change in kinetic energy of the fluid surrounding an accelerating bubble [68]. Milne-

F VM ¼

p 3 dub d r 12 b l dt

(15)

where FVM is the virtual mass force of a bubble and ub is the velocity of the bubble. Therefore, the virtual mass force per unit bubble volume, FVM,0, can be expressed as:

F VM;0 ¼ CVM rl aVM

(16)

where CVM is 0.5 and aVM is the acceleration of the bubble. Auton [70] linked CL and CVM by applying the frame indifference principle and found that CL and CVM are identical. Legendre et al. [29] pointed out that CVM¼CL cannot be deduced from the application of the frame indifference principle and the identity of CVM¼CL only holds for nearly steady flows with low shear rates. Finally, note that whereas adding a virtual mass force in CFD simulations may not have a significant impact on the predicted profiles, it can enhance their numerical stability [44,71]. 2.5. Stability of a single bubble A single bubble freely rising in a stagnant liquid may break up if it is larger than a critical size (the so-called maximum stable bubble size). Grace et al. [72] attributed bubble breakage to Rayleigh-Taylor instabilities. Kitscha et al. [73] linked the bubble breakage to the Kelvin-Helmholtz instabilities. Luo et al. [74] took the effect of the internal gas circulation into consideration and successfully proved that the resulting increased pressure can decrease the maximum stable bubble size. 2.6. Mass transfer of a single bubble When a bubble rises in a liquid, the mass transfer resistance is mainly on the liquid side. The two-film [75], the penetration [76] and the surface renewal [77] theories are three classical approaches to describe the liquid side mass transfer coefficient. The two-film theory assumes that the mass transfer is a steady-state process and there is a stagnant film near the interface. However, the mass transfer between a rising bubble and its surrounding liquid is unsteady. In this case, the Higbie’s penetration theory and its extension (the surface renewal theory) are more suitable for the description of mass transfer. Both the Higbie’s penetration and surface renewal theories assume that the mass transfer coefficient is controlled by the rate of surface renewal. The liquid-side mass transfer coefficient kl for a bubble with mobile surface can be expressed as:

sffiffiffiffiffi Dl p tc

2 kl ¼ pffiffiffi

(17)

where tc is the contact time and Dl is the molecular diffusivity. The penetration theory proposed by Higbie [76] assumes a nonsteady-state liquid film and a down-flowing laminar flow. The contact time is estimated as:

tc ¼

db Ub

(18)

where Ub is the rising velocity of the bubble. Danckwerts [77] further assumed that the main contribution of

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surface renewal is the creation of turbulence eddies and their positive effect on the mass transfer. The mass transfer rate on the liquid side based on surface renewal theory can be expressed as:

sffiffiffiffiffi Dl kl ¼ cr tr

(19)

where cr is a model parameter and tr is the mean time between renewal events. The latter variable is assumed to be proportional to the Kolmogorov time scale (vl/ε)1/2, where vl is the liquid kinematic viscosity and ε is the turbulent dissipation rate. The penetration theory assumes the flow is laminar while the surface renewal theory is based on the turbulent eddies. The mass transfer coefficient predicted by Higbie’s model is related to the bubble size and rising velocity. By contrast, the mass transfer coefficient predicted by the surface renewal model is independent of the bubble size and rising velocity. Alves et al. [78] found that the Higbie’s model and Eq. (18) could adequately predict the mass transfer coefficient of single bubbles up to db ¼ 6 mm in clean water when the turbulent dissipation rate, ε, is up to 0.04 m2 s3. However, the addition of a surfactant may lead to a totally rigid air-liquid interface so that the behavior of a bubble is altered, € ssling resulting in a decrease in the mass transfer coefficient, kl. Fro [79] proposed a theoretical prediction of mass transfer of a rigid bubble from the laminar boundary layer theory giving that:

kl ¼ c

qffiffiffiffiffiffiffiffiffiffiffiffiffi 2=3 1=6 Ub =db Dl vl

(20)

where c is 0.6.

improve the mass transfer rate on the gas side on the one hand and decrease the bubble size especially in high pressure systems. Flow around the bubble and bubble wake can not only improve liquid back-mixing but also enhance the heat transfer by breaking up the boundary layer near the wall. The shedding of bubble wake can also lead to the lateral motion of bubbles. However, this phenomenon is not well understood at the moment. In practical applications, the bubble interface can be covered with a certain amount of surfactant or solids. The presence of such impurities that fully cover the bubble interface can increase the drag coefficient and decrease the mass transfer rate. The dynamics of a partly covered bubble is more complicated and would need to be fully investigated. 3. Mesoscale phenomena The mesoscale phenomena such as complex bubble-bubble and bubble-liquid interactions make the hydrodynamic behaviors of a bubble swarm quite different from that of a single bubble. On the one hand, bubble-bubble interactions can either accelerate or hinder the terminal velocity of individual bubbles in a swarm. On the other hand, bubble-bubble and bubble-liquid interactions can lead to the breakage or coalescence of bubbles, resulting in the change in bubble polydispersity. This can further make the characterization of multiphase flow systems more difficult. This prohibits the direct application of single bubble dynamics theories to practical applications. Therefore, the mesoscale phenomena are the key to understand the multiscale multiphase flow phenomena in the BCRs. 3.1. Motion behavior of a bubble swarm

2.7. Summary of microscale phenomena The understanding of single bubble dynamics or mass transfer is key to that of complex multiphase transport phenomena prevailing in BCRs. One of the fundamental issues is to understand the forces governing the motion of a bubble. The drag is the dominating force and is directly related to the terminal velocity, mass transfer rate as well as the energy input of the bubble into the liquid. Free slip conditions and a deformable interface make the drag coefficient of a bubble more complex than that of a solid particle. A small bubble is spherical and the well-known Schiller-Naumann drag model is applicable. The Ishii-Zuber and Tomiyama drag models are suitable for a single bubble of spherical or deformable shape in quiescent liquid. However, the local flow of a bubble is complex and has a significant effect on the drag coefficient. Such effect is seldom reported and needs more attention. The Higbie’s mass transfer model, in which the contact time is determined by both the bubble size and rise velocity, is suitable for a clean bubble when the turbulent dissipation rate is up to 0.04 m2 s3. The lift force is important and may be in certain circumstances one order of magnitude larger than the drag force. However, the origin of the lift force is complex and needs more investigations. The effect of the virtual mass force or wall lubrication force may play a role when the bubble is accelerated or near the wall. The more complex hydrodynamic behavior of a single bubble with regards to that of a single solid particle stemming from the deformable and free slip interface. The shape of the bubble is related to the drag coefficient. In fact, the shape of a single bubble can be determined by the bubble size as well as the physical properties of the system. Three dimensionless numbers Reb, Mo and €, can be used to qualitatively characterize the bubble shape. Eo However, more work is needed to quantitatively characterize the bubble shape. Flow inside the bubble is also very important for the bubble dynamics. The circulation of gas inside the bubble can

The rising behavior of a bubble swarm is different and more complicated than that of a single bubble due to the collective effect. Richardson et al. [80] found that the sedimentation of a solid particle in a liquid-solid suspension is hindered. The drag force coefficient of a solid particle in a suspension, Cd, is expressed by the drag force coefficient of a single particle, Cd,0, modified with a factor with respect to the solid volume fraction, εs. This factor reflects the hindering effect of surrounding particles in the suspension. The effect of a bubble swarm on the motion of a bubble is always taken into account by a correction factor linked to the volume fraction of the bubbles. In a homogeneous and monodisperse bubble swarm, the rise of the bubbles is prohibited by the increasing so-called gas hold-up, εg, which represents the volume fraction of the gas phase [81e83]. The decrease of the rising velocity of a bubble swarm is partly due to the decrease of the buoyancy force. Indeed, the buoyancy force on a bubble in a swarm is smaller than that of a single bubble in a liquid, as the density difference between the gas phase and gas-liquid mixture is smaller. In this case, the drag force coefficient of a bubble swarm, Cd, is larger than that of a single bubble. Ishii et al. [41] partly attributed the increase of the drag coefficient to the increase of the pseudoviscosity of the suspension. Simonnet et al. [84] found that Cd increases with local gas hold-up when the gas hold-up is less than 0.15 and the bubble size is less than 7 mm. However, the average bubble velocity tends to increase with the gas hold-up in the heterogeneous flow regime, which indicates the decrease of Cd. Indeed, Ishii et al. [41] found that Cd decreases with the local gas hold-up in the heterogeneous flow regime. Furthermore, Simonnet et al. [84] found that Cd decreases with the local gas hold-up when the gas hold-up is larger than 0.15 and bubble size is larger than 7 mm. They attributed the decrease of Cd to the coalescence of bubbles, the acceleration of bubbles in the leading bubble wake and the bubble-induced turbulence. Table 2 lists the

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Table 2 Drag coefficient correlations for a bubble swarm reported in the literature. Authors

Correlations

Richardson et al. [80]

Cd ¼ Cd;0 ð1  εs Þ2n0 , n0 > 0 (liquid-solid system)

Wallis [90]

Cd ¼ Cd;0 ð1  εg Þ2

Marrucci [91] Ishii et al. [41]

5=3 2

Cd ¼ Cd0 ð1  εg Þ4 ð1  εg Þ

mm ¼ ð1  εg Þ2:5ðmg þ0:4ml Þ=ðmg þml Þ ml Stokes regime: Cd ¼

24 , Re ¼ rl Uslip db =mm Re

Viscous regime: Cd ¼

1 þ 17:67½f ðεg Þ6=7 24 ð1 þ 0:1Re0:75 ÞCd ¼ Cd;0 Re 18:67f ðεg Þ

!2

Heterogeneous flow regime: Cd ¼ Cd;0 ð1  εg Þ2 1=3 2

Garnier et al. [82]

Cd ¼ Cd0 ð1  εg Þ

Simonnet et al. [84]

Cd ¼ Cd0 ð1  εg Þ1 (db < 7 mm and εg<15%)   2=25 εg (db>7 mm and εg>15%) Cd ¼ Cd0 ð1  εg Þ ð1  εg Þ25 þ 4:8 1  εg

Riboux et al. [92]

Þ Cd ¼ Cd0 ð1  ε0:49 g

experimental correlations for the effect of gas hold-up on the drag coefficient of a swarm. In CFD simulations, the drag coefficient of a single bubble is in practice usually corrected to account for the swarm effect. For heterogeneous flows, the correction factor is chosen to be smaller than one, which represents the accelerating effect of bubble swarms [85]. Conversely, the correction factor is usually set to be larger than one in the homogenous flow regime, to model the hindering effect of bubble swarms [86e88]. Nevertheless, further developments are needed to obtain a more general correction factor that can be applied across various flow regimes and situations. Few studies have dealt with the swarm effect on the lift force. Behzadi et al. [89] proposed an empirical correlation for the lift coefficient of a bubble swarm, which is positive and inversely proportional to the gas hold-up. However, Kulkarni [65] pointed out that more work is needed as the correlation from Behzadi et al. [89] has limited applicability since it is only valid for gas-liquid flows in small-diameter pipes. 3.2. Bubble breakage and coalescence The constant coalescence and breakage of the bubbles have been observed in a large bubble swarm by many investigators [93]. These phenomena can lead to a change of bubble size distribution, which may then affect bubble dynamics and mass transfer. The origin of bubble breakage comes from the interaction between the liquid phase and bubbles. The mechanism that governs the bubble breakage and coalescence phenomena have been the topic of a large body of literature. The turbulent fluctuations are the main reason for bubble breakage in practical applications, though different mechanisms such as viscous shear, interface instability and turbulent fluctuations can lead to the breakage of bubbles [94]. One set of theories attributes the bubble breakage mechanism to the competition between the force stabilizing the bubble and the force destabilizing the bubble. The former refers to surface tension and the latter is regarded as the turbulent stress force. Kolmogorov [95] and Hinze [96] separately suggested that a bubble would break up if the ratio between the average stress force caused by turbulent fluctuations and the surface tension force, the Weber number (We), is above a critical value, Wecrit. Their theories rely on the basic assumptions that only the inertia-dominated eddies larger than the Kolmogorov scale and the velocity fluctuations within a distance of

2

the bubble diameter are capable of causing bubble breakage. In this case, a critical bubble diameter dmax beyond which breakup occurs is, hence, postulated. It should be emphasized that the dmax can be determined from the Wecrit. Delichatsios [97] suggested that the calculation of the Weber number should be based on the maximum stress force or fluctuation. Many studies focusing on determining the Wecrit have been conducted for liquid-liquid systems [95,96,98,99]. It has been observed that a strong turbulent dissipation rate or a smaller surface tension leads to a smaller dmax. Experimental investigations for the determination of dmax in gasliquid systems are scarce. Sevik et al. [100] found that the Wecrit of air bubbles in a turbulent jet flow is 1.26 and attributed the difference between the Wecrit values for bubbles and droplets to the difference in density ratios. Risso et al. [101] found that Wecrit is 5 if the breakage is only caused by the turbulence. They attributed the increase of Wecrit or dmax to the absence of other causes of breakage. Lehr et al. [102] used much lower Wecrit ¼ 0.06 in their simulations. Zhao et al. [103] took into account the turbulent eddy efficiency on bubble breakage. Another set of bubble or droplet breakage theories is based upon the competition between the turbulent kinetic energy and the surface energy. Coulaloglou et al. [104] suggested that an oscillating deformed droplet may break up if the turbulent kinetic energy of eddies colliding with this droplet is greater than its surface energy. Luo et al. [105] assumed that collisions between the bubbles and the surrounding eddies can lead to breakage if the energy inherent to the surrounding turbulent eddies is larger than the increase of surface energy due to breakage. Wang et al. [106] combined both the critical Weber number, Wecrit, and the bubble surface energy to investigate breakage. The smaller the surface tension and the stronger the turbulence, the more important the occurrence of bubble breakage. Most of breakage rate models assume that instant breakage occurs whenever an eddy of sufficiently high energy collides with the bubble. Coulaloglou et al. [104] assumed that the breakage frequency is proportional to the fraction of eddies containing a turbulent kinetic energy greater than the droplet surface energy. Narsimhan et al. [107] assumed that instant breakage occurs whenever an eddy of sufficiently high energy collides with the droplets. Luo et al. [105], Lehr et al. [102] and Wang et al. [106] used in their work this assumption. However, Risso et al. [101] pointed out that while intense enough turbulent eddies may cause instant breakup of non-deformed bubbles, the breakage of deformed bubbles takes some time due to the bubble oscillation damping.

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The breakage of a bubble into two daughter bubbles is widely assumed in the literature [104,105]. However, Risso et al. [101] reported that two fragments were obtained in 48% of cases, 3 to 10 fragments were obtained in 37% of cases and over 10 fragments were created in the remaining 15% of cases after bubble breakage. Bubbles coalescence only occurs when collisions between bubbles are efficient. These collisions are related to the number density and the relative displacements of neighbouring bubbles. Collisions of bubbles are usually considered to come from the turbulent motion [108]. Therefore, the basis of the bubble collision frequency is the turbulent theory. The efficiency of coalescence can be determined by the energy model [109], critical approach model [102] or film drainage model from Das et al. [110]. Currently, coalescence models largely depend on uncertain assumptions and adjustable parameters. The reader is referred to Liao et al. [108] for more details about them. A large surface tension favors bubble coalescence [108]. The addition of a surfactant, alcohol [111] or electrolyte solutions in the air-water system [112] can hinder bubble coalescence [113] and then favor smaller bubbles. The breakage and coalescence theories, including breakage and coalescence criteria as well as the concept of daughter bubble size distribution, can be incorporated into Population Balance Equations (PBEs), which can be coupled with CFD simulation to obtain macroscale properties such as the bubble size distribution (BSD). Due to the complexity of integro-partial differential PBE, numerically accurate and computationally efficient methods for solving these equations are hardly available [114,115]. Among available methods as shown in Table 3 are: (1) Direct Discretization Methods (DDMs) [116], (2) Monte Carlo Methods (MCMs) [117], and (3) Method of Moments (MOMs) [118,119]. DDMs or MOMs coupled with CFD codes are available in both commercial and open-source platforms such as Ansys Fluent and OpenFOAM [120]. DDMs can directly give the BSD but requires a large amount of computational resources [115]. MOMs are more computational efficient but the downside is that the actual BSDs are not preserved [121]. Both DDM [122,123] and MOM [124] have been used to model the gas-liquid flows to predict the global gas hold-up, radial gas hold-up distribution, Sauter-mean diameter radial distribution or BSD in different axial locations in the developed flow region [114,125]. However, experimental data for large-scale BCRs are still missing and this prevents accurate validations to be performed to assess the applicability of these models [126]. Recently, Yang et al. [127] proposed a Dual Bubble Size (DBS) model, which provides new insight into the mesoscale phenomena in bubble columns. In the DBS model, the energy related to mesoscale bubble breakage and coalescence is regarded as a portion of the total energy dissipation and determined by a stability criterion, as explained by Li [128] in the context of EnergyMinimization Multi-Scale (EMMS) approach [128e131]. The DBS model is supported by experimental observations [132] and predict the flow regime transition at the macroscale [133]. It can help CFD models to improve their accuracy when predicting gas hold-up [134] and BSDs [135,136]. 3.3. Summary of mesoscale phenomena The bubble rising behavior at the mesoscale level is quite

different from that of a single bubble at the microscale level. In fact, a rising bubble in a swarm is either hindered or accelerated. In homogeneous flows, the rise of a bubble seems to be hindered. One of the reasons is that the driving buoyancy force decreases due to the reduction of density difference between bubbles and gas/liquid mixture. Therefore, the drag force coefficient of a bubble in a swarm is always expressed as a single bubble drag coefficient modified by a hindering effect factor related to the volume fraction of the bubble swarm. However, the hindering effect factor is non-linearly related to volume fraction of the swarm. On the other hand, the hindering effect may come from the change of the apparent “mixture” viscosity. In heterogeneous flows, the rise velocity of a bubble would be accelerated in a bubble swarm with wide bubble size distribution. In this case, large bubble wakes can accelerate the bubbles entrained in them. Quantification of the hindering or accelerating effect of a swarm would require further investigations. Breakage and coalescence are two important phenomena at the mesoscale. Precise mechanisms or criteria triggering these two phenomena and the rate at which they occur is currently lacking and would deserve further research. In addition, a better prediction of the daughter size distribution resulting from bubble breakage would also deserve more fundamental understanding of the phenomena. In practical applications, bubble breakage and coalescence are largely caused by the turbulence. Bubbles can break up with either a strong enough turbulent stress force or incoming turbulent eddies containing sufficient energy. Regarding bubble coalescence, bubble collisions might also result from the turbulent fluctuations. Therefore, a better understanding of the turbulence, especially in multiphase turbulence, is key to improve the breakage and coalescence theories. However, our knowledge of turbulent phenomena in multiphase flows is still limited and the behavior of turbulent eddies in such cases is not fully understood. Thus, further work on this subject is needed. The instant and binary breakage are basic assumptions in most current theories. However, experimental evidences show that there are some resonance-like mechanisms that lead to bubble oscillations so that more than two daughter bubbles can be obtained in some cases. More work is still needed to shed light on the importance of these resonance-like mechanisms, and the occurrence of multiple daughter bubbles during breakage. Finally, the effects of the presence of surfactant, impurities and solid particles still need to be better understood for both breakage and coalescence phenomena. 4. Macroscale phenomena The macroscale phenomena directly influence the performance of BCRs. Large-scale liquid recirculation patterns affect liquid backmixing, which in turn impacts the space-time yields and selectivity of the process. Gas hold-up directly affects gas-liquid mass transfer rate. Gas hold-up along with bubble sizes determine the gas-liquid interface area. Gas hold-up radial distribution is one of the crucial parameters as it affects both the liquid recirculation and the local mass transfer rate. Gas hold-up radial distribution is closely related to the bubble size, which is governed by the bubble coalescence and breakage at the mesoscale. In fact, two distinct flow regimes, i.e. homogeneous and heterogeneous, can be encountered at the reactor scale. In the former, gas bubbles are uniformly distributed

Table 3 Advantages and disadvantages of different numerical methods for PBE. Methods

Advantages

Disadvantages

Direct Discretization Methods (DDMs) Monte Carlo Methods (MCMs)

   

   

Method of Moments (MOMs)

Directly predict the BSD Simple and straightforward method Able to deal with high-dimensional problem Computational efficient

High computational cost High computational cost Exhibit statistical noise Reconstruction of the actual BSDs required

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with identical sizes. The latter is characterized by a bimodal bubble size distribution with higher gas hold-up in the center and lower one near the wall.

4.1. Flow regime Flow regime is closely related to the superficial gas velocity. The mesoscale bubble coalescence or breakage phenomena do not usually occur at low superficial gas velocities. In this case, bubble sizes are determined by the gas distributor design and do not change after their detachment from the gas distributor. This is the so-called bubbly flow or the homogeneous flow regime. For airwater systems, a bubbly flow with 2e10 mm identical bubbles occurs when the superficial gas velocity is less than 0.05 m s1 [137]. For homogenous flows, the uniform radial distribution of gas volume fraction is due to the uniform bubbles and leads to relatively weak back-mixing of the liquid phase or higher space-time yields. The homogenous flow regime is preferred for slow reactions, in which the mass transfer is not the limiting step, with more space for liquid reactions. Mesoscale phenomena such as bubble breakage and coalescence are intensified at high superficial gas velocity, and the bubble size distribution becomes wider. In this case, the flow transits into a heterogeneous or churn turbulent regime. A heterogeneous flow is characterized by a bi-modal volume-based bubble size distribution, consisting of fast large bubbles and slow rising small bubbles [138]. Large bubbles tend to rise at the center, whereas small bubbles are either entrained upward in the wake of large bubbles at the center or downward near the wall, due to strong liquid back-mixing. Strong liquid circulation leads to higher back-mixing or lower space-time yields. However, intense back-mixing is beneficial to heat transfer, which keeps the temperature uniform and helps preclude the occurrence of “hot spot”. Large bubbles have higher rising velocities, leading to a lower residence time of the gas phase. Therefore, the choice of flow regime in the BCR depends on the industrial application. If the superficial gas velocity continues to increase in a small diameter bubble column, the heterogeneous flow will transit into a slug flow. The relationship between the superficial gas velocity and the flow regime is shown in Fig. 8. The hydrodynamics in a BCR can vary a lot from homogenous to heterogeneous flows. However, the mechanisms for the flow regime transition is not clearly understood. The drift flux analysis

623

and the dynamic gas disengagement technique (DGD) are two classical methods used to identify the flow regime transition in BCRs. In the former, the flow regime transition occurs at the point where the slope of UG/εg against UG þ UL or the drift flux UG(1εg) ± ULεg against the gas hold-up εg changes drastically. While the DGD technique assumes a bi-modal volume-based bubble size distribution in the heterogeneous flow regime, the dispersion height is measured after the sudden shut-down of the gas feeding system and the resulting temporal variation of dispersion height is used to identify the flow regime. The superficial gas velocity at which the flow regime transition occurs, UG,trans, is closely related to the gas distributor, the dimension of the BCR and the operating conditions such as the gas density, the presence of solvent, the liquid viscosity, the temperature and the pressure. The homogeneous flow regime cannot be obtained in most cases with a perforated plate distributor of hole diameters do > 1 mm. Increasing the free plate area can stabilize the homogeneous flow regime [139]. The increase of the column height, diameter or the liquid viscosity tend to destabilize the homogeneous regime and decrease the value of UG,trans [139,140]. The increase of gas density by increasing pressure or molar weight can always help to stabilize the homogeneous flow regime [141e143]. The presence of electrolytes, alcohol or carboxyl methyl cellulose (CMC) can change the coalescing behavior of the liquid phase and then change UG,trans. Electrolytes and alcohol can suppress the bubble coalescence and delay the transition [139,144]. However, CMC solution can enhance the coalescing behavior and thus advance the regime transition [145]. The effect of temperature on UG,trans is still unclear. Grover et al. [146] found that UG,trans decreases with the increase of temperature as a result of increased coalescence for the air-water system. Lin et al. [147] concluded that the increase of temperature could increase UG,trans. Wilkinson et al. [148] correlated UG,trans with the density of the gas and liquid phases, the surface tension, the liquid viscosity and the gravitational acceleration constant based on experiments.

4.2. Macroscale flow structure The flow structure depends on the flow regime. In homogeneous flows, the gas phase uniformly disperses in the radial or transverse directions. In heterogeneous flows, there is a radial profile of the time-averaged gas hold-up with high values in the

Fig. 8. Various flow regimes in bubble column reactors (adapted from Ref. [137]).

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center and lower ones near the wall. Such a non-uniform distribution of the time-averaged gas hold-up leads to an intensive timeaveraged liquid circulation. However, the transient flow structure is more complex than that of the time-averaged one. In particular, there are multiple dynamic circulation cells in the axial direction. These will be analysed in more details in the next sub-sections.

4.2.1. Radial distribution of time-averaged gas hold-up In the homogeneous flow regime, the radial distribution of the gas hold-up in the BCRs is flat and uniform and the profile does not change in the axial direction. However, the radial profile of gas hold-up may vary in the axial direction in the heterogeneous flow regime. In fact, the shape of the time-averaged radial gas hold-up profile stops changing above a certain height, Hcrit, in the column [149]. The zone above Hcrit is the so-called fully developed region whereas the zone below is referred to as the inlet zone. The gas distributor configuration has a large impact on the gas hold-up profile in the inlet zone and Hcrit. As shown in Fig. 9, the effect of the gas distributor diminishes above Hcrit. Chabot et al. [150] found that Hcrit is greater than 3.05✕DT, where DT is the diameter of the BCR. Guan et al. [151] demonstrated the impact of the gas distributor on Hcrit. Kumar et al. [152] underline the impact of bubble column diameter on the actual shape of the gas hold-up profile in the fully developed region. Nassos et al. [154] proposed a correlation for the radial profile of the gas hold-up:

εg ðr=RÞ ¼ ~εg

m þ 2 1  ðr=RÞm m

(21)

where r/R is the dimensionless radial position, ~εg is the chordal average gas hold-up and m is a model parameter related to the steepness of the gas hold-up profile with a larger m leading to a flatter profile. The correlation given by Eq. (21) implies that the gas hold-up reaches zero near the wall. Ueyama et al. [155] proposed a revised relation for the profile such as:

εg ðr=RÞ ¼ ~εg

m þ 2 1  cðr=RÞm m

(22)

where c is another model parameter allowing to flatten the profile as c decreases toward zero. Luo et al. [156] rewrote Eq. (22) taking into account the crosssectional average gas hold-up and obtained:

εg ðr=RÞ ¼ εg

mþ2 1  cðr=RÞm m þ 2  2c

(23)

Wu et al. [149] later correlated the model parameters c and m from experimental data. 4.2.2. Time-averaged liquid circulation In the early stage of liquid circulation investigations, the average circulating velocity or the time-averaged axial liquid velocity were investigated theoretically. Joshi et al. [157] developed a model based on multiple circulation cells within the bubble column reactor, in which the height of each cell was equal to the bubble column diameter. Based on an energy dissipation assumption, they expressed the average liquid circulation velocity as:



1=3 Vc ¼ 1:4 gDT UG  εg Ub

(24)

where Vc is the average circulation velocity, UG is the superficial gas velocity and Ub is the bubble rise velocity in stagnant liquid. This relation implies that there is no liquid circulation when UG ¼ εg Ub , which is quite questionable. García-Calvo et al. [158] derived an energy balance-based model and expressed the liquid velocity distribution (u) by:

 r m  u UL;max ¼ 1  2m=2 R

(25)

where m is a model parameter related to the location where the liquid flow direction reverses. Wu et al. [159] modified Eq. (25) on the basis of experimental data:

 0:44 u UL;max ¼ 1  2:65m0:44 cðr=RÞ2:65m c

(26)

where c and m here are model parameters fitted by Wu et al. [159].

Fig. 9. Effect of the gas distributor on the axial development of the hold-up profile: A) single point sparger; B) multipoint sparger (taken from Ref. [153] with permission).

4.2.3. Transient flow structures Transient flow in a bubble column reactor is significantly more complicated than the time-averaged one. Transient vortical structures in the large scale, which are interpreted as coherent structures by Joshi et al. [6], are prominent features of BCRs. The dynamic evolution of these coherent structures can not only enhance the heat transfer rate but also the intensity of radial mixing, which can improve, for instance, the light utilization efficiency in microalgae cultivation applications [160]. In the pioneering work of Becker et al. [161], bubble plume oscillations were observed experimentally and compared successfully to CFD simulations results, the latter of which were also used to investigate the dynamics of such coherent structures. Later, various macroscopic flow structures in a 3D bubble column were characterized using particle image velocimetry (PIV) [13]. As shown in Fig. 10, there are three different flow patterns that can be identified in this bubble column: dispersed bubble, vortical-spiral flow and turbulent flow. In the dispersed flow regime, bubbles rise upwards rectilinearly in chains. The liquid phase falls down between the bubble chains but flows upward in the vicinity of these chains. In the turbulent flow regime, large discrete bubbles can be found and the liquid phase is

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Fig. 10. Flow regimes in a 3D bubble column (taken from Ref. [13] with permission).

transported by the bubble wakes. In the vortical spiral flow regime, bubbles move in clusters and forms a central bubble stream that moves in a spiral manner. The liquid phase is carried upward by the bubble cluster in the center and flows downward in a spiral-like manner. Mudde et al. [162] measured the velocity fluctuations in a BCR with Laser Doppler Anemometry (LDA) and discovered that they were of same order as the mean velocity and that turbulence was anisotropic. Mudde et al. [163] found that the normal stresses dominated by large-scale structures are one order of magnitude higher than the shear stress in a 2D bubble column. However, these optical measurement methods such as PIV and LDA are only applicable to transparent systems such as air-water. For opaque systems, positron emission particle tracking (PEPT) and radioactive particle tracking (RPT) technologies can be used [164e166].

4.3. Bubble size Bubble size determines the mass transfer rate as well as the residence time of the gas phase. Bubble size is of significance for the design and scale-up of BCRs. In fact, both the initial values of bubble size and the mesoscale breakage and coalescence phenomena determine the evolution of bubble sizes. In the homogeneous flow regime, these bubble sizes are determined at the gas distributor. In the heterogeneous flow regime, breakage and coalescence of bubbles govern the bubble size. However, the influence of the initial bubble size and the gas distributor ceases at a distance from it which is one order of magnitude larger than the BCR diameter [167]. Describing a poly-dispersed bubble system can be challenging despite the fact that several techniques available to measure bubble size such as chemical absorption, fiber-optical probes [74,168e171], high-speed cameras [172e174], electrical conductivity device [175] or the DGD technique [176]. The volume-to-surface ratio diameter (also known as Sauter diameter), dvs or d32, and the maximum stable bubble size, dmax, are widely used as characteristic dimensions to represent a poly-dispersed bubble system. The former is closely related to mass transfer and widely used in design and scale-up of BCRs. The latter represents the diameter above which a bubble will break up. dmax itself can sometimes be used in place of d32 for estimating mass transfer rates [98]. The largest stable bubble diameter, dmax, along with a statistical model such as the lognormal distribution is another approach to estimate the bubble size distribution [177].

4.3.1. Initial bubble size The gas distributor configuration plays an important role in the initial bubble sizes. Polli et al. [167] pointed out that the hole size, the number of holes, the gas density and the gas flow rate are the main factors contributing to the initial bubble sizes. In general, finer hole sizes tend obviously to generate smaller bubbles. Porous plates tend to generate smaller bubbles than perforated ones [178,179]. The addition of holes might lead to a decrease in the flow rate through each hole and hence result in smaller bubbles [167,180]. However, a bubble might tend to coalesce with its neighbor if the distance between two adjacent holes is less than the initial bubble size. As the initial bubble size increases with the flow rate, the pressure fluctuations of the gas phase can lead to the variation of gas flow rate with time and hence non-uniform initial bubbles can be generated [181]. The installation of a gas chamber is thus of great importance for a single orifice gas distributor as it diminishes the pressure fluctuations [182]. The impact of a gas chamber decreases as the number of orifices increases and disappears as the number of orifices gets larger than 15 [183]. Moreover, poor wetting properties of the gas distributor were found to lead to larger initial bubble size [174,184]. The initial bubble size tends to decrease when the pressure or gas density are increased. Furthermore, the operating conditions such as temperature and physical properties can also impact the initial bubble size. The increase in temperature tends to decrease the viscosity of the liquid, which may lead to smaller initial bubbles. Smaller bubble sizes are also observed in smaller surface tension systems. Finally, the effect of the presence of solid particles on the initial bubbles sizes is still controversial [185]. 4.3.2. Sauter diameter Table 4 summaries the correlations of the bubble Sauter diameter in the bulk region reported by several groups in the literature. When the BCRs diameter and superficial gas velocity are small (DT < 0.15 m and UG < 0.05 m s1), the increase of the BCR diameter leads to a decrease of the bubble sizes [182]. However, when the BCR diameter is large, the increase of the BCR diameter is expected to result in a larger mean bubble size. In fact, the impact of an increase of the BCR diameter on the mean bubble size depends on the BCR diameter. If the BCR diameter is small, increasing the diameter would lead to a decrease of the bubble size, whereas the opposite is expected if the BCR diameter is large. In particular, it has been shown that the mean diameter of bubbles in a BCR of DT ¼ 5.5 m was almost twice as large as that of a BCR of DT ¼ 0.6 m (UG ¼ 0.02e0.04 m s1) [186]. Besides, increase of pressure or gas

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Table 4 Correlations for the bubble Sauter diameter in the bulk region of a BCR as reported in the literature. Authors

Configuration

Measurement Technique

Correlations

Koide et al. [113]

 Porous plate  DT ¼ 10 cm  UG < 5 cm s1

Imaging technique

Akita et al. [182]

 H ¼ 250 cm  Square cross section of BCRs:  7.7  7.7 cm2, 15  15 cm2, 30  30 cm2;  UG < 4.2 cm s1  do ¼ 0.1e0.5 cm  DT ¼ 25 cm  Air-pure liquid (deionized water, mono-ethylene glycol, n-heptane)  UG < 10 cm s1  DT ¼ 5e10 cm  do ¼ 0.087e0.309 cm  N ¼ 21-49  UG ¼ 0.14e14.01 cm s1  Air-water/glycerol/kerosene

Imaging technique

dvs ¼ 26Bo0:5 Ga0:12 Fr0:12 DT

Imaging technique (measurement location: 60 cm above the distributor)

dvs ¼ 3g 0:44 s0:34 m0:22 r0:45 r0:11 UG0:02 g l l



dvs

Wilkinson et al. [188]

Kumar et al. [190]

0:1   Fr do s 1=3 (solution of surfactants) gr We0:5  0:160  1=3 Fr do s ¼ 1:65 (pure water and organic solvents) gr We0:5

dvs ¼ 0:64

Chemical reaction method (fitting based on data from experiments)

dvs ¼ 1:56Re0:058 o 0:32Re0:425 o 100Re0:4 o

Fukuma et al. [191]

Esmaeili et al. [168]

    

DT ¼ 15 cm UG ¼ 1e10 cm s1 Slurry BCRs DT ¼ 29.2 cm H ¼ 270 cm (unaerated liquid height ¼ 110 cm)  UG ¼ 3e22 cm s1  Air-water/glucose/CMC/ xanthan gum

Dual electro-resistivity probe (measurement location: 56 cm above the distributor) Fiber optic probes

sd2o Drg sd2o Drg

sd2o Drg

!0:25 1 < Reo < 10 dvs ¼

!0:25

100 < Reo < 2100 dvs ¼

!0:25 4; 000 < Reo < 70; 000

dvs ¼ 0:59ðUb∞ ð1  εG Þn0 Þ2 =g with n0 is an adjustable parameter ’

dvs =DT ¼ 18:7Ga0:15 Fr 0:53 e0:1ð1G =G

’’

Þ

density can increase bubble breakage and lead to smaller bubble size [187e189]. Furthermore, the decrease of operating temperature may increase bubble size as reported by Chabot et al. [150].

the “small” bubbles away from the center. The difference between the bubble sizes in central region and that at the wall increases with the superficial gas velocity [199,201].

4.3.3. Bubble size distribution A unimodal distribution function of bubble diameter well-fits the overall normalized bubble number density in both homogeneous and heterogeneous flows. In the homogeneous flows, a Gaussian curve [112] or a log-normal distribution [182,192] can represent the bubble size distribution well in the gas-liquid systems. Kagumba et al. [193] found that the bubble chord length, which is directly related to the bubble size, can be represented by a log-normal distribution in both homogeneous and heterogeneous flow regimes. With addition of solid particles, the number-based bubble size distribution also follows a log-normal distribution [194,195]. A bimodal volume-based bubble size distribution is often found in the heterogeneous flow regime. Vermeer et al. [196] proposed a bimodal bubble size distribution of fast rising “large” bubbles and slow rising “small” bubbles based on the DGD technique. Krishna et al. [197] also classified the bimodal distribution as “large” and “small” bubble populations. Rabha et al. [198] measured the volume-based bubble size distribution with the advanced noninvasive ultrafast electron beam X-ray tomography technique in the slurry bubble column reactor and found the volume-based bubble size distribution is unimodal in homogeneous flows (UG < 3.4 cm s1) and bimodal in heterogeneous flows (UG > 3.4 cm s1). Bubbles sizes are also reported to vary with the spatial locations in the BCR. It is generally acknowledged that the bubble sizes near the wall are much smaller than those in the center [176,199,200]. One reason may be that the lift force pushes

4.4. Summary of the macroscale phenomena At the macroscale, the flow pattern directly determines the space-time yields and selectivity. The large-scale flow pattern is closely related to the mesoscale bubble breakage and coalescence phenomena. Weak bubble breakage or coalescence occurs in homogenous flows, which leads to a flat radial gas hold-up distribution. The value of gas hold-up is influenced by the effect of a bubble swarm on the motion of individual bubbles. In this case, the liquid circulation in the BCR is weak, resulting a weak liquid back-mixing and higher space-time yields. Weak bubble breakage and coalescence can also lead to a uniform bubble size distribution. The bubble sizes are determined by the gas distributor. Finer holes, smaller gas flow rates, and more holes tend to generate smaller bubbles. However, the increase of the number of holes on a gas distributor means a shorter distance between adjacent holes, which leads to an increase in the coalescence of gas bubbles. A flat radial gas volume fraction distribution along with a uniform bubble size distribution indicates a relatively uniform mass transfer rate in a BCR. By contrast, bubble breakage and coalescence are strong in heterogeneous flows, resulting in a profile of radial gas hold-up distribution with large bubbles in the center and small bubbles uniformly distributed. In this case, the liquid circulation is strong, leading to strong back-mixing, a uniform temperature distribution and thus, to a better selectivity. The profile of gas hold-up or axial liquid velocity does not change if the location is higher than a

S. Shu et al. / Renewable Energy 141 (2019) 613e631

critical height, in the so-called fully developed zone. In this zone, bubble breakage and coalescence reach an equilibrium and dominate the bubble size distribution. The bubble size distribution based on the number density can be characterized with a unimodal distribution function. However, the bubble size distribution based on the volume fraction in the heterogeneous flow regime is bimodal. In these cases, both the radial gas hold-up distribution and the bubble sizes in the zone below the critical height are largely influenced by the gas distributor, the so-called inlet zone. Transient flow is more complicated. When the superficial gas velocity is low, the dispersed bubbles rise in chains and carry the liquid upwards in the vicinity of the bubbles. The liquid phase flows downwards between these bubble chains. In turbulent or heterogeneous flows, large discrete bubbles can be found and the liquid phase is transported by the bubble wakes. Between them, the bubbles move in clusters and form the central bubble stream that moves in a spiral-like manner. The liquid phase is carried upward by the bubble cluster in the center and flows downward in a spirallike manner. Measurements of liquid velocity fluctuations show that the turbulence in both 2D and 3D bubble column is anisotropic. Bubble size is key to the performance of the bubble column reactor. Instead of using a bubble size distribution, the Sauter bubble size, which is closely related to mass transfer, has been widely adopted to characterize the bubble size. 5. Conclusion Bubble column reactors have lots of attractive advantages such as simple construction, good heat transfer and lower shear rate. However, the rational design, scale-up and optimization of BCRs are still challenging due to complexity of the multiscale multiphase flow in such reactors. A large body of research has been conducted to give a better understanding of the phenomena at different scales, i.e., the microscale or bubble scale, the mesoscale or bubble swarm scale, and the macroscale or reactor scale. For the moment, a large fraction of this literature has been devoted to the microscale and macroscale phenomena. However, knowledge of the bubble dynamics at the microscale cannot be used directly in models for the prediction of macroscale phenomena. On the one hand, the effect of the bubble swarm may lead to a change in bubble rising velocity and, consequently, to a change in gas hold-up at the macroscale. One the other hand, strong bubble breakage and coalescence can lead to a change in the bubble size distribution and, hence, to the flow pattern at the macroscale. These phenomena at the mesoscale make the macroscale phenomena more complex. Therefore, a better understanding of the mesoscale phenomena is key to bridge the gap in our knowledge between the microscale and macroscale. With a better knowledge of the effect of bubble swarms, the rise or slip velocity of individual bubbles could be predicted, adequately, as could the energy transferred to the liquid phase and the intensity of the circulation. Bubble breakage and coalescence at the mesoscale lead to an increase in the bubble polydispersity and a change in flow regime transition. Therefore, a breakthrough in the understanding of mesoscale is needed. Both numerical modeling methods and experimental techniques should be developed to investigate mesoscale phenomena. More stable, robust, and accurate DNS methods for gas-liquid two-phase flows, which can not only track the evolution of the front of bubbles but also deal with topological change such as bubble breakage and coalescence, need to be developed. Advanced computational techniques such as GPUbased parallel computations [202] that accelerate the computational speed of intensive DNS simulations need to be deployed to investigate mesoscale phenomena. Moreover, the rapid development of machine learning techniques might be used for instance to calibrate the interphase force models and the breakage and

627

coalescence rules inherent to such DNS simulations. Experimental measurements that can provide the three-dimensional evolution of the bubble front and capture the breakage and coalescence phenomena needs to be developed to provide a better understanding of the mesoscale phenomena as well as validation data for DNS models. With a better understanding of mesoscale phenomena, socalled scale bridging techniques to fill the gap between the different scales of a multiscale model are then needed. The Dual Bubble Size (DBS) model [127], which is an extension of the Energy Minimization Multi-Scale (EMMS) approach, is one way to achieve such scale bridging.

Acknowledgement The financial support from Total, Setec, Sytcom and the Natural Sciences and Engineering Research Council (NSERC) are gratefully acknowledged. The authors would like to thank Mr. El Mahdi
mi Demol for their advice and comments. Lakhdissi and Mr. Re

Nomenclature A Cd Cd,0 CL cr c CVM CW1 CW2 CW3 Dl DT do db dh dvs dmax € Eo Fd FL FVM FW g Hcrit J Kb Kb0 kl N n n0 m Mo r R Rb Reb tc tr Ub Ul UL

Projected area of the bubble Drag coefficient of a bubble swarm Drag coefficient of a single bubble Lift force coefficient Model parameter Model parameter Virtual mass force coefficient Model parameter Model parameter Model parameter Molecular diffusivity Bubble column reactor diameter Hole diameter of a perforated gas distributor Bubble equivalent diameter Hydraulic diameter Sauter bubble diameter Maximum bubble size € tvo €s number Eo Drag force Lift force Virtual mass force Wall lubrication force Gravity acceleration constant Critical height Dimensionless number Model parameter Model parameter Liquid-side mass transfer coefficient Model parameter Unit normal to the wall Model parameters Model parameters Morton number Radial location Radius of bubble column reactor Frontal radius of the bubble curvature Reynolds number based on bubble equivalent diameter Contact time Mean time between renewal events Bubble velocity Liquid velocity Superficial liquid velocity

628

Uslip UG UG,trans UT u v Vb Vc We Wecrit Y

S. Shu et al. / Renewable Energy 141 (2019) 613e631

Relative velocity between the bubble and surrounding liquids Superficial gas velocity Superficial gas velocity at flow regime transition point Bubble terminal velocity Liquid velocity at different radial locations Kinematic viscosity Bubble volume Liquid circulation velocity Weber number Critical Weber number Distance between bubble and wall

Abbreviations BCR Bubble Column Reactor CFD Computational Fluid Dynamics CMC Carboxyl Methyl Cellulose DBS Dual Bubble Size Model DGD Dynamic Gas Disengagement technique DNS Direct Numerical Simulation EMMS Energy-Minimization Multi-Scale ERT Electrical Resistance Tomography F-T Fischer-Tropsch PIV Particle Image Velocimetry LDA Laser Doppler Anemometry

ε εG εs

rl rg ml mref s t qm c U

[13]

[14]

[15]

[16]

[17]

[18]

[19] [20]

[21]

[22]

Greek

b

[12]

Dimensionless number Turbulent dissipation rate Gas hold-up Volume fraction of solid phase Liquid density Gas density Liquid viscosity Reference viscosity Surface tension Stress tensor Maximum angle Ratio of the cross-stream axis to the parallel axis Local shear rate

References [1] W.D. Deckwer, R.W. Field, Bubble Column Reactors, Wiley, New York, 1992. [2] P. Rollbusch, M. Bothe, M. Becker, M. Ludwig, M. Grunewald, M. Schluter, R. Franke, Bubble columns operated under industrially relevant conditions current understanding of design parameters, Chem. Eng. Sci. 126 (2015) 660e678. https://doi.org/10.1016/j.ces.2014.11.061. [3] P. Zehner, M. Kraume, Bubble Columns, Ullmann’s Encyclopedia of Industrial Chemistry, 2000. [4] S. Lier, J. Riese, G. Cvetanoska, A.K. Lesniak, S. Muller, S. Paul, L. Sengen, M. Grunewald, Innovative scaling strategies for a fast development of apparatuses by modular process engineering, Chem. Eng. Process 123 (2018) 111e125. https://doi.org/10.1016/j.cep.2017.10.026. [5] W.D. Deckwer, On the mechanism of heat transfer in bubble column reactors, Chem. Eng. Sci. 35 (6) (1980) 1341e1346. https://doi.org/10.1016/00092509(80)85127-X. [6] J.B. Joshi, V.S. Vitankar, A.A. Kulkarni, M.T. Dhotre, K. Ekambara, Coherent flow structures in bubble column reactors, Chem. Eng. Sci. 57 (16) (2002) 3157e3183. https://doi.org/10.1016/S0009-2509(02)00192-6. [7] C. Posten, Design principles of photo-bioreactors for cultivation of microalgae, Eng. Life Sci. 9 (3) (2009) 165e177. https://doi.org/10.1002/elsc. 200900003. [8] J.C. Merchuk, Shear Effects on Suspended Cells, Bioreactor Systems and Effects, Springer Berlin Heidelberg, Berlin, Heidelberg, 1991, pp. 65e95. [9] N. Deen, R. Mudde, J. Kuipers, P. Zehner, M. Kraume, Bubble Columns, Ullmann’s Encyclopedia of Industrial Chemistry, 2010. [10] M. Bothe, Experimental Analysis and Modeling of Industrial Two-phase Flows in Bubble Column Reactors, Institute of Multiphase Flows, TU €ttingen, 2016, p. 148. Hamburg-Harburg, Go [11] J. Degen, A. Uebele, A. Retze, U. Schmid-Staiger, W. Trosch, A novel airlift

[23] [24]

[25] [26] [27]

[28]

[29]

[30] [31]

[32]

[33]

[34]

[35] [36] [37]

[38]

[39] [40]

photobioreactor with baffles for improved light utilization through the flashing light effect, J. Biotechnol. 92 (2) (2001) 89e94. https://doi.org/10. 1016/S0168-1656(01)00350-9. J. Huang, Y. Li, M. Wan, Y. Yan, F. Feng, X. Qu, J. Wang, G. Shen, W. Li, J. Fan, W. Wang, Novel flat-plate photobioreactors for microalgae cultivation with special mixers to promote mixing along the light gradient, Bioresour. Technol. 159 (2014) 8e16. https://doi.org/10.1016/j.biortech.2014.01.134. R.C. Chen, J. Reese, L.S. Fan, Flow structure in a three-dimensional bubble column and three-phase fluidized bed, AIChE J. 40 (7) (1994) 1093e1104. https://doi.org/10.1002/aic.690400702. A.A. Kendoush, K.W. Gaines, C.W. White, Theory and indirect measurements of the drag force acting on a rising ellipsoidal bubble, J. Fluid Flow Heat Mass Transf. 3 (1) (2016) 92e98. D. Qian, J.B. McLaughlin, K. Sankaranarayanan, S. Sundaresan, K. Kontomaris, Simulation of bubble breakup dynamics in homogeneous turbulence, Chem. Eng. Commun. 193 (8) (2006) 1038e1063. https://doi.org/10.1080/ 00986440500354275. I. Roghair, M.W. Baltussen, M.V. Annaland, J.A.M. Kuipers, Direct numerical simulations of the drag force of bi-disperse bubble swarms, Chem. Eng. Sci. 95 (2013) 48e53. https://doi.org/10.1016/j.ces.2013.03.027. E. Loth, Quasi-steady shape and drag of deformable bubbles and drops, Int. J. Multiph. Flow 34 (6) (2008) 523e546. https://doi.org/10.1016/j. ijmultiphaseflow.2007.08.010. M.K. Tripathi, K.C. Sahu, R. Govindarajan, Why a falling drop does not in general behave like a rising bubble, Sci. Rep. 4 (2014) 4771. https://doi.org/ 10.1038/srep04771. R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops, and Particles, Academic Press, 1978. D.W. Moore, The rise velocity of distorted gas bubbles in a liquid of small viscosity, J. Fluid Mech. 23 (1965) 749. https://doi.org/10.1017/ S0022112065001660. R.M. Davies, G. Taylor, The mechanics of large bubbles rising through extended liquids and through liquids in tubes, Proc. R. Soc. A 200 (1062) (1950) 375. https://doi.org/10.1098/rspa.1950.0023. S. Takagi, A. Prosperetti, Y. Matsumoto, Drag coefficient of a gas bubble in an axisymmetric shear flow, Phys. Fluids 6 (9) (1994) 3186e3188. https://doi. org/10.1063/1.868097. F.H. Garner, D. Hammerton, Circulation inside gas bubbles, Chem. Eng. Sci. 3 (1) (1954) 1e11. https://doi.org/10.1016/0009-2509(54)80001-7. K. Hayashi, A. Tomiyama, Effects of surfactant on lift coefficients of bubbles in linear shear flows, Int. J. Multiph. Flow 99 (2018) 86e93. https://doi.org/10. 1016/j.ijmultiphaseflow.2017.10.003. M.J.M. Hill, On a spherical vortex, Philos. Trans. R. Soc. A 185 (1894) 213e245. https://doi.org/10.1098/rsta.1894.0006. L.S. Fan, K. Tsuchiya, Bubble Wake Dynamics in Liquids and Liquid-Solid Suspensions, Butterworth-Heinemann, 1990. €ulen, Int. J. Heat Mass W. Kast, Analyse des w€ armeübergangs in blasensa Transf. 5 (3e4) (1962) 329e336. https://doi.org/10.1016/0017-9310(62) 90022-4. C. Brücker, Structure and dynamics of the wake of bubbles and its relevance for bubble interaction, Phys. Fluids 11 (7) (1999) 1781e1796. https://doi.org/ 10.1063/1.870043. D. Legendre, J. Magnaudet, The lift force on a spherical bubble in a viscous linear shear flow, J. Fluid Mech. 368 (1998) 81e126. https://doi.org/10.1017/ S0022112098001621. A. Serizawa, I. Kataoka, Dispersed flow - I, Multiphas. Sci. Technol. 8 (1e4) (1994) 125e194. https://doi.org/10.1615/MultScienTechn.v8.i1-4.40. A. Tomiyama, H. Tamai, I. Zun, S. Hosokawa, Transverse migration of single bubbles in simple shear flows, Chem. Eng. Sci. 57 (11) (2002) 1849e1858. https://doi.org/10.1016/s0009-2509(02)00085-4. J. Magnaudet, I. Eames, The motion of high-Reynolds-number bubbles in inhomogeneous flows, Annu. Rev. Fluid Mech. 32 (1) (2000) 659e708. https://doi.org/10.1146/annurev.fluid.32.1.659. T. Sanada, M. Watanabe, T. Fukano, A. Kariyasaki, Behavior of a single coherent gas bubble chain and surrounding liquid jet flow structure, Chem. Eng. Sci. 60 (17) (2005) 4886e4900. https://doi.org/10.1016/j.ces.2005.04. 010. S.L. Shu, N. Yang, Direct numerical simulation of bubble dynamics using phase-field model and lattice Boltzmann method, Ind. Eng. Chem. Res. 52 (33) (2013) 11391e11403. https://doi.org/10.1021/Ie303486y. G. Mougin, J. Magnaudet, Path instability of a rising bubble, Phys. Rev. Lett. 88 (1) (2001), 014502. https://doi.org/10.1103/PhysRevLett.88.014502. P.G. Saffman, The lift on a small sphere in a slow shear flow, J. Fluid Mech. 22 (2) (2006) 385e400. https://doi.org/10.1017/S0022112065000824. W. Dijkhuizen, M. van Sint Annaland, J.A.M. Kuipers, Numerical and experimental investigation of the lift force on single bubbles, Chem. Eng. Sci. 65 (3) (2010) 1274e1287. https://doi.org/10.1016/j.ces.2009.09.084. J.R. Grace, T. Wairegi, T.H. Nguyen, Shapes and velocities of single drops and bubbles moving freely through immiscible liquids, Trans. Inst. Chem. Eng. 54 (3) (1976) 167e173. L. Schiller, A. Naumann, A drag coefficient correlation, VDI Zeitung 77 (318) (1935) 51. A. Tomiyama, Struggle with computational bubble dynamics, Multiphas. Sci. Technol. 10 (4) (1998) 369e405. https://doi.org/10.1615/MultScienTechn. v10.i4.40.

S. Shu et al. / Renewable Energy 141 (2019) 613e631 [41] M. Ishii, N. Zuber, Drag coefficient and relative velocity in bubbly, droplet or particulate flows, AIChE J. 25 (5) (1979) 843e855. https://doi.org/10.1002/ aic.690250513. [42] R. Torvik, H.F. Svendsen, Modeling of slurry reactors - a fundamental approach, Chem. Eng. Sci. 45 (8) (1990) 2325e2332. https://doi.org/10.1016/ 0009-2509(90)80112-R. [43] A. Sokolichin, G. Eigenberger, Gas-liquid flow in bubble columns and loop reactors: part I. detailed modelling and numerical simulation, Chem. Eng. Sci. 49 (24b) (1994) 5735e5746. https://doi.org/10.1016/0009-2509(94)002894. [44] M.V. Tabib, S.A. Roy, J.B. Joshi, CFD simulation of bubble column - an analysis of interphase forces and turbulence models, Chem. Eng. J. 139 (3) (2008) 589e614. https://doi.org/10.1016/j.cej.2007.09.015. [45] C. Laborde-Boutet, F. Larachi, N. Dromard, O. Delsart, D. Schweich, CFD simulation of bubble column flows: investigations on turbulence models in RANS approach, Chem. Eng. Sci. 64 (21) (2009) 4399e4413. https://doi.org/ 10.1016/j.ces.2009.07.009. [46] H.P. Luo, M.H. Al-Dahhan, Verification and validation of CFD simulations for local flow dynamics in a draft tube airlift bioreactor, Chem. Eng. Sci. 66 (5) (2011) 907e923. https://doi.org/10.1016/j.ces.2010.11.038. [47] D. Zhang, N.G. Deen, J.A.M. Kuipers, Numerical simulation of the dynamic flow behavior in a bubble column: a study of closures for turbulence and interface forces, Chem. Eng. Sci. 61 (23) (2006) 7593e7608. https://doi.org/ 10.1016/j.ces.2006.08.053. [48] X.D. Jiang, N. Yang, B.L. Yang, Computational fluid dynamics simulation of hydrodynamics in the riser of an external loop airlift reactor, Particuology 27 (2016) 95e101. https://doi.org/10.1016/j.partic.2015.05.011. [49] R.M.A. Masood, A. Delgado, Numerical investigation of the interphase forces and turbulence closure in 3D square bubble columns, Chem. Eng. Sci. 108 (2014) 154e168. https://doi.org/10.1016/j.ces.2014.01.004. [50] S. Shu, N.E. Bahraoui, F. Bertrand, J. Chaouki, A bubble-induced turbulence model for gas-liquid flows, to be submitted to, Chem. Eng. Sci. (2019). [51] T.R. Auton, The lift force on a spherical body in a rotational flow, J. Fluid Mech. 183 (1987) 199e218. https://doi.org/10.1017/S002211208700260X. [52] D.A. Drew, R.T. Lahey, The virtual mass and lift force on a sphere in rotating and straining inviscid flow, Int. J. Multiph. Flow 13 (1) (1987) 113e121. https://doi.org/10.1016/0301-9322(87)90011-5. [53] I. Ẑun, The transverse migration of bubbles influenced by walls in vertical bubbly flow, Int. J. Multiph. Flow 6 (6) (1980) 583e588. https://doi.org/10. 1016/0301-9322(80)90053-1. [54] M. Lance, M.L. de Bertodano, Phase distribution phenomena and wall effects in bubbly two-phase flows, Multiphas. Sci. Technol. 8 (1e4) (1994) 69e123. https://doi.org/10.1615/MultScienTechn.v8.i1-4.30. [55] M. Rastello, J.L. Marie, M. Lance, Drag and lift forces on clean spherical and ellipsoidal bubbles in a solid-body rotating flow, J. Fluid Mech. 682 (2011) 434e459. https://doi.org/10.1017/jfm.2011.240. [56] M. Rastello, J.L. Marie, N. Grosjean, M. Lance, Drag and lift forces on interfacecontaminated bubbles spinning in a rotating flow, J. Fluid Mech. 624 (2009) 159e178. https://doi.org/10.1017/S0022112008005399. [57] P. Ern, F. Risso, D. Fabre, J. Magnaudet, Wake-induced oscillatory paths of bodies freely rising or falling in fluids, Annu. Rev. Fluid Mech. 44 (1) (2012) 97e121. https://doi.org/10.1146/annurev-fluid-120710-101250. [58] R.M. Wellek, A.K. Agrawal, A.H.P. Skelland, Shape of liquid drops moving in liquid media, AIChE J. 12 (5) (1966) 854e862. https://doi.org/10.1002/aic. 690120506. [59] H.M. Prasser, M. Beyer, H. Carl, S. Gregor, D. Lucas, H. Pietruske, P. Schütz, F.P. Weiss, Evolution of the structure of a gaseliquid two-phase flow in a large vertical pipe, Nucl. Eng. Des. 237 (15) (2007) 1848e1861. https://doi. org/10.1016/j.nucengdes.2007.02.018. [60] M. Elena Díaz, F.J. Montes, M.A. Gal
an, Influence of the lift force closures on the numerical simulation of bubble plumes in a rectangular bubble column, Chem. Eng. Sci. 64 (5) (2009) 930e944. https://doi.org/10.1016/j.ces.2008.10. 055. [61] H.F. Svendsen, H.A. Jakobsen, R. Torvik, Local flow structures in internal loop and bubble column reactors, Chem. Eng. Sci. 47 (13) (1992) 3297e3304. https://doi.org/10.1016/0009-2509(92)85038-D. [62] J. Xue, F. Chen, N. Yang, W. Ge, EulerianeLagrangian simulation of bubble coalescence in bubbly flow using the spring-dashpot model, Chin. J. Chem. Eng. 25 (3) (2017) 249e256. https://doi.org/10.1016/j.cjche.2016.08.006. [63] J. Xue, F. Chen, N. Yang, W. Ge, A study of the soft-sphere model in eulerianLagrangian simulation of gas-liquid flow, Int. J. Chem. React. Eng. 15 (1) (2017). https://doi.org/10.1515/ijcre-2016-0064. [64] R. Rzehak, T. Ziegenhein, S. Kriebitzsch, E. Krepper, D. Lucas, Unified modeling of bubbly flows in pipes, bubble columns, and airlift columns, Chem. Eng. Sci. 157 (2017) 147e158. https://doi.org/10.1016/j.ces.2016.04. 056. [65] A.A. Kulkarni, Lift force on bubbles in a bubble column reactor: experimental analysis, Chem. Eng. Sci. 63 (6) (2008) 1710e1723. https://doi.org/10.1016/j. ces.2007.10.029. [66] S.P. Antal, R.T. Lahey, J.E. Flaherty, Analysis of phase distribution in fully developed laminar bubbly two-phase flow, Int. J. Multiph. Flow 17 (5) (1991) 635e652. https://doi.org/10.1016/0301-9322(91)90029-3. € tvo €s [67] A. Tomiyama, A. Sou, I. Zun, N. Kanami, T. Sakaguchi, Effects of Eo number and dimensionless liquid volumetric flux on lateral motion of a bubble in a laminar duct flow A2 - serizawa, Akimi, in: T. Fukano, J. Bataille

629

(Eds.), Multiphase Flow 1995, Elsevier, Amsterdam, 1995, pp. 3e15. [68] D. Drew, L. Cheng, R.T. Lahey, The analysis of virtual mass effects in twophase flow, Int. J. Multiph. Flow 5 (4) (1979) 233e242. https://doi.org/10. 1016/0301-9322(79)90023-5. [69] L.M. Milne-Thomson, Theoretical Hydrodynamics, 5-th ed., The Mcmillan Company, New York, 1968. [70] T.R. Auton, Dynamics of Bubbles, Drops, and Particles in Motion in Liquids, University of Cambridge, 1984. [71] N.G. Deen, T. Solberg, B.H. Hjertager, Large eddy simulation of the gaseliquid flow in a square cross-sectioned bubble column, Chem. Eng. Sci. 56 (21e22) (2001) 6341e6349. https://doi.org/10.1016/S0009-2509(01)00249-4. [72] J.R. Grace, T. Wairegi, J. Brophy, Break-up of drops and bubbles in stagnant media, Can. J. Chem. Eng. 56 (1) (1978) 3e8. https://doi.org/10.1002/cjce. 5450560101. [73] J. Kitscha, G. Kocamustafaogullari, Breakup criteria for fluid particles, Int. J. Multiph. Flow 15 (4) (1989) 573e588. https://doi.org/10.1016/03019322(89)90054-2. [74] X.K. Luo, D.J. Lee, R. Lau, G.Q. Yang, L.S. Fan, Maximum stable bubble size and gas holdup in high-pressure slurry bubble columns, AIChE J. 45 (4) (1999) 665e680. https://doi.org/10.1002/aic.690450402. [75] W.K. Lewis, W.G. Whitman, Principles of gas absorption, Ind. Eng. Chem. Res. 16 (12) (1924) 1215e1220. https://doi.org/10.1021/ie50180a002. [76] R. Higbie, The rate of absorption of a pure gas into a still liquid during short periods of exposure, Trans. AIChE 31 (1935) 365e389. [77] P.V. Danckwerts, Significance of liquid-film coefficients in gas absorption, Ind. Eng. Chem. 43 (6) (1951) 1460e1467. https://doi.org/10.1021/ ie50498a055. [78] S.S. Alves, J.M.T. Vasconcelos, S.P. Orvalho, Mass transfer to clean bubbles at low turbulent energy dissipation, Chem. Eng. Sci. 61 (4) (2006) 1334e1337. https://doi.org/10.1016/j.ces.2005.08.001. € ssling, Evaporation of falling drops (Über die verdünstung fallenden [79] N. Fro tropfen), Gerlands Beitage Geophys. 52 (1938) 170e216. [80] J.F. Richardson, W.N. Zaki, Sedimentation and fluidisation: Part I, Trans. Inst. Chem. Eng. 32 (1954) 35e53. [81] D. Colombet, D. Legendre, F. Risso, A. Cockx, P. Guiraud, Dynamics and mass transfer of rising bubbles in a homogenous swarm at large gas volume fraction, J. Fluid Mech. 763 (2014) 254e285. https://doi.org/10.1017/jfm. 2014.672.
, Measurement of local flow characteristics in [82] C. Garnier, M. Lance, J.L. Marie buoyancy-driven bubbly flow at high void fraction, Exp. Therm. Fluid Sci. 26 (6) (2002) 811e815. https://doi.org/10.1016/S0894-1777(02)00198-X. [83] I. Roghair, Y.M. Lau, N.G. Deen, H.M. Slagter, M.W. Baltussen, M.V. Annaland, J.A.M. Kuipers, On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers, Chem. Eng. Sci. 66 (14) (2011) 3204e3211. https://doi.org/10.1016/j.ces.2011.02.030. [84] M. Simonnet, C. Gentric, E. Olmos, N. Midoux, Experimental determination of the drag coefficient in a swarm of bubbles, Chem. Eng. Sci. 62 (3) (2007) 858e866. https://doi.org/10.1016/j.ces.2006.10.012. [85] Y.F. Liu, O. Hinrichsen, Study on CFDePBM turbulence closures based on keε and Reynolds stress models for heterogeneous bubble column flows, Comput. Fluids 105 (2014) 91e100. https://doi.org/10.1016/j.compfluid.2014.09. 023. [86] H. Rusche, R.I. Issa, The Effect of Voidage on the Drag Force on Particles, Droplets and Bubbles in Dispersed Two-phase Flow, Japanese European TwoPhase Flow Meeting, Tshkuba, Japan, 2000. [87] G.Y. Yang, H.H. Zhang, J.J. Luo, T.F. Wang, Drag force of bubble swarms and numerical simulations of a bubble column with a CFD-PBM coupled model, Chem. Eng. Sci. 192 (2018) 714e724. https://doi.org/10.1016/j.ces.2018.07. 012. [88] R. Demol, D. Vidal, S. Shu, F. Bertrand, J. Chaouki, Mass transfer in the homogeneous flow regime of a bubble column, Submitted to Chem. Eng. Process. (2019). [89] A. Behzadi, R.I. Issa, H. Rusche, Modelling of dispersed bubble and droplet flow at high phase fractions, Chem. Eng. Sci. 59 (4) (2004) 759e770. https:// doi.org/10.1016/j.ces.2003.11.018. [90] G.B. Wallis, Some hydrodynamic aspects of two-phase flow and boiling, in: Int. Heat Transfer Conference, Boulder, Colorado USA, 1961, pp. 319e325. [91] G. Marrucci, Communication. Rising velocity of swarm of spherical bubbles, Ind. Eng. Chem. Fundam. 4 (2) (1965) 224e225. [92] G. Riboux, F. Risso, D. Legendre, Experimental characterization of the agitation generated by bubbles rising at high Reynolds number, J. Fluid Mech. 643 (2009) 509e539. https://doi.org/10.1017/S0022112009992084. [93] R. Krishna, M.I. Urseanu, J.M. van Baten, J. Ellenberger, Rise velocity of a swarm of large gas bubbles in liquids, Chem. Eng. Sci. 54 (2) (1999) 171e183. https://doi.org/10.1016/S0009-2509(98)00245-0. [94] Y.X. Liao, D. Lucas, A literature review of theoretical models for drop and bubble breakup in turbulent dispersions, Chem. Eng. Sci. 64 (15) (2009) 3389e3406. https://doi.org/10.1016/j.ces.2009.04.026. [95] A.N. Kolmogorov, On the disintegration of drops in a turbulent flow, Akad. Doklady, Nauk (USSR) 66 (1949). [96] J.O. Hinze, Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes, AIChE J. 1 (3) (1955) 289e295. https://doi.org/10.1002/ aic.690010303. [97] M.A. Delichatsios, Model for the breakup rate of spherical drops in isotropic turbulent flows, Phys. Fluids 18 (6) (1975) 622e623. https://doi.org/10.1063/

630

S. Shu et al. / Renewable Energy 141 (2019) 613e631

1.861201. [98] C.A. Sleicher, Maximum stable drop size in turbulent flow, AIChE J. 8 (4) (1962) 471e477. https://doi.org/10.1002/aic.690080410. [99] F.B. Sprow, Distribution of drop sizes produced in turbulent liquiddliquid dispersion, Chem. Eng. Sci. 22 (3) (1967) 435e442. https://doi.org/10.1016/ 0009-2509(67)80130-1. [100] M. Sevik, S.H. Park, The splitting of drops and bubbles by turbulent fluid flow, J. Fluids Eng. 95 (1) (1973) 53e60. https://doi.org/10.1115/1.3446958. [101] F. Risso, J. Fabre, Oscillations and breakup of a bubble immersed in a turbulent field, J. Fluid Mech. 372 (1998) 323e355. https://doi.org/10.1017/ s0022112098002705. [102] F. Lehr, D. Mewes, A transport equation for the interfacial area density applied to bubble columns, Chem. Eng. Sci. 56 (3) (2001) 1159e1166. https://doi.org/10.1016/S0009-2509(00)00335-3. [103] H. Zhao, W. Ge, A theoretical bubble breakup model for slurry beds or threephase fluidized beds under high pressure, Chem. Eng. Sci. 62 (1e2) (2007) 109e115. https://doi.org/10.1016/j.ces.2006.08.008. [104] C.A. Coulaloglou, L.L. Tavlarides, Description of interaction processes in agitated liquid-liquid dispersions, Chem. Eng. Sci. 32 (11) (1977) 1289e1297. https://doi.org/10.1016/0009-2509(77)85023-9. [105] H. Luo, H.F. Svendsen, Theoretical model for drop and bubble breakup in turbulent dispersions, AIChE J. 42 (5) (1996) 1225e1233. https://doi.org/10. 1002/aic.690420505. [106] T.F. Wang, J.F. Wang, Y. Jin, A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow, Chem. Eng. Sci. 58 (20) (2003) 4629e4637. https://doi.org/10.1016/j.ces.2003.07.009. [107] G. Narsimhan, J.P. Gupta, D. Ramkrishna, A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersions, Chem. Eng. Sci. 34 (2) (1979) 257e265. https://doi.org/10.1016/0009-2509(79)87013-X. [108] Y.X. Liao, D. Lucas, A literature review on mechanisms and models for the coalescence process of fluid particles, Chem. Eng. Sci. 65 (10) (2010) 2851e2864. https://doi.org/10.1016/j.ces.2010.02.020. [109] H. Sovova, Breakage and coalescence of drops in a batch stirred vesseldII comparison of model and experiments, Chem. Eng. Sci. 36 (9) (1981) 1567e1573. https://doi.org/10.1016/0009-2509(81)85117-2. [110] P.K. Das, R. Kumar, D. Ramkrishna, Coalescence of drops in stirred dispersion. A white noise model for coalescence, Chem. Eng. Sci. 42 (2) (1987) 213e220. https://doi.org/10.1016/0009-2509(87)85051-0. [111] Y.T. Shah, S. Joseph, D.N. Smith, J.A. Ruether, On the behavior of the gas phase in a bubble column with ethanol-water mixtures, Ind. Eng. Chem. Process Des. Dev. 24 (4) (1985) 1140e1148. https://doi.org/10.1021/i200031a041. [112] G. Marrucci, L. Nicodemo, Coalescence of gas bubbles in aqueous solutions of inorganic electrolytes, Chem. Eng. Sci. 22 (9) (1967) 1257e1265. https://doi. org/10.1016/0009-2509(67)80190-8. [113] K. Koide, S. Kato, Y. Tanaka, H. Kubota, Bubbles generated from porous plate, J. Chem. Eng. Jpn. 1 (1) (1968) 51e56. https://doi.org/10.1252/jcej.1.51. [114] D. Ramkrishna, M.R. Singh, Population balance modeling: current status and future prospects, Annu. Rev. Chem. Biomol. Eng. 5 (1) (2014) 123e146. https://doi.org/10.1146/annurev-chembioeng-060713-040241. [115] A. Falola, A. Borissova, X.Z. Wang, Extended method of moment for general population balance models including size dependent growth rate, aggregation and breakage kernels, Comput. Chem. Eng. 56 (2013) 1e11. https://doi. org/10.1016/j.compchemeng.2013.04.017. [116] S. Kumar, D. Ramkrishna, On the solution of population balance equations by discretizationdI. A fixed pivot technique, Chem. Eng. Sci. 51 (8) (1996) 1311e1332. https://doi.org/10.1016/0009-2509(96)88489-2. [117] Y.L. Lin, K. Lee, T. Matsoukas, Solution of the population balance equation using constant-number Monte Carlo, Chem. Eng. Sci. 57 (12) (2002) 2241e2252. https://doi.org/10.1016/S0009-2509(02)00114-8. [118] D.L. Marchisio, R.O. Fox, Solution of population balance equations using the direct quadrature method of moments, J. Aerosol Sci. 36 (1) (2005) 43e73. https://doi.org/10.1016/j.jaerosci.2004.07.009. [119] R. McGraw, Description of aerosol dynamics by the quadrature method of moments, Aerosol Sci. Technol. 27 (2) (1997) 255e265. https://doi.org/10. 1080/02786829708965471. [120] A. Passalacqua, F. Laurent, E. Madadi-Kandjani, J.C. Heylmun, R.O. Fox, An open-source quadrature-based population balance solver for OpenFOAM, Chem. Eng. Sci. 176 (2018) 306e318. https://doi.org/10.1016/j.ces.2017.10. 043. [121] K. Hutton, N. Mitchell, P.J. Frawley, Particle size distribution reconstruction: the moment surface method, Powder Technol. 222 (2012) 8e14. https://doi. org/10.1016/j.powtec.2012.01.029. [122] R.B. Li, S.B. Kuang, T.A. Zhang, Y. Liu, A.B. Yu, Numerical investigation of gaseliquid flow in a newly developed carbonation reactor, Ind. Eng. Chem. Res. 57 (1) (2017) 380e391. https://doi.org/10.1021/acs.iecr.7b04026. [123] P. Chen, J. Sanyal, M.P. Dudukovic, CFD modeling of bubble columns flows: implementation of population balance, Chem. Eng. Sci. 59 (22e23) (2004) 5201e5207. https://doi.org/10.1016/j.ces.2004.07.037. [124] A. Buffo, D.L. Marchisio, M. Vanni, P. Renze, Simulation of polydisperse multiphase systems using population balances and example application to bubbly flows, Chem. Eng. Res. Des. 91 (10) (2013) 1859e1875. https://doi. org/10.1016/j.cherd.2013.06.021. [125] K.Y. Guo, T.F. Wang, Y.F. Liu, J.F. Wang, CFD-PBM simulations of a bubble column with different liquid properties, Chem. Eng. J. 329 (2017) 116e127. https://doi.org/10.1016/j.cej.2017.04.071.

[126] T.F. Wang, Simulation of bubble column reactors using CFD coupled with a population balance model, Front. Chem. Sci. Eng. 5 (2) (2010) 162e172. https://doi.org/10.1007/s11705-009-0267-5. [127] N. Yang, J.H. Chen, H. Zhao, W. Ge, J.H. Li, Explorations on the multi-scale flow structure and stability condition in bubble columns, Chem. Eng. Sci. 62 (24) (2007) 6978e6991. https://doi.org/10.1016/j.ces.2007.08.034. [128] J. Li, Multi-scale Modelling and Method of Energy Minimization for ParticleFluid Two Phase Flow, Institute of Chemical Metallurgy, Chinese Academy of Sciences, Beijing, 1987. [129] J.H. Li, C.L. Cheng, Z.D. Zhang, J. Yuan, A. Nemet, F.N. Fett, The EMMS model its application, development and updated concepts, Chem. Eng. Sci. 54 (22) (1999) 5409e5425. https://doi.org/10.1016/S0009-2509(99)00274-2. [130] W. Ge, J.H. Li, Physical mapping of fluidization regimes - the EMMS approach, Chem. Eng. Sci. 57 (18) (2002) 3993e4004. https://doi.org/10.1016/S00092509(02)00234-8. [131] J. Li, M. Kwauk, Particle-fluid Two-phase FlowdThe Energy-Minimization Multi-Scale Method, Metallurgical Industry Press, Beijing, 1994. [132] C. Han, X.P. Guan, N. Yang, Structure evolution and demarcation of small and large bubbles in bubble columns, Ind. Eng. Chem. Res. 57 (25) (2018) 8529e8540. https://doi.org/10.1021/acs.iecr.8b00703. [133] N. Yang, J.H. Chen, W. Ge, J.H. Li, A conceptual model for analyzing the stability condition and regime transition in bubble columns, Chem. Eng. Sci. 65 (1) (2010) 517e526. https://doi.org/10.1016/j.ces.2009.06.014. [134] N. Yang, Z.Y. Wu, J.H. Chen, Y.H. Wang, J.H. Li, Multi-scale analysis of gasliquid interaction and CFD simulation of gas-liquid flow in bubble columns, Chem. Eng. Sci. 66 (14) (2011) 3212e3222. https://doi.org/10.1016/j. ces.2011.02.029. [135] Q. Xiao, J. Wang, N. Yang, J.H. Li, Simulation of the multiphase flow in bubble columns with stability-constrained multi-fluid CFD models, Chem. Eng. J. 329 (2017) 88e99. https://doi.org/10.1016/j.cej.2017.06.008. [136] N. Yang, Q. Xiao, A mesoscale approach for population balance modeling of bubble size distribution in bubble column reactors, Chem. Eng. Sci. 170 (2017) 241e250. https://doi.org/10.1016/j.ces.2017.01.026. [137] A. Shaikh, M.H. Al-Dahhan, A review on flow regime transition in bubble columns, Int. J. Chem. React. Eng. 5 (1) (2007). https://doi.org/10.2202/15426580.1368. [138] R. Krishna, P.M. Wilkinson, L.L. Van Dierendonck, A model for gas holdup in bubble columns incorporating the influence of gas density on flow regime transitions, Chem. Eng. Sci. 46 (10) (1991) 2491e2496. https://doi.org/10. 1016/0009-2509(91)80042-W. [139] J. Zahradnik, M. Fialova, M. Ruzicka, J. Drahos, F. Kastanek, N.H. Thomas, Duality of the gas-liquid flow regimes in bubble column reactors, Chem. Eng. Sci. 52 (21e22) (1997) 3811e3826. https://doi.org/10.1016/S0009-2509(97) 00226-1. [140] M.C. Ruzicka, J. Drahos, P.C. Mena, J.A. Teixeira, Effect of viscosity on homogeneous-heterogeneous flow regime transition in bubble columns, Chem. Eng. J. 96 (1e3) (2003) 15e22. https://doi.org/10.1016/j.cej.2003.08. 009. [141] H.C.J. Hoefsloot, R. Krishna, Influence of gas-density on the stability of homogeneous flow in bubble-columns, Ind. Eng. Chem. Res. 32 (4) (1993) 747e750. https://doi.org/10.1021/ie00016a024. [142] H.M. Letzel, J.C. Schouten, R. Krishna, C.M. van den Bleek, Gas holdup and mass transfer in bubble column reactors operated at elevated pressure, Chem. Eng. Sci. 54 (13e14) (1999) 2237e2246. https://doi.org/10.1016/ S0009-2509(98)00418-7. [143] H.M. Letzel, J.C. Schouten, C.M. vandenBleek, R. Krishna, Influence of elevated pressure on the stability of bubbly flows, Chem. Eng. Sci. 52 (21e22) (1997) 3733e3739. https://doi.org/10.1016/S0009-2509(97)00219-4. [144] C.P. Ribeiro Jr., D. Mewes, The influence of electrolytes on gas hold-up and regime transition in bubble columns, Chem. Eng. Sci. 62 (17) (2007) 4501e4509. https://doi.org/10.1016/j.ces.2007.05.032. [145] B.N. Thorat, J.B. Joshi, Regime transition in bubble columns: experimental and predictions, Exp. Therm. Fluid Sci. 28 (5) (2004) 423e430. https://doi. org/10.1016/j.expthermflusci.2003.06.002. [146] G.S. Grover, C.V. Rode, R.V. Chaudhari, Effect of temperature on flow regimes and gas hold-up in a bubble column, Can. J. Chem. Eng. 64 (3) (1986) 501e504. https://doi.org/10.1002/cjce.5450640321. [147] T.J. Lin, K. Tsuchiya, L.S. Fan, On the measurements of regime transition in high-pressure bubble columns, Can. J. Chem. Eng. 77 (2) (1999) 370e374. https://doi.org/10.1002/cjce.5450770224. [148] P.M. Wilkinson, A.P. Spek, L.L. Van Dierendonck, Design parameters estimation for scale-up of high pressure bubble columns, AIChE J. 38 (1992) 544. https://doi.org/10.1002/aic.690380408. [149] Y.X. Wu, B.C. Ong, M.H. Al-Dahhan, Predictions of radial gas holdup profiles in bubble column reactors, Chem. Eng. Sci. 56 (3) (2001) 1207e1210. https:// doi.org/10.1016/S0009-2509(00)00341-9. [150] J. Chabot, H.I. Delasa, Gas holdups and bubble characteristics in a bubblecolumn operated at high-temperature, Ind. Eng. Chem. Res. 32 (11) (1993) 2595e2601. https://doi.org/10.1021/ie00023a023. [151] X.P. Guan, N. Yang, Z.Q. Li, L.J. Wang, Y.W. Cheng, X. Li, Experimental investigation of flow development in large-scale bubble columns in the churn-turbulent regime, Ind. Eng. Chem. Res. 55 (11) (2016) 3125e3130. https://doi.org/10.1021/acs.iecr.5b04015. [152] S.B. Kumar, D. Moslemian, M.P. Dudukovic, Gas-holdup measurements in bubble columns using computed tomography, AIChE J. 43 (6) (1997)

S. Shu et al. / Renewable Energy 141 (2019) 613e631 1414e1425. https://doi.org/10.1002/aic.690430605. [153] J.B. Joshi, Computational flow modelling and design of bubble column reactors, Chem. Eng. Sci. 56 (21) (2001) 5893e5933. https://doi.org/10.1016/ S0009-2509(01)00273-1. [154] G.P. Nassos, S.G. Bankoff, Slip velocity ratios in an air-water system under steady-state and transient conditions, Chem. Eng. Sci. 22 (4) (1967) 661e668. https://doi.org/10.1016/0009-2509(67)80049-6. [155] K. Ueyama, T. Miyauchi, Properties of recirculating turbulent two -phase flow in gas bubble columns, AIChE J. 25 (2) (1979) 258e266. https://doi.org/ 10.1002/aic.690250207. [156] H. Luo, H.F. Svendsen, Turbulent circulation in bubble-columns from eddy viscosity distributions of single-phase pipe-flow, Can. J. Chem. Eng. 69 (6) (1991) 1389e1394. https://doi.org/10.1002/cjce.5450690622. [157] J.B. Joshi, M.M. Sharma, A circulation cell model for bubble columns, Trans. Inst. Chem. Eng. 57 (4) (1979) 244.
n, Prediction of fluid dynamics and liquid mixing in [158] E. García-Calvo, P. Leto bubble columns, Chem. Eng. Sci. 49 (21) (1994) 3643e3649. https://doi.org/ 10.1016/0009-2509(94)00171-5. [159] Y.X. Wu, M.H. Al-Dahhan, Prediction of axial liquid velocity profile in bubble columns, Chem. Eng. Sci. 56 (3) (2001) 1127e1130. https://doi.org/10.1016/ S0009-2509(00)00330-4. [160] X. Gao, B. Kong, R.D. Vigil, Simulation of algal photobioreactors: recent developments and challenges, Biotechnol. Lett. 40 (9e10) (2018) 1311e1327. https://doi.org/10.1007/s10529-018-2595-3. [161] S. Becker, A. Sokolichin, G. Eigenberger, Gas-liquid flow in bubble columns and loop reactors: Part II. Comparison of detailed experiments and flow simulations, Chem. Eng. Sci. 49 (24) (1994) 5747e5762. [162] R.F. Mudde, J.S. Groen, H.E.A. Van Den Akker, Liquid velocity field in a bubble column: LDA experiments, Chem. Eng. Sci. 52 (21e22) (1997) 4217e4224. https://doi.org/10.1016/S0009-2509(97)88935-X. [163] R.F. Mudde, D.J. Lee, J. Reese, L.S. Fan, Role of coherent structures on Reynolds stresses in a 2-D bubble column, AIChE J. 43 (4) (1997) 913e926. https://doi. org/10.1002/aic.690430407. [164] J. Chaouki, F. Larachi, M.P. Dudukovi
c, Noninvasive tomographic and velocimetric monitoring of multiphase flows, Ind. Eng. Chem. Res. 36 (11) (1997) 4476e4503. https://doi.org/10.1021/ie970210t. [165] J. Chaouki, F. Larachi, M.P. Dudukovic, Non-invasive Monitoring of Multiphase Flows, Elsevier, 1997. [166] S.K. Xu, Y.H. Qu, J. Chaouki, C. Guy, Characterization of homogeneity of bubble flows in bubble columns using RPT and fibre optics, Int. J. Chem. React. Eng. (2005). [167] M. Polli, M. Di Stanislao, R. Bagatin, E. Abu Bakr, M. Masi, Bubble size distribution in the sparger region of bubble columns, Chem. Eng. Sci. 57 (1) (2002) 197e205. https://doi.org/10.1016/S0009-2509(01)00301-3. [168] A. Esmaeili, S. Farag, C. Guy, J. Chaouki, Effect of elevated pressure on the hydrodynamic aspects of a pilot-scale bubble column reactor operating with non-Newtonian liquids, Chem. Eng. J. 288 (Supplement C) (2016) 377e389. https://doi.org/10.1016/j.cej.2015.12.017. [169] A. Esmaeili, C. Guy, J. Chaouki, The effects of liquid phase rheology on the hydrodynamics of a gaseliquid bubble column reactor, Chem. Eng. Sci. 129 (2015) 193e207. https://doi.org/10.1016/j.ces.2015.01.071. [170] A. Esmaeili, C. Guy, J. Chaouki, Local hydrodynamic parameters of bubble column reactors operating with non-Newtonian liquids: experiments and models development, AIChE J. 62 (4) (2016) 1382e1396. https://doi.org/10. 1002/aic.15130. [171] M. Mokhtari, J. Chaouki, New technique for simultaneous measurement of the local solid and gas holdup by using optical fiber probes in the slurry bubble column, Chem. Eng. J. 358 (2019) 831e841. https://doi.org/10.1016/j. cej.2018.10.067. [172] X.P. Guan, N. Yang, Bubble properties measurement in bubble columns: from homogeneous to heterogeneous regime, Chem. Eng. Res. Des. 127 (Supplement C) (2017) 103e112. https://doi.org/10.1016/j.cherd.2017.09.017. [173] Y.M. Lau, N.G. Deen, J.A.M. Kuipers, Development of an image measurement technique for size distribution in dense bubbly flows, Chem. Eng. Sci. 94 (0) (2013) 20e29. https://doi.org/10.1016/j.ces.2013.02.043. €fer, C. Merten, G. Eigenberger, Bubble size distributions in a bubble [174] R. Scha column reactor under industrial conditions, Exp. Therm. Fluid Sci. 26 (6e7) (2002) 595e604. https://doi.org/10.1016/S0894-1777(02)00189-9. [175] Y.T. Shah, S. Joseph, D.N. Smith, J.A. Ruether, Two-bubble class model for churn turbulent bubble-column reactor, Ind. Eng. Chem. Process Des. Dev. 24 (4) (1985) 1096e1104. https://doi.org/10.1021/i200031a034. [176] S.A. Patel, J.G. Daly, D.B. Bukur, Bubble-size distribution in Fischer-Tropschderived waxes in a bubble column, AIChE J. 36 (1) (1990) 93e105. https:// doi.org/10.1002/aic.690360112. [177] L.S. Fan, G.Q. Yang, D.J. Lee, K. Tsuchiya, X. Luo, Some aspects of high pressure phenomena of bubbles in liquid and liquid-solid suspensions, Chem. Eng. Sci. 54 (1999) 4681. https://doi.org/10.1016/S0009-2509(99)00348-6. [178] M.R. Bhole, S. Roy, J.B. Joshi, Laser Doppler anemometer measurements in

[179]

[180]

[181]

[182]

[183]

[184]

[185]

[186]

[187]

[188]

[189]

[190]

[191]

[192]

[193]

[194]

[195]

[196]

[197]

[198]

[199]

[200]

[201]

[202]

631

bubble column: effect of sparger, Ind. Eng. Chem. Res. 45 (26) (2006) 9201e9207. https://doi.org/10.1021/ie060745z. G. Hebrard, D. Bastoul, M. Roustan, Influence of the gas sparger on the hydrodynamic behavior of bubble-columns, Chem. Eng. Res. Des. 74 (3) (1996) 406e414. A.A. Kulkarni, J.B. Joshi, Bubble formation and bubble rise velocity in gasliquid systems: a review, Ind. Eng. Chem. Res. 44 (16) (2005) 5873e5931. https://doi.org/10.1021/Ie049131p. H. Tsuge, S.I. Hibino, Bubble formation from a submerged single orifice accompanied by pressure fluctuations in gas chamber, J. Chem. Eng. Jpn. 11 (3) (1978) 173e178. https://doi.org/10.1252/jcej.11.173. K. Akita, F. Yoshida, Bubble size, interfacial area, and liquid-phase mass transfer coefficient in bubble columns, Ind. Eng. Chem. Process Des. Dev. 13 (1) (1974) 84e91. https://doi.org/10.1021/i260049a016. T. Miyahara, Y. Matsuba, T. Takahashi, Size of bubbles generated from perforated plates, Int. Chem. Eng. 23 (3) (1983) 8. https://doi.org/10.1252/ kakoronbunshu.8.13. S. Gnyloskurenko, A. Byakova, T. Nakamura, O. Raychenko, Influence of wettability on bubble formation in liquid, J. Mater. Sci. 40 (9e10) (2005) 2437e2441. https://doi.org/10.1007/s10853-005-1971-2. G.Q. Yang, B. Du, L.S. Fan, Bubble formation and dynamics in gas-liquid-solid fluidization- A review, Chem. Eng. Sci. 62 (1e2) (2007) 2e27. https://doi.org/ 10.1016/j.ces.2006.08.021. K. Ueyama, S. Morooka, K. Koide, H. Kaji, T. Miyauchi, Behavior of gas bubbles in bubble columns, Ind. Eng. Chem. Process Des. Dev. 19 (4) (1980) 592e599. https://doi.org/10.1021/i260076a015. Y. Kang, Y.J. Cho, K.J. Woo, K.I. Kim, S.D. Kim, Bubble properties and pressure fluctuations in pressurized bubble columns, Chem. Eng. Sci. 55 (2) (2000) 411e419. https://doi.org/10.1016/S0009-2509(99)00336-X. P.M. Wilkinson, H. Haringa, L.L. Van Dierendonck, Mass transfer and bubble size in a bubble column under pressure, Chem. Eng. Sci. 49 (9) (1994) 1417e1427. https://doi.org/10.1016/0009-2509(93)E0022-5. P.M. Wilkinson, L.L. Von Dierendonck, Pressure and gas-density effects on bubble break-up and gas hold-up in bubble-columns, Chem. Eng. Sci. 45 (8) (1990) 2309e2315. https://doi.org/10.1016/0009-2509(90)80110-Z. A. Kumar, T.E. Degaleesan, G.S. Laddha, H.E. Hoelscher, Bubble swarm characteristics in bubble columns, Can. J. Chem. Eng. 54 (6) (1976) 503e508. https://doi.org/10.1002/cjce.5450540604. M. Fukuma, K. Muroyama, A. Yasunishi, Properties of bubble swarm in a slurry bubble column, J. Chem. Eng. Jpn. 20 (1) (1987) 28e33. https://doi.org/ 10.1252/jcej.20.28. R. Parthasarathy, N. Ahmed, Size distribution of bubbles generated by finepore spargers, J. Chem. Eng. Jpn. 29 (6) (1996) 1030e1034. https://doi.org/ 10.1252/jcej.29.1030. M. Kagumba, M.H. Al-Dahhan, Impact of internals size and configuration on bubble dynamics in bubble columns for alternative clean fuels production, Ind. Eng. Chem. Res. 54 (4) (2015) 1359e1372. https://doi.org/10.1021/ ie503490h. A. Matsuura, L.S. Fan, Distribution of bubble properties in a gas-liquid-solid fluidized bed, AIChE J. 30 (6) (1984) 894e903. https://doi.org/10.1002/aic. 690300604. A. Yasunishi, M. Fukuma, K. Muroyama, Measurement of behavior of gas bubbles and gas holdup in a slurry bubble column by a dual electroresistivity probe method, J. Chem. Eng. Jpn. 19 (5) (1986) 444e449. https://doi.org/10. 1252/jcej.19.444. D.J. Vermeer, R. Krishna, Hydrodynamics and mass transfer in bubble columns in operating in the churn-turbulent regime, Ind. Eng. Chem. Process Des. Dev. 20 (3) (1981) 475e482. https://doi.org/10.1021/i200014a014. R. Krishna, J. Ellenberger, Gas holdup in bubble column reactors operating in the churn turbulent flow regime, AIChE J. 42 (1996) 2627. https://doi.org/10. 1002/aic.690420923. S. Rabha, M. Schubert, M. Wagner, D. Lucas, U. Hampel, Bubble size and radial gas hold-up distributions in a slurry bubble column using ultrafast electron beam X-Ray tomography, AIChE J. 59 (5) (2013) 1709e1722. https:// doi.org/10.1002/aic.13920. S. Grevskott, B.H. Sannaes, M.P. Dudukovic, K.W. Hjarbo, H.F. Svendsen, Liquid circulation, bubble size distributions, and solids movement in twoand three-phase bubble columns, Chem. Eng. Sci. 51 (10) (1996) 1703e1713. https://doi.org/10.1016/0009-2509(96)00029-2. A.C. Saxena, N.S. Rao, S.C. Saxena, Bubble size distribution in bubble columns, Can. J. Chem. Eng. 68 (1) (1990) 159e161. https://doi.org/10.1002/cjce. 5450680119. Y.H. Yu, S.D. Kim, Bubble properties and local liquid velocity in the radial direction of cocurrent gasdliquid flow, Chem. Eng. Sci. 46 (1) (1991) 313e320. https://doi.org/10.1016/0009-2509(91)80140-T. S.L. Shu, N. Yang, GPU-accelerated large eddy simulation of stirred tanks, Chem. Eng. Sci. 181 (2018) 132e145. https://doi.org/10.1016/j.ces.2018.02. 011.