# Two-Phase Bubble Columns: A Comprehensive Review

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## Abstract

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## 1. Introduction

_{G}, the mean bubble residence time, …) fluid dynamic properties. In this respect, the correct quantification and estimation of these fluid dynamics parameters is very important for avoiding overestimations of investment costs (i.e., with respect to the downstream processing units and to the reactor itself) and design failures. The review is structured as follows: In Section 2, we discuss how the operating parameters, operating modes, liquid and gas phase properties, and design parameters influence the flow regime transitions and the fluid dynamics parameters. In Section 3, we present the experimental techniques to study the “bubble-scale” and the “reactor-scale” fluid dynamic properties. In Section 4, we present modeling approaches to study the “bubble-scale” and the “reactor-scale” bubble column fluid dynamics.

## 2. Bubble Column Fluid-Dynamics

_{c}, the operating temperature, T

_{c}, and the superficial velocities, U

_{G}and U

_{L}), operating modes (i.e., the co-current, the counter-current and the batch modes), liquid and gas phase properties (i.e., liquid viscosity, μ

_{L}, the gas density, ρ

_{G}, the presence of active agents, …), and design parameters (i.e., the gas sparger design, the column diameter, d

_{c}, and the aspect ratio, AR influence the flow regime transitions and the global and local fluid dynamics parameters(it is worth noting that the correct definition of AR relies on the initial liquid level (AR = H

_{0}/d

_{c}), rather the column height, H

_{c}, as discussed by Sasaki et al. [6]). It is worth noting that the proposed discussion considers an air-water bubble column operating at ambient conditions as the “baseline” case; thus, the variations owing to the design and operation parameters are discussed starting from this the “baseline” case.

#### 2.1. The Flow Regimes and the Flow Regime Tranistions in Bubble Columns

#### 2.1.1. The Flow Regimes in Bubble Columns

_{G}) using a fixed system design and phase properties: (i) the mono-dispersed homogeneous flow regime; (ii) the pseudo-homogeneous flow regime; (iii) the transition flow regime; (iv) the heterogeneous flow regime; (v) the slug flow regime; and (vi) the annular flow regimes. The first flow regime observed is the “mono-dispersed homogeneous” flow regime, which is characterized by mono-dispersed BSDs, according to the change in the sign of the lift force coefficient (see ref. [7,8,9,10,11,12]); an example of this flow regime is the one observed by Mudde et al. [13], in which a “fine” gas sparger was used, or by Besagni et al. [14]. When increasing U

_{G}, the BSDs shift towards higher bubble diameters (according to the change in the sign of the lift force coefficient, and, thus, the “pseudo-homogeneous” homogeneous flow regime is observed; of course, if large bubbles are aerated [7] by using a “coarse” gas sparger, the lower boundary of the former flow regime moves towards a low U

_{G}and eventually, disappears. Generally speaking, the mono-dispersed and poly-dispersed BSDs are characterized by limited interactions between the bubbles and only “non-coalescence-induced” bubbles exist, as defined and observed by [7]. In general, one may refer to the homogeneous flow regime as the complete range of both mono-dispersed and pseudo-homogeneous flow regimes. With an increase in U

_{G}, the number of bubbles having a negative lift force coefficient increases and these continue migrating towards the center of the bubble column; at a certain point, the rate of this process leads to the formation of “coalescence-induced” bubbles [7] (as demonstrated by Besagni et al. [11,12]) when “coalescing” solutions are employed. Conversely, if “non-coalescing” solutions are employed, the transition flow regime is identified by the appearance of “clusters of bubbles”, as defined in refs. [15,16] and observed by [7]. The flow regime transitions from the homogeneous flow regime towards the transition flow regime concept are displayed in Figure 3.

_{G}= 0.02 m/s and, by increasing the gas flow rate, both “non-coalescence-induced” and “coalescence induced” bubbles increased until the non-coalescence-induced” bubbles reached a maximum, and then decreased toward a constant value. Conversely, Scumbe and Grund [18], Krishna et al. [19,20] and Besagni et al. [7] reported that the non-coalescence-induced” bubbles increase until the flow regime transition, and then only “coalescence induced” bubbles increase. The difference is probably caused by the different gas sparger design (“fine-gas sparger” and “coarse-gas sparger”). At higher U

_{G}, the heterogeneous flow regime is achieved. This flow regime is associated with high coalescence and breakage rates and a wide variety of bubble sizes; in the heterogeneous flow regime, the liquid phase flows in an oscillatory manner, upwards and downwards, because the U

_{G}is not high enough to carry the liquid continuously upward. The reader should refer to the recent review proposed by Montoya et al. [21] for a complete discussion on this flow regime. When U

_{G}is increased, the slug flow regime may occur; however, in industrial applications, this flow regime is not observed, owing to Rayleigh–Taylor instabilities [22] which arise because of the large diameter effects. The quantification of Rayleigh–Taylor instabilities at the “reactor-scale” is quantified through the dimensionless diameter, D*

_{H}:

_{H}is the hydraulic diameter, d

_{c}is the bubble column (inner) diameter, σ is the surface tension, g is the acceleration due to gravity and ρ

_{L}− ρ

_{G}is the density difference between the two phases. Bubble columns with D*

_{H}greater than the critical value, D*

_{H}

_{,cr}= 52 [23]—d

_{c}≳ 0.13–0.15 m (at ambient temperature and pressure, as a circular bubble column)—are considered to be large diameter bubble columns. When D*

_{c}> D*

_{H}

_{,cr}, the cap bubbles can no longer be sustained, and “coalescence-induced” bubbles (or, cluster of bubbles) appear instead of the slug flow regime. At very high U

_{G}, the annular flow regime may occur; this is characterized by a high-speed gas core containing entrained liquid droplets. Generally, this flow regime cannot be observed in industrial-scale reactors because the gas velocity requested is too high. In practical applications (i.e., in industrial applications), large-diameter and large-scale bubble columns are used. In addition, in these applications, the slug and annular flows are not observed (see the air-water flow regime maps in ref. [24]; this flow map is very important as it describes the relationships between the flow regimes and the bubble column diameter, which is related to Rayleigh–Taylor instabilities).

#### 2.1.2. The Flow Regime Transitions

_{G}

_{,trans}≈ 0.02–0.04 m/s [25]. Depending on the many variables characterizing the system, U

_{G}

_{,trans}either reduces (“homogeneous flow regime destabilization”) or increases (“homogeneous flow regime stabilization”). In the following sections, the influence of the operating parameters, operating modes, liquid and gas phase properties, and design parameters on the flow regime transitions are discussed

#### Influence of the Bubble Column Design

**Column diameter.**It is not clear whether the increase in d

_{c}stabilizes [29,30] or destabilizes [8,31,32,33] the homogeneous flow regime. Sarrafi et al. [29] (d

_{c}= 0.08–0.155 m—also taking data from the literature), Ohki and Inoue [30] (d

_{c}= 0.04–0.16 m and H

_{c}= 2.0–3.0 m) and Urseanu [19] (d

_{c}= 0.051–0.63 m, H

_{c}= 2.0–4.0 m) found that an increase in d

_{c}, increased U

_{G}

_{,trans}. Sarrafi et al. [29] also reported that, for d

_{c}> 0.15 m, U

_{G}

_{,trans}does not depend on d

_{c}. Conversely, Ruzicka et al. [31] (d

_{c}= 0.14, 0.29 and 0.4 m, H

_{c}= 0.1–1.2 m) and Zahradnik et al. [8] (d

_{c}= 0.14, 0.15, and 0.29, H

_{c}= 0.25–1.5 m) found that an increase in d

_{c}reduced U

_{G}

_{,trans}. Al-Oufi et al. [32,33] observed that the annular gap layout destabilizes the homogenous flow regime. Conversely, Besagni et al. [34] found that the annular gap layout stabilized the homogeneous flow regime, possibly because of the gas sparger in the open tube configuration. In the current authors’ opinion, the influence of the bubble column diameter is related to Rayleigh–Taylor instabilities [22], as previously discussed.

**Aspect ratio.**In low-AR bubble columns, the flow circulation patterns are not completely developed, and thus, the end-effects (i.e., top column effects and near gas sparger) are more relevant; in particular, the gas sparger effects tend to destabilize the homogeneous flow regime in low-AR bubble columns, for example, Xue et al. [35] observed a strong influence of the gas sparger parameters in local void fractions up to AR = 5. Thorat and Joshi (d

_{c}= 0.385 m, H

_{c}= 0.385–3.08 m) and Ruzicka et al. [31] observed that an increase in AR destabilized the homogeneous flow regime. Sarrafi et al. [29] observed no effect beyond 4 m. Of course, these results are related to the evalution of the BSDs inside the bubble columns [36]. Besagni et al. [37] found that, when using a “coarse-gas sparger”, an increase in AR destabilized the homogeneous flow regime up to a critical value equal to approximately 5 in batch mode (Figure 2) and 10 in counter-current mode (Figure 3). Figure 4 summarizes the main results from the literature concerning the influence of the aspect ratio on the flow regime transitions; Figure 4 considers both (a) the regime transition between the pseuso-homoegenous flow regime and the transition flow regime and (b) the regime transition between the transition flow regime and the heterogeneous flow regime.

**Gas sparger design.**It is known that the gas sparger design highly influences the flow regime transitions in bubble column. Generally, gas spargers with a hole diameter, d

_{0}, smaller than 1 mm (the “fine distributors”, i.e., porous [32,33,38,39], membrane [40], needles [13,14,41] or sieve tray/perforated plates with small openings [37] see Figure 5 for a representation of these gas spargers) can maintain a stable mono-dispersed homogenous flow regime also at a high U

_{G}; hence, the homogeneous flow regime is stabilized [13] because the “mono-dispersed homogeneous” flow regime exists [37] and the “pseudo-homogeneous” flow regime takes place after the former is destabilized. In contrast, using a “coarse gas sparger” (d

_{0}> 1 mm—see Figure 6 for a representation of these gas spargers) the “mono-dispersed homogeneous” flow regime may not exist and a “pseudo-homogeneous” flow regime may be observed at lower gas superficial velocities; finally, when using a “very coarse gas sparger” (d

_{0}>> 1 mm), the homogeneous flow regime may not be established and the transitional/heterogeneous flow regime may take place [42]. A well-designed “fine distributor” may stabilize the homogeneous flow regime far beyond expectations, as demonstrated by Mudde et al. [13]. An early study was conducted by Zahradnik [43], demonstrating that, in perforated plates, the smaller the gas sparger opening, the more the homogeneous flow regime is stabilized. Sarrafi et al. [29] found that the transition velocity decreases as d

_{0}increases, up to a value of d

_{0}= 0.0015 m. Thorat and Joshi [44] found that the homogeneous flow regime is stabilized with a decrease in d

_{hole}(d

_{0}= 0.0008–0.05 m). It is also worth noting that they observed that an increase in AR destabilized the homogeneous flow regime. Sal et al. [45] found that U

_{G}

_{,trans}decreases while increasing the gas sparger openings (d

_{0}= 0.001–0.003 m). Besagni et al. [46] compared two different “coarse-gas spargers” and found no significant differences between them; conversely, in more recent studies, Besagni et al. [14,37] compared the flow regime transitions in both “coarse-gas spargers” and “fine-gas spargers” (a needle gas sparger [14] or a perforated gas sparger [37]); they found that a uniform bubble bed stabilized the homogeneous flow regime, owing to the presence of the monodispersed homogenous flow regime.

#### Influence of the Bubble Column Operation Mode

_{c}= 0.152 m, H

_{c}= 2.5 m) observed that the superficial liquid velocity (U

_{L}up to 0.04 m/s) had no influence in both the counter-current and the co-current modes. Jin et al. [48] (d

_{c}= 0.160 m, H

_{c}= 2.5 m) reported that the transition point is approximately the same for the three working modes if U

_{L}< 0.04 m/s, whereas, for higher U

_{L}(in co-current and counter-current modes), the transition velocity decreases when the U

_{L}increases. Otake et al. [49] (d

_{c}= 0.05 m, H

_{c}= 1.5 m) observed earlier flow regime transitions with an increased liquid flow rate in counter-current operations (U

_{L}up to −0.15 m/s). A similar conclusion was also drawn by Yamaguchi and Yamazaki [50] (d

_{c}= 0.04 m and 0.08 m). Extensive studies on the influence of the liquid velocity in the counter-current mode were performed by Besagni et al. [28,34,46,51,52,53,54], who found that the counter-current mode destabilized the homogeneous regime in all of the different configurations studied (viz. the pipe sparger in the annular gap and in the open tube configuration [34] and the spider sparger [46]). Figure 7 and Figure 8 summarize the main experimental results obtained by Besagni et al. [28,34,46,51,52,53,54]. Recently, Trivedi et al. [55] performed a detailed study on the hydrodynamics of counter-current bubble columns and found results in agreement with Besagni and co-authors. It is worth noting that Besagni et al. [28] also studied the influence of the bubble column aspect ratio in the counter-current mode; the results of this study are presented in Figure 8, where it can be observed that an increase in AR destabilized the homogeneous flow regime up to a critical value, equal to approximately 5 in batch mode (Figure 2) and 10 in counter-current mode (Figure 3). Further explanation for this effect is provided in Section 2.2, when discussing the influence of the operation mode on the gas hold-up.

#### Influence of the Liquid Phase Viscosity

_{L}, depending on its value, either stabilizes or destabilizes the homogeneous flow regime compared to air-water systems (The Mono-Ethylene Glycol, MEG, concentration, c

_{MEG}= 0%: U

_{G}

_{,trans}= 0.0264 m/s, ε

_{G}

_{,trans}= 0.09; c

_{MEG}= 5%: U

_{G}

_{,trans}= 0.039 m/s, ε

_{G}

_{,trans}= 0.18; c

_{MEG}= 80%: U

_{G}

_{,trans}= 0.023 m/s, ε

_{G}

_{,trans}= 0.07). It is worth noting that the increased coalescence may also suppress the homogeneous flow regime and, for μ

_{L}> 8 mPa·s, it may not exist even with a ‘fine gas sparger’ [8,58,59,60].

_{L}values (μ

_{L}= 1–3 mPa·s) and that U

_{G}

_{,trans}increases with μ

_{L}in this range, whereas it decreases at moderate μ

_{L}values (μ

_{L}= 3–22 mPa·s). Olivieri et al. [61] observed a stabilization of the homogeneous flow regime up to μ

_{L}= 4.25 mPa·s, and then for higher μ

_{L}, a destabilization of the homogeneous flow regime. Finally, the above-mentioned results were also confirmed by ultrafast X-ray tomography investigations [62] (d

_{c}= 0.07 m, H

_{c}= 1.5 m, orifice gas sparger, d

_{0}= 1 mm). In this study, it was observed that, beyond μ

_{L}= 5.18–8.95 mPa·s, the homogeneous flow regime is no longer observed. It is worth noting that, in these systems, a non-Newtonian fluid behavior may exist; in this respect, we propose some speculations based on the conclusions of Olivieri et al. [61]; they interpreted the stabilization of the homogeneous flow regime as the relaxation time by considering the negative wake phenomena occurring in the rear of a bubble in the case of viscoelastic fluids. Please refer to Besagni et al. [11] for further discussion concerning this point.

#### Influence of the Active Compounds: Inorganic Compounds (“positive surfactants”)

_{t}, is defined as the concentration, c, of the non-coalescent media above which bubble coalescence is drastically reduced. Depending on the concentration, n, we can define a “coalescent regime” (n* ≤ 1) and a “non-coalescent regime” (n* > 1), through the dimensionless concentration, n*—following the formulation of n

_{t}, proposed by Prince and Blanch [72] (a modified version of the formulation of Marrucci [63]):

^{−28}J m), R

_{g}is the gas constant, T is the temperature of the system, r

_{b}is the bubble radius, and σ and ∂σ/∂n are the surface tension and the surface tension gradient with electrolyte concentration, respectively. This concentration has been found to be unique for each salt [73], valid for a swarm of bubbles and does not depend on U

_{G}[66,74]. Some of the relevant studies from the existing literature are here summarized, i.e., Besagni et al. [75] (d

_{c}= 0.24 m, H

_{0}= 3.0 m; kitchen quality NaCl—Figure 6a,b), Thorat and Joshi [44] (d

_{c}= 0.385 m, H

_{c}= 0.385–3.08 m, NaCl), Ribeiro and Mewes [76] (d

_{c}= 0.12 m, H

_{c}= 1.25 m, NaCl, Na

_{2}SO

_{4}, NaI), Kelkar et al. [77] (d

_{c}= 0.154 m, H

_{C}= 3.25 m), Grover et al. [78] (d

_{c}= 0.1 m, H

_{C}= 1.5 m. NaCl and CuCl

_{2}), Rucizka et al. [70] (d

_{c}= 0.14 m, H

_{0}= 0.4 m; CaCl

_{2}), Orvalho et al. [69] (d

_{c}= 0.14 m, H

_{0}= 0.4 m; Na

_{2}SO

_{4}, NaCl and kitchen quality NaCl). It is worth nothing that Rucizka et al. [70] and Orvalho et al. [69] observed a dual effect of the electrolyte concentration on the flow regime transitions. Figure 9a,b display the influence of NaCl concentration on the flow regime transition points; conversely, Figure 10a,b display the influence of AR and n* on the flow regime transition points, supporting the above-mentioned discussion on the influence of the concentration of the inorganic substances in the “coalescent regime” (n* ≤ 1) and the “non-coalescent regime” (n* > 1).

#### Influence of the Active Compounds: Organic Compounds (“negative surfactants”)

_{c}= 0.1, 0.15 and 0.38 m, H

_{c}= 4 m, EtOH up to 1%

_{vol}), Al-Oufi et al. [32] (d

_{c}= 0.102 m, H

_{c}= 2.25 m, EtOH up to 300 ppm

_{wt}), Dargar et al. [83] (d

_{c}= 0.127 m, H

_{c}= 2.75 m, EtOH up to 5%

_{wt}) and Besagni et al. [7] (d

_{c}= 0.24 m, H

_{0}= 3.0 m; 0.05%

_{wt}—Figure 6a,b). Figure 9a,b display the influence of EtOH concentration on the flow regime transition points; conversely, Figure 11a,b display the influence of AR and EtOH concentration on flow regime transition points.

#### Pressure and Temperature

_{c}= 0.1–15.2 MPa, T

_{c}= 298–350 K) showing a stabilization of the homogeneous flow regime; it can be stated that many industrial processes may be run in the pseudo-homogeneous flow regimes. Generally, at high pressures and temperatures and for p

_{c}> 10 MPa, the flow regime transition is mainly governed by the temperature, properties of the liquid phase, liquid velocity and gas sparger design [1].

#### 2.1.3. The Flow Regime Maps

#### 2.2. The Gas Hold-Up

_{G}, determines the residence time and, in combination with the BSD, the interfacial area (related to the reactor size); it is related to the column design, operation, phase properties and operating conditions. It is worth noting that, generally, the homogeneous flow regime is more sensitive than the heterogeneous flow regime to the operating parameters.

#### 2.2.1. Influence of the Bubble Column Operation

#### Superficial Gas Velocity and Gas Holdup Curve Structure

_{G}and U

_{G}) summarizes the complexity of the bubble column fluid dynamics, from the “bubble-scale” to the “reactor-scale” (see, for example, the discussion in ref. [69], Section 3.2). For example, “coarse gas spargers” lead to monotonic gas hold-up curves (Figure 13a), whereas, when using “fine gas spargers”, a peak in the gas hold-up curve can appear (Figure 13b) owing to the mono-dispersed bubble size distribution, which leads to the hindrance effect (related to the Ledinegg instability) that is physically manifested by a peak on the gas hold-up curve, leading to a reversed S-shaped curve, as discussed and commented on in refs. [14,37] and displayed in Figure 3b.

_{G}= (U

_{G}/U

_{swarm}); theoretically, if the bubbles travel at their terminal velocity, ɛ

_{G}will increase linearly with the gas flow rate. However, the coupling between the phases causes deviations from linearity (see ref. [60]). In the homogeneous flow regime, the hindrance reduces the bubble velocity, thus increasing ɛ

_{G}, whereas in the transition flow regime the “coalescence-induced” bubbles result in a decrease in ɛ

_{G}and cause ɛ

_{G}to decrease proportionally less than U

_{G}[34]. The presence of “non-coalescence induced” and “coalescence induced” bubbles has been investigated by different authors (i.e., [17,18,19,46]). For the sake of clarity, Figure 14 and Figure 15 provide a comparison of gas hold-up curves from different studies proposed in the literature and summarized in Table 1 and Table 2.

#### Superficial Liquid Velocity

_{L}< 0.01 m/s), in co-current or in counter-current mode [1,2,3]. Low liquid velocities generally do not affect ɛ

_{G}—as found by several investigators [24,47,113,114,115,116,117,118,119]—because, if U

_{L}is low compared to the bubble rise velocities, the acceleration of the bubbles (caused by the non-stagnant operation) will be negligible [120]. For example, Akita and Yoshida [47] reported a negligible effect of liquid velocities up to 0.04 m/s, either in gas–liquid counter-current or con-current operations. Of course, at higher liquid velocities, the column operation influences ɛ

_{G}; it is generally admitted that co-current operation tends to reduce ɛ

_{G}and counter-current operation increases ɛ

_{G}, as bubbles are either accelerated or decelerated by liquid motion [1,114]. Baawain et al. [121], showed that the counter-current or co-current operating mode influenced ɛ

_{G}for about 5% of its weight, and less than 1% of its bubble size, showing that the effect observed is mainly caused by the bubble rise velocity and not only caused by the bubble size. In agreement with this explanation concerning the influence of U

_{L}on bubble motion, different studies have reported ɛ

_{G}decreasing in co-current operations [49,90,122,123,124,125,126,127,128] and increasing in counter-current operations [48,49,128]. Biń et al. [128] compared the three-operation mode showing that ɛ

_{G}increases with U

_{L}in counter-current mode and decreases or remains constant in co-current mode. The effect is more pronounced at a high gas velocity and the difference in ɛ

_{G}between co-current and counter-current modes is around 10%. The same trends were observed by Jin et al. [48], with a maximum difference of 2% between counter-counter and co-current operations and it was found that the influence of working mode is lower at high ɛ

_{G}. Similar trends were found by Otake et al. [49], as already discussed when considering the flow regime transition. Besagni et al. [34,46,51,52,53,54] found that when increasing the liquid flowrate, a faster increase in the hold-up is observed at low U

_{G}, and the transition point also moves toward lower superficial gas velocities (Figure 16, Figure 17 and Figure 18). This change was explained by the effect of the liquid flow, which slows down the rise of the bubbles, leading to higher hold-up—a more compact arrangement of bubbles leads to an earlier flow regime transition. Above a certain hold-up (depending on U

_{L}), the liquid superficial velocity has no more influence on the hold-up; this is possibly caused by the limited influence of the operation mode on the fully established heterogeneous flow regime. In agreement with the above-mentioned study, Jin et al. [48] also observed that the influence of working mode is lower at high ɛ

_{G}.

#### 2.2.2. Influence of the Bubble Column Design

#### Column Size

_{c}, ɛ

_{G}may decrease [20,30,129,130,131]. The increased recirculation was shown by Krishna et al. [130]. Urseanu et al. [131] observed the decrease in ɛ

_{G}(d

_{c}between 0.15 and 0.23 m) while working with viscous fluids, in agreement with Behkish et al. [132], who noticed that the effect of the column diameter on viscous fluid is higher. The data collected by Fai et al. [105] and Yoshida and Akita [112] show that the effect of the column diameter on ɛ

_{G}is negligible for columns larger than d

_{c}= 0.15 m. Hughmark [133] found an effect of column size on ɛ

_{G}up to a diameter of 0.10 m. Kata [134] conducted measurements in 0.066 m, 0.122 m and 0.214 m columns and found that ɛ

_{G}increases with a decreasing d

_{c}. Koide et al. [135] measured ɛ

_{G}in an 0.55 m column and found no significant difference from the literature values reported for columns less than d

_{c}= 0.60 m. Deckwer [136] found a difference in ɛ

_{G}between a d

_{c}= 0.041 m column and a d

_{c}= 0.10 m column. Hikita et al. [80] measured ɛ

_{G}in a d

_{c}= 0.10 m column and compared their results with the ones reported in the literature for columns with d

_{c}> 0.10 m, finding no appreciable effect of d

_{c}on ɛ

_{G}. Gopal et al. [106] measured the ɛ

_{G}in d

_{c}= 0.2 m, d

_{c}= 0.6 m and d

_{c}= 1.0 m columns and concluded that d

_{c}and the gas sparger do not significantly influence ɛ

_{G}. Nottemkamper et al. [137] measured ɛ

_{G}in d

_{c}= 0.19 m, d

_{c}= 0.45 m and d

_{c}= 1.0 m columns and obtained comparable results for the d

_{c}= 0.19 m and d

_{c}= 0.45 m columns but lower ɛ

_{G}values for the d

_{c}= 1 m column at high gas rates, which they attributed to its larger diameter. Koide et al. [138] observed smaller ɛ

_{G}values in columns smaller than d

_{c}= 0.2 m. Accordingly to the model proposed by Lemoine et al. [139], d

_{c}has an influence in columns of up to d

_{c}= 0.70 m in either the homogeneous flow regime or in the heterogeneous flow regime. Despite some contradictory results, a generally accepted rule of thumb is that d

_{c}= 0.15 m is large enough for the results to be scalable [1,24,98,140], as supported by different investigators [8,31,141]. This scale-up criteria relies on the fluid dynamics phenomena and the coupling between the gas and liquid phases, which can be interpreted by considering the instabilities previously described in Equation (1).

#### Aspect Ratio

_{G}and are more evident in low AR bubble columns (see the experimental data summarized in Figure 19, Figure 20 and Figure 21). In particular, in systems where the bubbles are not at their maximum equilibrium size (and where coalescence may occur), the liquid height (H

_{0}) will influence the extent of the coalescence. Hence, ɛ

_{G}decreases with H

_{0}, because the higher the column, the longer the time the bubbles have to coalesce and thus, the lower the mean residence time of the dispersed phase. In one of the very first studies, Yoshida and Akita [112] and Patil et al. [142] did not observe any significant effects of AR on ɛ

_{G}. Wilkinson et al. [56]—when presenting the scale-up criteria—discussed the results obtained by Kastanek et al. [143]: the influence of AR is negligible for H

_{C}> 1–3 m and with AR > 5. Zahradnik et al. [8] found that ɛ

_{G}decreases and the homogeneous regime is destabilized while increasing the initial liquid level up to a critical aspect ratio, AR

_{Cr}; the authors concluded that their results support the assumption of a negligible influence of AR on ɛ

_{G}and flow regime transitions for AR > 5. These assumptions were also confirmed by Thorat et al. [111], who found a negligible influence of AR on ɛ

_{G}for AR > 5 (air-water) or AR > 8 (non-coalescing liquid phase). Sarrafi et al. [29] compared their experimental results with other experimental data and excluded any effect of the initial liquid level on the flow regime transition when H

_{0}> 3 m. Ruzicka et al. [31] found that an increase in liquid height decreases ɛ

_{G}and destabilizes the homogeneous regime up to critical values. Sasaki et al. [6] have found that an increase in liquid height destabilizes the homogeneous regime and decreases ɛ

_{G}(AR up to 5). The reader may also refer to the recent discussion by Besagni et al. [37], who compared values from a large experimental dataset to demonstrate that AR

_{Cr}= 5 (Figure 19a) is ensured only for pure liquid phase systems with very large sparger openings operated in the batch mode (i.e., AR

_{Cr}increases to 10 in counter-current mode (Figure 19b) and in pure liquid phase systems, with small gas sparger openings operated in batch mode (Figure 21)). The relationship between AR and the gas hold-up in batch mode is displayed in Figure 19a; conversely, Figure 19b displays the relationship between AR and the gas hold-up in counter-current mode. Other data concerning the relationship between the gas hold-up and AR for different liquid phases are discussed in the next sections, concerning the influence of the active compounds.

#### Gas Sparger

_{G}value and, specifically, the ɛ

_{G}curve (the relationship between U

_{G}and ε

_{G}). It is difficult to provide a general rule because of the many parameters involved (i.e., gas sparger design and operating conditions) and the controversial results presented in the literature. Despite some discrepancies, we can state that, when using a “fine distributor”, the ɛ

_{G}curve increases linearly in the homogeneous flow regime, then reaches a peak and then rises again [18,19,20,32,33,82,129,145]; conversely, when using a “coarse distributor”, the ɛ

_{G}curve grows continuously [19,49,111,112,131,146,147]. This was also proposed by Urseanu [19] and Deckwer [3]. The different behaviors are due to the bubble dynamics (i.e., bubble formation at the gas sparger and coalescence/break-up phenomena); in “coarse distributors” there is a continuous appearance of large bubbles [19], whereas, in “fine distributors” large bubbles start appearing after the flow transition [18]. In accordance with this discussion, “fine distributors” have higher ɛ

_{G}compared to “coarse distributors” because of their narrower BSD (mainly, a lower rise velocity) [8,129,148] and, for the same distributor, a decrease in d

_{0}leads to an increase in ɛ

_{G}(mainly, a lower number of large bubbles) [30,45,123,149]. In the current authors’ opinion, these considerations can be explained by the flow regimes described in Section 2.2.1 and displayed in Figure 3.

#### 2.2.3. Influence of the Liquid Properties

#### Viscous Media

_{G}were observed as the liquid viscosity, μ

_{L}, increased. All the apparently contradictory results can be explained by interpreting them in terms of the “dual effect of viscosity over ɛ

_{G}, as described by Besagni et al. [11]—they found that ɛ

_{G}continuously (and non-linearly) increases when the MEG concentration increase, up to c

_{MEG}= 5% − µ

_{L}= 1.01 mPa·s, along with the contribution of small bubbles (Figure 22a). Conversely, if the concentration is further increased from c

_{MEG}= 5% − µ

_{L}= 1.01 mPa·s to c

_{MEG}= 80% − µ

_{L}= 7.97 mPa·s, ɛ

_{G}decreases (Figure 22b). For this last concentration, the ɛ

_{G}curve lies even below that obtained for pure water. The authors explained this behavior occurs because at low viscosities, the coalescence is limited, and the large drag force reduces the bubble rise velocity, causing an increase in ɛ

_{G}. When the viscosity increases, the tendency to coalescence prevails, creating large bubbles rising the column at a higher velocity, thus reducing ɛ

_{G}. In particular, the “dual effect of the viscosity” has been clearly observed in other experimental investigations [60,61,62,150,151,152,153]: Eissa and Schugerl [150] (d

_{c}= 0.12 m, H

_{c}= 3.9 m, d

_{0}= 2 mm) showed that ɛ

_{G}first increases (μ

_{L}< 3 mPa·s), then decreases (3 < μ

_{L}< 11 mPa·s), and finally becomes roughly constant (μ

_{L}> 11 mPa·s). Bach and Pilhofer [153] (d

_{c}= 0.10 m, d

_{0}= 0.5 mm) showed that ɛ

_{G}first increases (μ

_{L}< 1.5–2 mPa·s), then decreases (3 < μ

_{L}< 11–12 mPa·s), and finally becomes roughly constant (μ

_{L}> 12 mPa·s). Godbole et al. [152] (d

_{c}= 0.305 m, H

_{c}= 2.44 m, d

_{0}= 1.66 mm) showed that ɛ

_{G}first increases up to μ

_{L}= 2.23–4.75 mPa·s, then it decreases (7.81 < μ

_{L}< 52.29 mPa·s, depending on U

_{G}), and finally becomes roughly constant (μ

_{L}> 52.29 mPa·s). Khare and Joshi [151] (d

_{c}= 0.20 m, H

_{c}= 3.0 m, d

_{0}= 2.0 mm) showed that ɛ

_{G}first increases up to μ

_{L}= 4 mPa·s, and then it decreases for 4 < μ

_{L}< 10 mPa·s. Ruzicka et al. [60] (d

_{c}= 0.14 m, H

_{0}= 0.2–0.8 m, d

_{0}= 0.5 mm) showed that ɛ

_{G}first increases for μ < 3 mPa·s, and it decreases for 3 < μ < 22 mPa·s. Olivieri et al. [61] (d

_{c}= 0.12 m, H

_{c}= 2 m, H

_{0}= 0.8 m, needle gas sparger, d

_{0}= 0.4 mm) showed that ɛ

_{G}first increases up to μ

_{L}= 4.25 mPa·s, and then it decreases at higher viscosities. Concerning the remaining literature surveyed, the main results are described here. Please note that these apparently contradictory results can be explained by the above-mentioned criterion. For example, Yoshida and Akita [47,112] (d

_{c}= 0.152 m) reported that ɛ

_{G}varies with μ

_{L}in an irregular manner. Wilkinson et al. [56] (d

_{c}= 0.15–0.23 m, H

_{0}= 1.2 m; d

_{0}= 7 mm, d

_{c}= 0.158 m, H

_{0}= 1.5 m, d

_{0}= 2 mm) found that a high viscous liquid phase (μ

_{L}= 21 mPa·s) causes a decrease in ɛ

_{G}. Kuncová and Zahradnik [59] (d

_{c}= 0.15 m, H

_{c}= 1 m, H

_{0}= 0.5 m, d

_{0}= 0.5 mm) investigated the effect of μ

_{L}on the dynamics of bubble bed formation using several aqueous solutions of saccharose (1 < μ

_{L}< 110 mPa·s)—they found a decrease in ɛ

_{G}when the viscosity increased. Hwang and Cheng [154] (d

_{c}= 0.19 m, H

_{c}= 2.5 m, d

_{0}= 1 mm) investigated the ɛ

_{G}structure in highly viscous Newtonian and non-Newtonian media and observed that ɛ

_{G}decreases when the viscosity increases. Zahradnik et al. [8] (d

_{c}= 0.15 m, H

_{0}= 0.53 m, d

_{0}= 0.5 mm) found that moderate/high viscosities (3 < μ

_{L}< 110 mPa·s) decrease ɛ

_{G}. Yang et al. [57] (d

_{c}= 0.15 m, H

_{c}= 1.7 m, d

_{0}= 0.7 mm) investigated the influence of the viscosity (1 < μ

_{L}< 31.5 mPa·s) on ɛ

_{G}by using a viscosity-increasing agent; they observed that ɛ

_{G}decreases when the viscosity increases. It is worth noting that the influence of the viscosity may be also described in terms of non-Newtonian behavior (and vice-versa). This point has been discussed by Besagni et al. [11] by coupling the results obtained by Olivieri et al. [61] with the “dual effect of viscosity”; the non-Newtonian related stabilization of the homogeneous flow regime proposed by Olivieri et al. [61], along with the “dual effect of viscosity” relationship between the flow regime transition and the gas hold-up, help to explain the higher ɛ

_{G}for the non-Newtonian solutions observed by Godbole et al. [152].

#### Active Compounds

_{G}increases mainly as a consequence of homogeneous flow regime stabilization for both electrolytes [8,76,77,79,80,155,156,157,158] and ethanol [19,20,32,82,83,114,125,159,160,161] solution. ɛ

_{G}also increases because the active material adsorbed at the surface is pushed towards the back of the bubble. This causes a surface tension gradient, opposing the tangential shear stress and thus increasing the drag and reducing the rise velocity. It is worth noting that some authors have also observed a dual effect for active compounds at high concentrations.

#### Inorganic Compounds

_{G}and the concentration ratio, c ≤ c

_{t}. Zahradník et al. [8] (d

_{c}= 0.14, 0.15 and 0.29 m, H

_{c}= 2.6 m) studied the influence of nine electrolytes and found that ɛ

_{G}grew continuously for c ≤ c

_{t}, but little change in ε

_{G}was observed for c > c

_{t}. These findings were also observed by Besagni et al. [28,162] in a large scale bubble column, operated both in batch mode and counter-current mode (see, for example, Figure 23). A similar dependence was found for all electrolytes at c = c

_{t}. Ribeiro and Mewes [76] (d

_{c}= 0.12 m, H

_{c}= 1.25 m) tested three electrolytes (NaCl, Na

_{2}SO

_{4}, NaI) with four concentrations in the “coalescent flow regime”, and ɛ

_{G}was found to increase up to the critical concentration. Kelkar et al. [77] (d

_{c}= 0.154 m, H

_{C}= 3.25 m) reported an increase in ɛ

_{G}and a negligible effect of electrolytes (NaCl, CaCl

_{2}and Na

_{2}SO

_{4}) above the critical concentration. Ruzicka and co-authors [69,70] (d

_{c}= 0.14 m, H

_{0}= 0.4, d

_{0}= 0.5 mm) observed a dual effect of inorganic compounds on ɛ

_{G}and flow regime transition with respect to the electrolyte concentration. The reader may refer to [75] as well as to refs. [69,70] for a further literature review.

#### Organic Compounds

_{G}values and the transition point (see refs. [19,20,32,82,83])—regardless of the ɛ

_{G}value, all the studies agree in regard to the increase in ɛ

_{G}using ethanol solution. On the other hand, a limited number of studies considered the foaming phenomena which may occur when using organic compounds [161]. Indeed, foaming phenomena may occur as discussed in the experimental studies of Besagni et al. [7,12,37] and as predicted by the theoretical approach of Shah et al. [161]; in this respect, Figure 24 displays the gas hold-up curve in batch mode for the different aspect ratios (5 ≤ AR ≤ 12.5) and EtOH concentrations. Compared with the air-water system, the gas hold-up increases with the addition of ethanol; starting from a low ethanol concentration, the gas hold-up increases compared with the air-water case. If the ethanol concentration is further increased, foam phenomena are observed (i.e., Figure 24b and Figure 25b—reference codes are in Table 3). It is also worth noting that foaming phenomena are not only related to the concentration, but to the aspect ratio itself. A discussion on these experimental results can be found in our previous studies.

_{c}= 0.154 and 0.3 m, H

_{C}= 2.44 and 3.35 m) studied the influence of ethanol (0.5–2.4%

_{wt}) on ɛ

_{G}. Zahradnik et al. [159] (d

_{c}= 0.15 m, H

_{C}= 1.8 m) observed that the increase in ɛ

_{G}was higher with longer hydrophobic chains of n-alcohols. Rollbush et al. [114] (d

_{c}= 0.16 m, H

_{C}= 1.8 m) reported an increase in ɛ

_{G}using ethanol (1%

_{vol}) up to 220%, compared with tap water. Hur et al. [163] investigated the effect of adding alcohol in bubble columns on ɛ

_{G}: The continuous mode decreased ɛ

_{G}in the homogeneous flow regime, whereas, in the heterogeneous flow regime, it increased ɛ

_{G}. The batch mode was less effective than the continuous mode in the heterogeneous flow regime. Pjoteng et al. [125] (d

_{c}= 0.10 m, H

_{C}= 1.8 m) reported an increase in ɛ

_{G}with the addition of ethanol (0.5%

_{wt}), up to a pressure of 9 MPa. Guo et al. [164] studied a small diameter (d

_{c}= 0.1 m) and small-scale (H

_{c}= 1.8 m) bubble column. They found that ɛ

_{G}first increases, and then decreases, with an increasing ethanol concentration, owing to the non-linear effect of the alcoholic solution on the “coalescence-induced” and “non-coalescence-induced” bubbles. Similar results were presented by Syeda et al. [165] and by Shah et al. [161]. These studies [161,164,165,166] explain the trend of ɛ

_{G}with respect to the mole fraction of one of the two components in the liquid mixture, based on the model proposed by Andrew [167].

#### 2.2.4. Influence of Gas Properties

_{G}is discussed in terms of the gas density (ρ

_{G}) [169]. A variation in the gas density may result from high pressure operation [2,56,84,89,94,114,169,170], from the use of different gases [47,80,89,94,107,166,168,169,171,172,173] or from a combination of both [89,94,169]. As the influence of the operating pressure is discussed in another section (Section 2.2.1), we here discuss the influence of the gas employed. Apart from Shulman and Molstadz [171]—who reported no effect of the gas properties—the literature agrees that, in columns of at least d

_{c}= 0.15 m, higher ρ

_{G}result in higher ε

_{G}[56]; a variation in ρ

_{G}means a variation in the residence time of the dispersed phase [114]. Also, the influence of ρ

_{G}over ɛ

_{G}has frequently been attributed to differences in bubble size distributions at the gas sparger [166,168,173]. Bagha et al. [166] (d

_{c}= 0.0382 m, H

_{C}= 1.14 m) and Koetsier et al. [172] reported that gases with higher density have higher ɛ

_{G}. Akita and Yoshida [47] (square column, 0.15 × 0.15 m) tested four gases (air, He, O

_{2}, CO

_{2}) with water and showed that the effect of ρ

_{G}on ɛ

_{G}can be neglected, although ɛ

_{G}with helium are slightly lower for higher gas velocities. Hikita et al. [80] (d

_{c}= 0.10 m, H

_{C}= 1.50 m) observed an effect of gas density by testing different gases (air, H

_{2}, CO

_{2}, CH

_{4}, C

_{3}H

_{8}and H

_{2}–N

_{2}mixtures). Sada et al. [107] (d

_{c}= 0.073 m, H

_{C}= 0.95 m) concluded that the influence of ρ

_{G}on ɛ

_{G}should be considered in gas–molten salt systems because of the low gas–liquid density ratio at high temperatures. Özturk et al. [173] (d

_{c}= 0.085 m, H

_{C}= 0.95 m) considered different gases (Air, H

_{2}, He, N

_{2}, and CO

_{2}) and observed that ɛ

_{G}increases when ρ

_{G}increases. Reilly et al. [94] (d

_{c}= 0.15 m, H

_{C}= 1.7 m) investigated various gases at atmospheric pressure (Air, Ar, He, N

_{2}, and CO

_{2}) and conducted experiments at higher pressures (d

_{c}= 0.15 m) using N

_{2}and air in isoparaffinic solvents—higher ρ

_{G}extended the homogeneous flow regime and increased ɛ

_{G}, especially in the heterogeneous flow regime. Krishna et al. [89] (d

_{c}= 0.16 m, H

_{C}= 1.2–4.0 m) observed that ɛ

_{G}increases with ρ

_{G}as a result of both higher pressures and higher gas molecular weights (He, N

_{2}, Ar, CO

_{2}, and sulfur hexafluoride gases) in water. Jordan and Schumpe [169] (d

_{c}= 0.1 m, H

_{C}= 2.4 m) adjusted ρ

_{G}by increasing the pressure and by using mixtures of He and N

_{2}; they concluded that the differences in ɛ

_{G}could be attributed only to differences in ρ

_{G}and not to the nature of the gas. Hech et al. [168] (d

_{c}= 0.15 m, H

_{C}= 2.2 m) tested different gases (air, He, N

_{2}, and CO

_{2}) and found that that higher gas densities lead to higher ɛ

_{G}.

#### 2.2.5. Influence of the Pressure and Temperature

#### Pressure

_{G}[1,20,56,84,85,86,88,89,90,92,93,94,95,96,97,98,116,125,131,132,169,170,174,175,176,177,178,179,180,181,182,183,184]. Moreover, the influence of the pressure over gas hold-up is non-linear with U

_{G}; whereas, in the heterogeneous flow regime, ɛ

_{G}always increases, in the homogeneous flow regime, some authors have observed an increase–even if less than in the heterogeneous flow regime–and others have observed no effect [1]. The literature also agrees when reporting a plateau for a certain value of pressure, above which, no significant effect of pressure on ɛ

_{G}is observed. The value of the plateau depends on the operating conditions [132], and typically, values are 6 MPa [185], 7 MPa [180] and 10 MPa [178]. The effect of pressure can be related to the liquid phase (i.e., µ

_{L}and ρ

_{L}) and gas phase (i.e., ρ

_{G}) parameters. The liquid phase has a limited influence [180]; the parameters of the gas phase have a higher influence [1]. It is worth noting that the gas density also has an influence over ɛ

_{G}at ambient pressure (Section 2.2.4). Ishiyama et al. [177] (CO

_{2}/water) reported a negative effect of pressure on ɛ

_{G}at pressures above 0.8 MPa in a heterogeneous flow regime; this behaviour was explained by an increase in μ

_{L}with dissolved CO

_{2}.

#### Temperature

_{c}, over ɛ

_{G}[56,86,117,132,177,180,181,186,187,188]; on the other hand, Pohorecki et al. [187] observed no effect of T

_{c}—possibly caused by the evaporation—and some authors observed a decrease of ɛ

_{G}T

_{c}increased [78,136,189,190]. It is worth noting that the studies reporting a decrease in ɛ

_{G}used at low U

_{G}and, in accordance with Yang et al. [190], turbulence is low under these conditions,. Under these conditions, an increase in T

_{c}, reduces the viscosity and weakly promotes turbulence; this promotes collision and increases film drainage speed and thus, coalescence. Grover et al. [78] (d

_{c}= 0.1 m, H

_{c}= 1.5 m, air-water) observed a negative effect of T

_{c}at atmospheric pressure (T

_{c}between 303 and 353 K), although for T

_{c}> 323 K, the effect of temperature is limited. Deckwer et al. [136] (d

_{c}= 0.041 m, N2/Paraffin system) observed a decrease in ɛ

_{G}until a plateau was reached. For the same range of U

_{G}and for the same d

_{c}as that of Deckwer et al. [136], Kölbel et al. [189] (d

_{c}= 0.041 m, H

_{c}= 4 m, H

_{2}, Ethylene/C

_{13–18}) observed the same decrease in the homogeneous flow regime, but no effect in the heterogeneous flow regime. Yang et al. [190] (d

_{c}= 0.041 m, H

_{2}, CO/Paraffin) observed a decrease in ɛ

_{G}with an increase in both T

_{c}(293–523 K) and p

_{c}(1–3 MPa).

#### 2.2.6. Gas Hold-Up Correlations

_{G}in two-phase bubble columns. This section summarizes and described some frequently used ɛ

_{G}correlations. In particular, Table 4 summarizes the range of applicability of the reported correlations. In particular, in the following sub-sections, the main gas hold-up correlations are presented grouped in six main schemes of correlation: (a) the scheme of correlations by Lockett and Kirkpatrick; (b) the scheme of correlations based on ε

_{G}; (c) the scheme of correlations by Akita and Yoshida [47]; (d) the scheme of correlations for Newtonian and non-Newtonian liquid phases; (d) the scheme of correlations based on the work of Syeda et al. [165]; (e) the scheme of correlations for bubble column scaling-up.

#### Scheme of Correlations by Lockett and Kirkpatrick

_{b}= 0.0005, none of the proposed equations are completely satisfactory. They concluded that the Richiardon and Zaki equation with exponent 2.39 is adequate if a correction factor is also included to take bubble deformation into account. Therefore, they proposed the following correlation:

#### Scheme of Correlations Based on ε_{G}: From Hugmark towards Reilly et al.

_{G}values in air-water bubble columns in both the homogeneous flow regime and the transition/heterogeneous flow regime:

_{G}(in the homogeneous flow regime) the denominator is approximately constant and ɛ

_{G}increases linearly with U

_{G}. At relatively high values of U

_{G}(in the transition/heterogeneous flow regime), ɛ

_{G}decreases as U

_{G}increases. A scheme of correlation, similar to the one proposed by Mashelkar [192], was proposed by Kato and Nishiwaki [193], as reported by [24,193] and shown in Equation (6):

_{G}is less than approximately 5 [cm/s], they almost agree with the values calculated by Kato et al. [134] when d

_{c}= 5.50 m:

_{G}, in air–various liquid systems with Equation (8):

^{−4}and 5 × 10

^{−4}, respectively. The vapor pressure (P

_{v}) of the solvent has been used as a correlating term to account for the temperature effect:

_{G}and suggested the following equation, which takes into account the liquid properties, the column dimensions and the operating variables:

_{b}), usually ranging between 0.002–0.004 m, has been found to have no significant effect on ɛ

_{G}. Therefore, d

_{b}= 0.0003 m can be used in the equation.

_{G}data for air in several liquids and found that their own and the previous investigators’ data can be correlated using Equation (11) as a function of the dimensionless gas velocity, ${U}^{\ast}={u}_{G}{\left[{{\rho}_{L}}^{2}/\sigma ({\rho}_{L}-{\rho}_{G})g\right]}^{1/4}$ as follows:

_{G}in bubble columns with various gases and pure liquids or aqueous solutions of non-electrolytes as well as for air in aqueous solutions of various electrolytes. These data have been used to study the effects of the physical properties of gas and liquid on the gas hold-up. As a result, a new dimensionless correlation has been presented and reads as follows:

_{G}is increased by the presence of electrolytes in water. It assumes different values depending on the ionic strength (I). As the exponents show, the physical properties of the gas phase are of little relevance, but, if omitted, the mean deviation of the correlated data rises to 15% as opposed to 4% when included. Such properties could affect the relative gas hold-up, especially under high pressure conditions, but no systematic research has yet been done on this particular aspect.

_{G}, σ, ρ

_{G}, ρ

_{L}, μ

_{G}, μ

_{L}) of both the gas and liquid involved in their study. The result of their analysis using a non-linear multi-regression technique is given by Equation (14), and is in good agreement with their experiment and also with thirteen literature datasets selected by the authors to correspond to its region of validity. For example, only data for columns of at least 0.15 m in diameter were considered:

^{3}/s] is used instead of the surface tension (σ):

#### Scheme of Correlations Based on ε_{G}/(1 − ε_{G})^{4}: The Akita and Toshida Scheme

_{1}= 0.20 for non-electrolyte systems, whereas c

_{1}= 0.25 has been suggested for electrolyte solutions which cause a larger ɛ

_{G}compared with non-electrolyte liquids (as discussed in the previous sections). Equation (6) is still considered the state of the art in gas hold-up correlations. Mersmann [199], as reported by [24], proposed a correlation similar to Equation (16) in [47]:

#### Scheme of Correlations for Newtonian and Non-Newtonian Liquid Phases

_{br}is the bubble rising velocity. It predicts a dependency of the gas hold-up on the column diameter which is similar to that in the model proposed by [204] but somewhat smaller:

_{a}is the apparent kinematic viscosity and the apparent dynamic viscosity is expressed in terms of the consistency index, K, as follows ${\mu}_{a}=K{\dot{\gamma}}^{n-1}=K{(2{U}_{G}/{d}_{c})}^{n-1}$:

_{G}/ε

_{G}°, where ε

_{G}° is the calculated values from Equation (32) for the electrolyte solutions. This correction factor has been found to depend on the operating temperature (Table 6). The vapor pressure of liquid (P

_{v}) was used to indicate the effect of temperature on the gas hold-up, since the change tendency of the gas hold-up with temperature is basically similar to the relationship between P

_{v}and temperature. An increase in temperature results in an increase in the gas hold-up, particularly above 75 °C, 65 °C and 80 °C for the air-water, air–alcohol and air–5% NaCl systems, respectively. Zou et al. [209] also observed that ε

_{G}(air–alcohol) > ε

_{G}(air–5%NaCl) > ε

_{G}(air-water). Finally, Kawase et al., in their review on bubble column reactors, [140] reported the correlation proposed by [210] and given by Equation (33):

#### Scheme of Correlations of Syeda et al.

_{1}) and the exponent (b) have been determined experimentally in combination with the model proposed for binary mixtures (Equation (35)), where We

_{1}and We

_{2}represent the Weber numbers of the pure liquids with lower and higher surface tensions, respectively, in the binary mixtures:

_{1}and b have been found to be 1.334 and 0.032, respectively. The Weber numbers used in the two equations have been derived strictly for pure liquids. Changes in bubble size and bubble mobility that might affect the gas hold-up of the mixtures are included in the frothing parameter, ${crk}^{2}/\sigma $.

#### Scheme of Correlations for Bubble Column Scaling-Up

_{“coalescence}

_{induced}

_{”}

_{bubbles}and U

_{“non-coalescence}

_{induced}

_{”}

_{bubbles}and are the rising velocity of the “coalescence induced” bubbles and the “non-coalescence induced” bubbles, respectively. The different terms in Equation (36) were derived by changing the equation of Wilkinson et al. [56] as follows:

_{H}–Equation (1)), the dimensionless bubble column height (AR, aspect ratio refs. [28,144]), and the dimensionless sparger opening (d*

_{o}); the last one is defined as the ratio between the bubble size produced at the gas sparger, estimated by the correlation in ref. [212], as the diameter of the gas sparger opening.

#### 2.3. The Bubble Size Distribution and Shapes

_{G}are caused by the modifications to the BSDs connected to the bubble interface properties [65,214,215]. Besides this concept, using a practical point of view, apart from ɛ

_{G}, another fundamental parameter for the bubble column fluid dynamics is the bubble size distribution (BSD) which provides, along with ɛ

_{G}, an evaluation of the interfacial area [140] and is used in computational luid dynamics (CFD) for model set-up and validation [9]. The variation in the BSD is among the main reasons for the effects of the operating parameters on ɛ

_{G}and the flow regime transition (i.e., stabilization of the homogeneous flow regime, increase in ɛ

_{G}and, eventually, the plateau effect). In addition to the BSD, the bubble shapes should be considered (i.e., the aspect ratio). Indeed, the interface shape and size is important for characterizing multi-phase flows properly (i.e., heat and mass transfer at the interface and the closures in computational fluid dynamics). Generally speaking, the bubble shape depends on the system variables (as discussed by Haberman et al. [216]): the bubble rising velocity (u

_{b}), the equivalent bubble diameter (d

_{eq}), the difference between the phase densities (ρ

_{L}–ρ

_{G}), the liquid viscosity (μ

_{L}), the gas phase viscosity (μ

_{G}), the surface tension (σ) and the gravity acceleration (g). These variables are coupled into non-dimensional numbers:

- the Eötvös number:$$Eo=\frac{g{\rho}_{L}{d}_{eq}^{2}}{\sigma}$$
- the Morton number (defined only by the properties of the phases):$$Mo=\frac{g\left({\rho}_{L}-{\rho}_{G}\right){\mu}_{L}^{4}}{{\rho}_{L}^{2}{\sigma}^{3}}$$
- the Reynolds number:$$Re=\frac{{\rho}_{L}{u}_{b}{d}_{eq}}{{\mu}_{L}}$$
- the Weber number:$$We=R{e}^{2}{\left(\frac{Mo}{Eo}\right)}^{-\frac{1}{2}}$$

_{b}) numbers. For a constant Morton number, the bubble shape evolves from spherical to ellipsoidal to cap-shaped when increasing its equivalent diameter (corresponding to the Eötvös number) (see, for example, the flow visualistaions in refs. [7,11]). The bubbles may be considered spheres when the surface tension and/or the viscous forces are larger than the inertial forces. Ellipsoidal bubbles are defined as oblate spheroid (an axisymmetric ellipse with respect to the minor axis). For example, Chao [218] argued that air bubbles in the water phase maintain a spherical shape up to Re = 400. When Re > 400, bubbles become flatted: first spheroidal and subsequently, spherical-cap bubbles. The rule of thumb is that bubbles having aspect ratio larger than 0.9 are spherical. Several attempts have been made in the literature to correlate the aspect ratio to dimensionless parameters (see refs. [219,220,221]): Eo [222,223], We [223,224,225], Tadaki number [217,226], the Mo and the Eo numbers [227] and the Re and the Eo numbers [228]. Unfortunately, most of these correlations were developed for single bubbles/drops, and they may not be suitable for dense bubbly flows, as discussed by Besagni et al. [11,53,229]. To this end, Besagni et al. [12,229] proposed a correlation for the bubble aspect ratio (E) for dense bubbly flows which reads as follows:

#### 2.3.1. Influence of the Bubble Column Design and Operation

#### Superficial Gas Velocity

_{b}is far from being understood and controversial results have been published. It is generally admitted that—in the homogeneous flow regime—d

_{b}increases [156] and, after the transition, “coalescence induced” bubbles appear, whereas the contribution of the “non-coalescence induced” bubbles remain constant [18]. Some authors reported an increase in d

_{b}with superficial gas velocity, both in the homogeneous and heterogeneous flow regimes [35,38,85,87,127,183,232,233,234,235,236,237,238,239], while other authors reported no effect of U

_{G}over d

_{b}[188,240] and some others reported a decrease in the bubble size [241]. In the literature, both unimodal [239] and bimodal [242,243,244,245] BSD have been found depending on the gas sparger design and operating conditions. Lau et al. [242] and Wongsuchoto et al. [244] observed a transition from unimodal BSD to bimodal BSD with increasing superficial velocity of the air. This change in the distribution of bubble size has been justified by increased coalescence [242] or break-up [244]. Besagni and Inzoli [46,52,53,229] described the BSDs in a polydispersed homogeneous flow regime in the batch and counter-current modes. The changes in the BSD (from bimodal to unimodal) in the counter-current mode were based on the force pushing the small bubbles toward the center of the pipe. This has been confirmed by the numerical studies of Lu et al. [246] and Lu and Tryggvason [247,248]. Generally, lot of parameters influence the bubble size distribution, and it is difficult to provide a general rule for BSD prediction. Similarly, the influence of the superficial gas velocity over the bubble shape (viz. the bubble aspect ratio, E) is far from being understood. For example, Figure 25, Figure 26 and Figure 27 present the experimental findings of the relationships between E and the gas and liquid superficial velocities, obtained by the PoliMi and HZDR research groups (see, for example, refs. [7,12,46,53,229,230,231,249,250,251]). These experimental findings concern air-water flow; experimental results concerning other working fluids are summarized in ref. [12].

_{eq}> 1 mm, E > 0.7). It can be observed that, in the different cases considered, changing U

_{L}and U

_{G}does not affect the E significantly. A detailed discussion of the above-mentioned experimental findings can be found in ref. [230] and thus, it is not repeated here. A valuable conclusion is that the relationship between bubble size and shape (in the boundaries of the homogeneous flow regime) mainly depends upon the liquid phase properties and not on the flow conditions (i.e., the gas and liquid flow rates). Generally, distinct differences are observed in smaller bubbles, whereas the aspect ratios approach each other with increasing bubble size. It is also worth noting that, in some flow conditions, it is found that as the bubble shape is not highly dependent on the flow conditions, it can be speculated that, in these cases, E mainly depends on the prevailing bubble size distribution, rather on the existing flow pattern [250].

#### Gas Sparger Design

#### 2.3.2. Influence of the Liquid Properties

#### Viscous Media

_{eq}< 1 mm and d

_{eq}> 20 mm [262], 1 mm < d

_{eq}< 10 mm and 10 < d

_{eq}< 150 mm [152], 0.7 mm < d

_{eq}< 10 mm and d

_{eq}> 10 mm [57]. Rahba et al. [62] reported bimodal BSD with large bubbles having d

_{eq}= 40–45 mm. For example, for viscosities beyond μ

_{L}> 30 mPa·s, the contribution of the small bubbles results in a further increase in the total ɛ

_{G}with increasing viscosity [59], instead of the levelling at a constant value, as reported by Eissa and Schugerl [150]. An exception to the above references is the study of La Rubia et al. [241], who observed a decrease of d

_{eq}from 4.6 to 4.2 mm while increasing U

_{G}. Besagni et al. reported a “dual effect of viscosity on BSDs”; the BSDs shifted towards low equivalent diameters for “low/moderate” viscosities and, for higher viscosities, cap-bubbles appeared in addition to the spherical–ellipsoidal bubbles. This behavior has been explained by two simultaneous effects: (i) at higher viscosities, coalescence is promoted and large bubbles appear; (ii) moderate viscosities stabilize and increase the boundary layer thickness between the bubbles, resulting in a decrease in the coalescence rate of the small bubbles (the momentum of the bubbles could not overcome the layer thickness). Similar behavior was observed by Yang et al. [57] in small scale bubble columns, using a “fine gas sparger”; they interpreted the increase in the number of small bubbles at moderate viscosities using the stabilization effect of the bubbly layers [60]. It is worth noting that increasing the liquid phase viscosity not only changes the prevailing bubble shape, but also the bubble dynamics (e.g., rising and horizontal velocities and bubble trajectories), owing to the balance between viscous and non-viscous forces [263]; the changes in the bubbly dynamics have been observed also from the point of view of aggregated bubbles [264]. Unfortunately, up to now, detailed experimental and numerical investigations on the bubble interface in viscous media have been quite limited, and there is a lack in fundamental knowledge, as reviewed in refs. [265,266]. Recently, Orvalho et al. [267] found that the bubble contact time increases with μ

_{L}, passing through three phases: (a) a rise at low μ

_{L}; (b) a jump at intermediate μ

_{L}; (c) a plateau at high μ

_{L}. This result seems to contradict with the above-mentioned findings and may be related to the diffuser flow conditions (i.e., the differences between bubble rising the bubble column and experiments on pair-wise bubbles); in regard to this concept, the interested reader may refer to the experimental study of Sun et al. [268]. Of course, future research studies should be devoted to performing 3D numerical simulations (e.g., see refs. [269]) and obtain local measurements, to study the bubble interface in viscous liquid phases and the bubble shapes and dynamics (e.g., see the pioneering study of Bhaga and Weber [166] and more recent papers [259,268]).

#### Active Compounds

_{b}is expected (along with an increase in ɛ

_{G}) [8,38,56,80,108,127,149,161,234,241,253,270,271]. Among the broader literature on the topic, we present some examples for the sake of clearness. Keitel et al. [270] found that the BSD shifts toward lower diameters when employing non-coalescing solutions. This was also observed by Shah et al. [161] (ethanol–water mixtures) and by Machon et al. [271], who observed a dependency on the active compound concentration. They found that, depending on the mixture concentration, the bubble diameter in water is much higher than in electrolyte and alcohol solutions (c > c

_{t}) or, if c ≈ c

_{t}, the diameter is midway between the one in water and the one for c > c

_{t}.

#### Influence of the Gas Properties

_{G}, when changing the type of gas, is due to changes in bubble size distribution [166,168,173]. Koetsier et al. [172] postulated that argon coalesces less than helium near gas spargers. Özturk et al. [173] linked the higher ɛ

_{G}to smaller bubbles formed at the gas sparger. Hecht et al. [168] concluded that the most probable causes of the differences in ɛ

_{G}are the changes in bubble size distributions (caused by the changes in the break-up and coalescence rate) as well as dissimilar bubble rise velocities.

#### 2.3.3. Influence of the Pressure and Temperature

#### Pressure

_{b}) [1], and some authors have reported a decreased bubble velocity and an increased number of bubbles [87,98,272]. The multiple effects of pressure were also found by Jiang et al. [178], who observed a plateau for ɛ

_{G}at approximately 10 MPa and a plateau for bubble diameter at approximately 1.5 MPa; this suggests that pressure has another effect that decreases bubble diameter. Lin et al. [180] (up to 16 MPa) observed a decrease in d

_{b}when pressure increased for a fixed U

_{G}(0.02 and 0.08 m/s): from d

_{b}= 2.7 mm (p

_{c}= 0.1 MPa) to 2 mm (p

_{c}= 3.5 MPa), 1 mm (p

_{c}= 7 MPa) and 0.8 mm (p

_{c}= 15.2 MPa). Schäfer et al. [183] (p

_{c}= 0.1–5.0 MPa) observed a decrease in d

_{b}while increasing pressure with different gas spargers at an ambient T

_{c}in the homogeneous flow regime. Letzel et al. [91,92,170] (p

_{c}= 0.1–1.3 MPa) reported that the pressure has an influence over large bubbles only, suggesting that, in the homogeneous flow regime, the influence of pressure may be also negligible. The influence of pressure over d

_{b}is related to the gas sparger employed—Maalej et al. [182] (p

_{c}= 0.1–5.0 MPa) reported a lower influence of pressure for porous gas spargers than for perforated gas spargers (because of the narrow BSD in “fine distributors”). Kölbel et al. [189] (p

_{c}= 0.1–0.6 MPa) attributed the absence of a pressure effect in both the homogeneous or heterogeneous flow regimes to the narrow BSD at the gas sparger. Oyevaar et al. [185] (p

_{c}= 0.1–8.0 MPa) observed higher effects of pressure in perforated gas spargers than in porous gas spargers. Idogawa et al. [176] (p

_{c}= 0.1–15.0 MPa) found that the influence of the gas sparger becomes lower as the pressure increases. The effect of the pressure is also influenced by the other operating parameters, such as liquid phase viscosity, the chemical reactions and high temperature. In these cases, the effect of pressure may be reduced by the opposite influence of the other parameters. Urseanu et al. [131] (p

_{c}= 0.1–1.0 MPa) reported a minor effect of pressure on ɛ

_{G}while working with viscous fluids, because of the opposite effect of pressure and viscosity on d

_{b}. Ishibashi et al. [273] (p

_{c}= 16.8–18.6 MPa) observed no effect of pressure in the presence of chemical reactions. Considering previous high pressure and temperature studies, it seems that there is a negligible influence of pressure, as reported by Soong et al. [188] (T

_{c}= 293–538 K, p

_{c}= 0.1–1.36 MPa), Sangnimnuan et al. [115] (T

_{c}= 437–537 K, p

_{c}= 4.5–15.0 MPa) and Pohorecki et al. [187,240] (T

_{c}= 303–433 K, p

_{c}= 0.1–1.1 MPa). These results may be explained as occurring due to the saturation of gas through the evaporation of liquid, resulting in an increase in d

_{b}[56,187,240].

#### Temperature

_{eq}is observed with an increase in temperature [180,183,188]. Lin et al. [180] observed a narrower BSD as temperature increased from 290 K to 351 K. Shafer et al. [183] (T

_{c}= 298–448 K) noticed a decrease in d

_{b}as long as p

_{c}was high compared to the saturation pressure and the evaporation was negligible. The role of evaporation was also discussed by Pohorecki et al [187], who found d

_{b}to not be dependent on T

_{c}(in the range T

_{c}= 303–433 K). Soong et al. [188] reported a decrease in d

_{b}when the temperature increased from 293 K (d

_{b}about 1.5 mm) to 538 K (d

_{b}about 0.4 mm). The reader should also refer to the section concerning viscous media for an insight into the influence of the liquid viscosity over d

_{b}.

#### 2.4. Local Flow Properties

_{G}, which is a function of both the radial and axial positions in the bubble column [275]. The axial and radial variation in ɛ

_{G}generate pressure changes, resulting in liquid flow recirculation. Therefore, the investigation of the local void fraction profiles provides an evaluation of the bubble column liquid phase recirculation [276]. The magnitude of the void fraction radial gradients and liquid velocity driven by the gas phase depend on U

_{G}, U

_{L}, the column design, the phase properties and the operating conditions. A review of the experimental data concerning the local void fraction profiles have been presented by Joshi et al. [276]. In general, three kind of profiles are found in the literature: wall peaked, flat and center peaked. In the homogeneous flow regime (and considering batch or co-current conditions), center peaked profiles occur because a large number of larger bubbles or agglomerates rise at a markedly higher velocity than smaller ones, and thus, they transport most of the gas, as reported by Besagni and Inzoli [34,46]. On the other hand, a wall peaked profile is due to a large number of small bubbles. This behavior is caused by the lift force, which changes its sign (for the air-water case, at d

_{b}= 5.8 mm), thus pushing the small bubbles towards the wall. This behavior is well known in the literature and was discussed by Tomiyama et al. [277] and Lucas et al. [278]. Increased gas flow rates resulted in greater profile curvature from the column wall towards the center as well as higher void fractions [34,46,125,276]. In particular, the existence of a pronounced radial ɛ

_{G}profile is among the main characteristics of the heterogeneous flow regime, which is characterized by strong liquid recirculation. Indeed, in this flow regime, the formation of “coalescence-induced” bubbles at higher gas flow rates leads to the above-mentioned increased curvature of the radial profiles. Conversely, flat void fraction profiles generally occur in the homogeneous flow regime using “fine gas spargers” under controlled conditions, as detailed by Mudde et al. [13]. In the literature, a large number of liquid circulation models have been proposed, for example, by Wu et al. [279], Nassos and Bankoff [280], Ueyama and Miyauchi [281] and Luo and Svendsen [282]. As stated before, the liquid velocity and liquid recirculation are strictly related to the local void fraction profiles. Also, the gas phase velocity is related to the local void fraction profile by means of the bubble size distribution.

#### 2.5. Interfacial Area

_{i}) is one of the most important parameter for multi-phase reactors. Like the gas holdup and the BSD, it depends on the geometry, operating conditions and the phase properties. The specific interfacial area and ε

_{G}are related with:

_{G}≤ 0.14, Equation (49) is simplified as follows:

#### 2.6. The Mass Transfer

_{L}a). It is worth noting that k

_{L}a is computed as the product of two parameters: (a) the liquid phase mass transfer coefficient k

_{L}; and (b) the gas–liquid interfacial area (a

_{i}). Accordingly, Martìn et al. [286] stated that the mass transfer rate is mainly determined by two mechanisms (viz. the bubble oscillations, related to the concentration profiles around the bubble, related to k

_{L}, and the contacting area, a). When considering previous literature, it is worth noting that bubble columns are mostly employed in slow reaction–absorption applications where the interphase mass transfer resistance on the gas side might be considered negligible compared with the gas–liquid side (k

_{L}) [287]. Generally speaking, up to now, only a few theoretical correlations for k

_{L}a prediction in two-phase bubble columns [288], numerical models [289,290,291,292] or correlation-based predictions [139,293] have been used.

#### 2.6.1. Physical Phenomena and Approaches

_{b}to bubble rise velocity; it characterizes the residence time of liquid elements at the interface and is used to estimate the liquid phase mass transfer coefficient (see, for example, refs. [300,301]); it is also worth mentioning that Nedeltchev et al. [288,296] assumed that the contact time depends on both the bubble surface and the rate of surface formation. Owing to the uncertainties in the estimation of the contact time, the penetration theory is inapplicable in different practical applications (e.g., turbulence may determine the renewal of the fluid layers surrounding the bubbles). It is also worth noting that Kawase et al. [302] proposed a semi-theoretical approach based on Higbie's penetration and Kolmogoroff's turbulence theories. Recently, Nedeltchev [297] proposed a modified version of the penetration theory that can be used to correctly predict the mass transfer coefficient in both the homogeneous flow regime and the transition flow regime.

#### Influence of the Bubble Column Design

**Gas sparger opening.**It is not surprising that the gas sparger has a large influence on k

_{L}a [24]; indeed, depending on the gas sparger openings, the prevailing flow regime changes (as previously discussed and observed by Besagni et al. [14,37]). The influence of k

_{L}a on the gas sparger openings has been observed, for example, by Koide et al. [138] and is displayed in Figure 29. k

_{L}a increases with a decrease in d

_{0}and increases (less than linearly) with an increase in ε

_{G}. The increase in the mass transfer with a decrease in the gas sparger opening is related to the prevailing flow regime determined by the gas sparger [14,37,45]—decreasing d

_{0}moves the prevailing BSD towards a lower bubble equivalent diameter and thus, increases the interfacial area. Hence, a good distribution of bubbles across the reactor increases the efficiency of the gas flow rate on k

_{L}a [286,303]. The relationship between k

_{L}a and ε

_{G}is related to the shape of the gas hold-up curve, depending on the flow regime; indeed, the bubble column fluid dynamics are highly influenced by the sparger design in the homogeneous flow regime, but not in the heterogeneous flow regime [28,111]. From these observations, it is clear that highly efficient bubble columns should be operated in the homogeneous flow regime, rather than in the heterogeneous flow regime, where the dispersed phase is transported through the column as large “coalescence-induced” bubbles. These “coalescence-induced” bubbles have a lower residence time, leading to a decreased conversion process. In other words, the further enhancement of ε

_{G}in the heterogeneous flow regime is marginal and certainly not commensurate with the imposed increase in gas flow rate.

**Column diameter.**Deckwer and Schumpe [287] found that the dependence of k

_{L}a on the column diameter exists only for column diameters smaller than 0.6 m. Conversely, Akita and Yoshida [47] (Figure 30 and Figure 31) found that the mass transfer coefficient is related to the bubble column diameter up to a diameter of approximately 0.15 m, which is in agreement with the “large diameter” concept described in Equation (1) and the experimental results of Vandu and Krishna [304] (Figure 32).

#### Influence of the Bubble Column Operating Conditions

_{L}a are discussed. In particular, the gas flow rate and the operating pressure are considered and commented on. It is known that the relationship between k

_{L}a and U

_{G}is well described by Equation (52):

_{L}a and U

_{G}is less than linear, in agreement with the considerations proposed in Section 2.6.1—the relationship between k

_{L}a and ε

_{G}is related to the shape of the gas hold-up curve, depending on the flow regime; indeed, in the transition/heterogeneous flow regimes, the dispersed phase is transported through the column as large “coalescence-induced” bubbles, having a lower residence time and thus, a lower mass transfer rate. This observation is in agreement with the experimental observations reported by Letzel et al. [170], Kang et al. [179] and Ozturk et al. [173] (Figure 33, Figure 34 and Figure 35). On the other hand, it has been observed that increasing the operating pressure increases k

_{L}a, owing to the decrease in bubble size (as previously stated). Figure 36 proposes a summary of previous literature concerning the relationship between k

_{L}a and U

_{G}.

#### Influence of the Liquid and Gas Phase Properties

_{L}a. Indeed, the liquid viscosity, if above the maximum value dictated by the “dual effect”, increases the mean diameter of the bubbles in the dispersion, thus reducing k

_{L}a. (as observed by Kang et al. [179]). In this condition, the lather bubbles are stable in the flow at higher viscosities. In addition, the liquid viscosity also decreases the liquid diffusivity (D

_{L}) [299]. In the case of oxygen transfer, D

_{L}is proportional to μ

_{L}

^{−0.57}[173]. It is worth noting that some authors have also considered the influence of the gas viscosity proposed by [306]. Finally, Ozturk et al. [173] (Figure 37 and Figure 38) and Gopal and Sharma [106] (Figure 39) studied the influence of the gas phase on k

_{L}a and observed a lower influence compared with the above-mentioned operating parameters.

#### 2.6.2. The Mass Transfer Correlations

_{2}SO

_{4}). Most of the correlations [24,47,285] for k

_{L}a prediction in gas–liquid bubble columns neglect the effect of the bubble column aspect ratio; in this regard, Zhao et al. [312] proposed empirical correlations that consider the effect of liquid height on k

_{L}a. The correlations proposed by Akita and Yoshida [47,285] (Equation (54)) and by Shah et al. [24] (Equation (55)) read as follows:

_{G}

_{,L}is the diffusivity coefficient between the gas/liquid phases. In their empirical correlation, Öztürk et al. [173] used the surface-to-volume mean bubble diameter as a characteristic length rather than the column diameter:

- Kawase and Moo-Young, 1987 [302]:$$\frac{({k}_{L}a){{d}_{c}}^{2}}{{D}_{G,L}}=0.452{\left(\frac{{\nu}_{L}}{{D}_{G,L}}\right)}^{1/2}{\left(\frac{{d}_{c}{u}_{G}}{\nu}\right)}^{3/4}{\left(\frac{g{{d}_{c}}^{2}{\rho}_{L}}{\sigma}\right)}^{3/5}{\left(\frac{{{U}_{G}}^{2}}{{D}_{T}g}\right)}^{7/60}$$
- Hikita et al., 1981 [306]:$$\frac{({k}_{L}a){u}_{G}}{g}=14.9{\left(\frac{{U}_{G}{\mu}_{L}}{\sigma}\right)}^{1.76}{\left(\frac{{{\mu}_{L}}^{4}g}{{\rho}_{L}{\sigma}^{3}}\right)}^{-0.248}{\left(\frac{{\rho}_{G}}{{\rho}_{L}}\right)}^{0.243}{\left(\frac{{\mu}_{L}}{{\rho}_{L}{D}_{G/L}}\right)}^{-0.604}$$
- Kang et al., 1999 [179]:$$({k}_{L}a)={d}_{c}{10}^{-3.08}{\left(\frac{{d}_{c}{U}_{G}{\rho}_{G}}{{\mu}_{L}}\right)}^{0.254}$$

## 3. The Experimental Techniques

#### 3.1. The Flow Regimes

_{G}-U

_{G}curve), the swarm velocity [7,11,34,46,51,52,53,75,76,89,91,229,315,316] and the drift flux [7,11,34,38,46,51,53,75,76,229,316,317,318] approaches. These methods are based on the analysis of ɛ

_{G}data. Other flow regime identification methods are the fractal [319], wavelet [320,321], Hurst’s [322,323], spectral and nonlinear chaos [93,320,324,325,326,327,328,329,330,331,332,333] and Kolmogorov entropy [332,333,334] approaches. Some authors have used methods based on the Shannon entropy, which measures the degree of indeterminacy in a system [335,336]. Nedeltchev identified the flow regimes based on the information entropy theory [26]. Finally, the detection of the flow regimes using local measurements was demonstrated by Shiea et al. [337]. The interested reader should refer to the recent study proposed by Tchowa Medjiade et al. [334] (viz. power spectral density, standard deviation, fractal analysis, Kolmogorov entropy); they found systematic differences between the various methods. In the current authors’ opinion, future studies should be devoted to proposing a comprehensive approach that is able to take into account all the possible flow regimes, without the possibility of subjective evaluation.

#### 3.2. The Gas Hold-Up

_{G}is the bed expansion technique [3]. This method is based on the measurement of the liquid free surface before and after aeration. Another well-known method is based on the measurement of the differential pressure between two or more points in the column [80,86,114,116,122,123,125,126,135,141,170,179,194,198,209], under the hypothesis of negligible acceleration and friction pressure losses, as discussed by Leonard et al. [1]. This method has been applied by a large number of authors. For example, Hikita et al. [80] determined ɛ

_{G}by measuring the static pressure at three points in the column at 0.25 m intervals, the lowest being 0.15 m above the gas sparger. Another method is the dynamic gas disengagement proposed by Schumpe et al. [18], which may also be used for investigating the structure of ɛ

_{G}(in terms of “coalescence induced” and “non-coalescence induced” bubbles) and for investigating the flow regime transition.

#### 3.3. The Bubble Size Distribution and Shape

_{G}exceeds 1%, more than 40% of the bubbles are overlapping [353,354]. Despite the proposed methods to deal with overlapping bubbles by different studies [242], no agreement has been found so far (i.e., some approaches may cause a reduction in the bubble sample size and/or an underestimation of the bubble diameter). Furthermore, by using the classical image analysis methods, only projected bubble images can be obtained; despite some proposals against this issue [355,356], a solution is far from being reached. For this reason, the use of 2D projected images is a common way of analyzing the bubble images [242,244,352,357,358,359]. Concerning the approximation of the ellipse of the bubbles, Lage and Espósito [358] have stated that the error in the measurement of each axis of the ellipse is approximately 6%. In addition, if considering the error introduced by the hypothesis of oblate spheroid and the optical distortion, Lage and Espósito have estimated that the experimental error in the determination of d

_{eq}is in the range of 10–15%. Finally, when investigating bubble size distributions, the number of bubbles that need to be sampled to achieve a reliable BSD should be considered [360]. Various studies have sampled different numbers of bubbles—in the range of 50 and 300 [38,244,358,361,362,363], and a sensitivity analysis on this issue has been presented in our previous paper [53].

#### 3.4. The Local Flow Properties

#### 3.5. The Mass Transfer

## 4. The Modeling Approaches

_{G}. It is worth noting that, at present, it is possible to model the flow regimes listed in Section 2.2.1 by using different modeling approaches (see the discussion below); unfortunately, a comprehensive model, able to simulate the whole range of operating conditions in bubble columns, is still missing.

#### 4.1. The Eulerian Multi-Fluid Approach

_{G}and the mean residence time of the disperse phase [420,439]. The lift force has three components, but the largest component acts perpendicular to the main bubble flow direction; in bubble columns, this means the radial direction. In particular, this force is responsible for the migration of small bubbles toward the column walls or the center of the pipe, depending on the bubble shape and size [277,278]. The turbulent dispersion force spreads out large bubbles from the pipe center and modulates peaks of small bubbles near the pipe walls [458]. Its magnitude is also high near distributor inlets, supporting the modeling of bubble dispersion near coarse gas spargers [429]. The wall lubrication force models the lift force close to the wall, thus pushing the bubbles away from it. Rzehak et al. [459] compared various formulations applied to vertical bubbly flow in a pipe and concluded that the inclusion of this force into the model is fundamental. The virtual mass force arises from the relative acceleration of an immersed moving object to its surrounding fluid; this force is often found to be negligible in bubble column simulations [421,428,437,439,449,460], and different studies have neglected it [420,430,440,441,446,450,458,461,462,463,464,465,466,467,468,469]. Recently, Ziegenhein et al. [10] demonstrated that the virtual mass force has an influence on the prediction of the turbulence intensity at higher flow rates. It is clear that an independent validation of each single force is not possible; for this reason, a complete set of interfacial forces should be used [9,10,443,444,445,447,459,470,471,472,473], as discussed by [443] and recently validated against a large set of experimental data [474].

#### 4.2. A Focus on the Population Balance Modeling

#### 4.2.1. The Approach

#### 4.2.2. The Governing Equations

_{b}and V

_{b}+ d(V

_{b}). u

_{b}is the local velocity of bubble volumes between V

_{b}and V

_{b}+ d(V

_{b}) at time t. Finally, $S\left(\overrightarrow{x},{V}_{b},t\right)$ is a source term, which reads as:

_{c}, S

_{b}S

_{ph}, S

_{p}, S

_{m}and S

_{r}are the bubble source/sink terms due to coalescence, break-up, phase change, pressure change, mass transfer and reaction, respectively.

_{c}, reads:

_{b}due to the coalescence of bubbles of volume, V

_{b}− V

_{b}′ and V

_{b}′; the second term is the death rate of bubbles of volume V

_{b}due to coalescence with other bubbles. Finally, Γ(V

_{b}, V

_{b}′) is the coalescence rate between bubbles of volume V

_{b}and V

_{b}′.

_{b}, reads:

_{b}due to the break-up of bubbles with volumes larger than V

_{b}; the second term is the death rate of bubbles of volume V

_{b}due to break-up. Finally, Ω(V

_{b}) is the break-up rate of bubbles of volume V

_{b}, m(V

_{b}′) is the mean number of daughter bubbles produced by break-up of a parent bubble of volume V

_{b}′ and P(V

_{b}, V

_{b}′) is the probability density function of daughter bubbles produced upon break-up of a parent bubble with volume V

_{b}′.

_{c}and S

_{b}are determined by modeling the bubble break-up and coalescence rates.

#### Bubble Coalescence Phenomena and Modeling

- collision between two bubbles trapping a liquid film;
- drainage of the liquid film;
- film rupture and coalescence.

#### Bubble Break-Up Phenomena and Modeling

_{b}

^{max}:

_{b}

^{max}= 10 cm in air-water systems in ambient conditions (therefore, interfacial instabilities are seldom considered). Finally, interfacial stresses must be considered—as the bubble rise velocity increases, the shearing-off events gain importance, and the interfacial shear force and drag force cause the shearing-off of small bubbles at the rim of large bubbles. Likewise, when the interphase relative velocity becomes too high and thus, the interfacial force is high enough, the bubbles become unstable and start to stretch in the direction of the destroying force (i.e., in counter-current mode). Generally speaking, in most cases, the carrier phase is turbulent; therefore, only the dominant break-up mechanism due to the turbulent fluctuations is considered [214]. Break-up models should also consider the distribution function of the daughter bubble size [501]. Some examples of daughter bubble size distributions are (i) statistical distribution (i.e., normal or the uniform); (ii) empirical distribution; or (iii) phenomenological distribution [214]. In this regard, the reader may refer to the experimental high-speed imaging of Solsvik et al. [502]: they conducted in an attempt to understand the break-up cascade; in particular, they found that the assumption of binary breakup in the PBM approach is acceptable only if the initial breakup is analyzed.

## 5. Conclusions, Guidelines and Outlooks

#### 5.1. Summary of the Literature Survey

_{G}. The bubble column design parameters influence the fluid dynamics because of the boundary conditions, as well as the end-effects (i.e., wall or outlet effects). Similarly, appropriate modeling approaches should be selected by taking into account the prevailing bubble size distributions in the system.

#### 5.2. Guidelines

- Different studies have investigated, in the few last decades, either local or global fluid dynamics properties; unfortunately, the studies concerning both local and global fluid dynamics properties are still limited in number. In this respect, experimental studies should always provide a multi-scale evaluation of bubble column fluid dynamics (i.e., by studying, at least, both gas hold-up and bubble size distribution);
- Experimental studies should always provide detailed information concerning the main bubble column design criteria (as listed in Section 2.3.2) in order to relate the experimental results to the geometrical scale of the bubble column (i.e., “large-diameter”, “small-diameter”, “coarse gas sparger”, fine gas sparger”, etc.); in particular, information on non-dimensional bubble column diameters, aspect ratios and gas sparger openings should be always provided
- In comparisons involving the flow regime transition, detailed information concerning the flow regime transition criteria should be specified; in addition, authors should carefully evaluate if the flow regime transition method is suitable with the applied definition of flow regime transition;
- When presenting a numerical approach, sensitivity studies on (a) interfacial closures; (b) time and (c) space discretization should be always be performed;
- The modeling approach of the dispersed phase (i.e., mono-dispersed, bi-dispersed, PBM) should always be related to the prevailing flow regime observed in the bubble column;

#### 5.3. Outlooks

- Proposing a precise mathematical description of the flow regime transitions, as listed and described in Section 2.2.2. This approach should take into account the role of instabilities in flow regime transitions, to support the mathematical description of the boundaries of the flow regimes; in particular, this approach should be applied to develop a comprehensive flow regime map;
- Performing comprehensive and multi-scale experimental studies to propose complete datasets for large-scale bubble columns (viz. gas holdup, bubble size distributions and shape data, and local flow properties); such datasets would provide a valuable basis to validate numerical approaches for scaling-up purposes;
- Understanding the influence of interfacial properties on the “bubble-scale” and clarifying how the “bubble-scale” influences the “reactor-scale”;
- Proposing a unified theory to explain all the dual effects observed in the literature (e.g., viscous liquid phases, organic compounds, inorganic compounds, etc.);
- Performing experimental studies concerning the influences of the operating conditions and phase properties on bubble shape in dense bubbly flow conditions; in this regard, approaches for studying the three-dimensional bubble shape should be developed;
- Performing a comprehensive comparison of the experimental techniques for both the global and the local fluid dynamic properties;

- Extending the validation of the “baseline” approaches, to establish a common numerical framework for both small scale and large scale bubble columns; the validation should consider both the local and the global fluid dynamics properties in bubble columns;
- Extending the validation of numerical approaches to model the mass transfer phenomena in bubble columns;
- Extending the validation of the numerical approaches to the heterogeneous flow regime.

## Author Contributions

## Conflicts of Interest

## Nomenclature

$AR=\frac{{H}_{0}}{{d}_{c}}$ | Aspect ratio |

$Eo=\frac{g\left({\rho}_{L}{-\rho}_{G}\right){d}_{eq}^{2}}{\sigma}$ | Eötvös number |

$Mo=\frac{g\left({\rho}_{L}{-\rho}_{G}\right){\mu}_{L}^{4}}{{\rho}_{L}^{2}{\sigma}^{3}}$ | Morton number |

$Re=\frac{{\rho}_{L}{Ud}_{eq}}{{\mu}_{L}}$ | Reynolds number |

${We=Re}^{2}{\left(\frac{Mo}{Eo}\right)}^{0.5}$ | Weber number |

${crk}^{2}/\sigma $ | Frothing parameters |

AR | Aspect Ratio |

BSD | Bubble Size Distribution |

CMC | Carboxymethyl cellulose |

DNS | Direct numerical simulations |

EtOH | Ethanol |

HZDR | Helmholtz–Zentrum Dresden–Rossendorf |

MEG | Monoethylene glycol |

NaCl | Sodium chloride |

PBE | Population Balance Equation |

PBM | Population Balance Model |

PoliMi | Politecnico di Milano |

RANS | Reynolds-averaged Navier–Stokes |

RSM | Reynolds Stress Models |

$\overline{u}$ | Local velocity | [m/s] |

a | Major axis of the bubble | [m] |

a_{i} | Interfacial area | [1/mm] |

B | Retarded Hamaker constant | [J m] |

c | Molar concentration | [mol/L] |

C | Parameter in Equation (31) | [-] |

c* | Equilibrium concentration | [mol/L] |

c_{EtOH}_{,wt} | Mass concentration of EtOH | [%] |

c_{MEG}_{,wt} | Mass concentration of MEG | [%] |

c_{t} | Molar transition concentration of NaCl | [mol/L] |

c_{wt} | Mass concentration | [kg/L] |

D*_{H} | Non-dimensional diameter (Equation (1)) | [-] |

D*_{H}_{,Cr} | Critical non-dimensional diameter | [-] |

d_{b}^{max} | Maximum stable bubble size | [mm] |

d_{c} | Diameter of the bubble column | [m] |

d_{eq} | Bubble equivalent diameter | [mm] |

D_{H} | Hydraulic diameter | [m] |

d_{o} | Gas sparger holes diameter | [mm] |

e | Parameter in Equation (7) | [-] |

E | Bubble aspect ratio | [-] |

f | Function | [-] |

F_{D} | Drag force | [kg/m^{2} s^{2}] |

F_{L} | Lift force | [kg/m^{2} s^{2}] |

F_{TD} | Turbulent dispersion force | [kg/m^{2} s^{2}] |

F_{VM} | Virtual mass force | [kg/m^{2} s^{2}] |

F_{WL} | Wall force | [kg/m^{2} s^{2}] |

g | Acceleration due to gravity | [m/s^{2}] |

h | Height along the bubble column | [m] |

H_{0} | Height of the free surface before aeration | [m] |

H_{c} | Height of the bubble column | [m] |

H_{D} | Height of the free surface after aeration | [m] |

K | Consistency index | [Pa s] |

k_{i} (i = 1, 2) | Coefficients in the aspect ratio correlation (Equation (44)) | [-] |

k_{L} | Volumetric mass transfer coefficient | [m/s] |

M_{I} | Momentum exchanges | [kg/m^{2} s^{2}] |

n* | Dimensionless concentration (Equation (2)) | [-] |

P | Probability density function in Equation (70) | [-] |

P_{v} | Vapor pressure in Equation (9) | [Pa] |

r_{b} | Bubble radius in Equation (2) | [mm] |

R_{b} | Gas constant | [J/mol K] |

S | Total source/sink term in the population balance equation | [m^{3}/s] |

S_{b} | Source/sink term due to break-up | [m^{3}/s] |

S_{c} | Source/sink term due to coalescence | [m^{3}/s] |

S_{m} | Source/sink term due to mass transfer | [m^{3}/s] |

S_{p} | Source/sink term due to pressure change | [m^{3}/s] |

S_{ph} | Source/sink term due to phase change | [m^{3}/s] |

S_{r} | Source/sink term due to reaction | [m^{3}/s] |

t | Time | [s] |

T | Temperature | [K] |

t_{G} | Mean residence time of the dispersed phase | [s] |

u | Mean rise velocity | [m/s] |

U | Superficial velocity | [m/s] |

U* | Dimensionaless gas velocity | [-] |

u_{b} | Local velocity of bubble volumes | [m/s] |

u_{br} | Bubble rising velocity in Equation (26) | [m/s] |

V | Volume | [m^{3}] |

V_{b} | Bubble volume in population balance equations | [m^{3}] |

z_{i} (i = 1, …, 5) | Coefficients in the aspect ratio correlation (Equation (44)) | [-] |

$\alpha $ | Void fraction in the Eulerian–Eulerian constitutive equations | [-] |

$\dot{\gamma}$ | Shear rate | [1/s] |

$\overline{\tau}$ | Viscous and Reynolds stresses | [kg/m s^{2}] |

β | Coefficient in Equations (6) and (7) | [-] |

γ | Coefficient in Equation (6) | [-] |

Γ | Coalescence rate | [m^{3}/s] |

ε | Hold-up | [-] |

ε | Parameter in Equation (12) | [-] |

μ | Dynamic viscosity | [kg/m s] |

μ_{a} | Apparent dynamic viscosity | [kg/m s] |

μ_{eff} | Effective viscosity | [kg/m s] |

ν | Bubble terminal velocity | [m/s] |

ν_{L} | Bubble terminal velocity | [m^{2}/s] |

ρ | Density | [kg/m^{3}] |

σ | Surface tension | [N/m] |

τ | Time scale | [1/s] |

χ | Kinematic surface tension | [m^{3}/s] |

Ω | Break-up rate | [m^{3}/s] |

c | Parameter related to the bubble column |

coalescence induced bubbles | Coalescence induced bubbles |

cr | Critical parameter |

G | Gas phase |

j | j-th dispersed phase in governing equations |

k | k-th continuous phase in governing equations |

L | Liquid phase |

Local | Local parameter |

non-coalescence induced bubbles | Non-coalescence induced bubbles |

swarm | Swarm parameter |

trans | Transition point (it refers at the homogeneous flow regime) |

trans,I | First transition point (end of the homogeneous flow regime, defined when considering both the first and the second regime transitions) |

trans,II | Second transition point (end of the transition flow regime, defined when considering both the first and the second regime transitions) |

wt | Mass concentration |

z | Generic phase in governing equations |

→ | Vector quantity |

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**Figure 3.**Flow regime transitions from the homogeneous flow regime towards the transition flow regime.

**Figure 4.**Influence of the aspect ratio (AR) on flow regime transition points—data obtained by [28].

**Figure 11.**Influence of AR and n* (as defined in Equation (2) on flow regime transition points—data obtained from ref. [12].

**Figure 16.**Pipe sparger in open tube configuration—data from ref. [34].

**Figure 17.**Pipe sparger in annular gap configuration—data from ref. [34].

**Figure 18.**Spider sparger—data from ref. [46].

**Figure 19.**Influence of AR in large diameter bubble columns with spider sparger—data from ref. [28].

**Figure 20.**Influence of the aspect ratio in large diameter bubble columns—studies from the literature.

**Figure 21.**Influence of AR in large diameter bubble columns with spider sparger—data from ref. [37].

**Figure 23.**Influence of NaCl on gas hold-up (batch mode)—data from ref. [28].

**Figure 24.**Influence of EtOH on gas hold-up—data from ref. [12].

**Figure 25.**Influence of the gas superficial velocity (HZDR, Helmholtz-Zentrum Dresden-Rossendorf, bubble columns) on the E-batch mode only.

**Figure 26.**Influence of the gas and liquid superficial velocities on E (PoliMi bubble columns [53]).

**Figure 28.**Influence of the liquid phase properties on the interfacial area—data from ref. [12].

**Figure 30.**Influence of the bubble column diameter on k

_{L}a—data from Akita and Yoshida [47]−0.15 M Na

_{2}SO

_{3}—AIR 20 °C.

**Figure 31.**Influence of the bubble column diameter on k

_{L}a—data from Akita and Yoshida [47]—water and O

_{2}, 30 °C.

**Figure 34.**Influence of the operating pressure on k

_{L}a—data from Kang et al. [179]—even distribution experimental data.

**Figure 35.**Influence of the operating pressure on k

_{L}a—data from Kang et al. [179]—even distribution experimental data.

**Table 1.**Literature studies (reference codes for Figure 14).

Code | Reference | d_{c} [m] | Aspect Ratio [-] | Sparger |
---|---|---|---|---|

R1 | [46] | 0.24 | 22.1 | Spider Sparger—d_{0} = 1–3.5 mm |

R2 | [105] | 0.46 | 6.63 | Ring Sparger—d_{0} = 0.5 mm |

R3 | [105] | 1.07 | 2.85 | Ring Sparger—d_{0} = 0.76 mm |

R4 | [47] | 0.152 | 26.32 | Single Hole—d_{0} = 5 mm |

R5 | [49] | 0.05 | 30 | Single Nozzle—d_{0} = 5 5 mm |

R6 | [49] | 0.05 | 30 | Multiple Nozzle—d_{0} = 0.65 mm |

R7 | [106] | 0.2 | 4 | Single Nozzle—d_{0} = 6 mm |

R8 | [106] | 0.2 | 4 | Ring Sparger—d_{0} = 1 mm |

R9 | [106] | 0.6 | 1 | Ring Sparger—d_{0} = 2 mm |

R10 | [106] | 0.6 | 1 | Ring Sparger—d_{0} = 3 mm |

R11 | [107] | 0.073 | 13 | Single Nozzle—d_{0} = 1.5–2.7–5.7 mm |

R12 | [108] | 0.3 | 16.7 | Perforated Plate—d_{0} = 1.5 mm |

R13 | [108] | 0.3 | 16.7 | Single Sparger—d_{0} = 25.4 mm |

R14 | [109] | 0.23 | 5.3 | Multiple Nozzle—d_{0} = 1 mm |

**Table 2.**Literature studies (reference codes for Figure 15).

Code | Reference | d_{c} [m] | Aspect Ratio [-] | Sparger |
---|---|---|---|---|

R1 | [46] | 0.24 | 22.1 | Spider Sparger—d_{0} = 1–3.5 mm |

R2 | [52] | 0.24—Annular gap | 22.1 | Pipe Sparger—d_{0} = 3.5 mm |

R3 | [47] | 0.152 | 26.32 | Single Hole—d_{0} = 5 mm |

R4 | [110] | 0.305 | - | Single Hole—d_{0} = 1.66 mm |

R5 | [110] | 0.127 | - | Single Hole—d_{0} = 1.66 mm |

R6 | [49] | 0.05 | 30 | Single Nozzle—d_{0} = 5 5 mm |

R7 | [111] | 0.385 | 7 | Sieve Plate—d_{0} = 1 mm |

R8 | [111] | 0.385 | 7 | Sieve Plate—d_{0} = 1.5 mm |

R9 | [111] | 0.385 | 7 | Sieve Plate—d_{0} = 3.0 mm |

R10 | [111] | 0.385 | 7 | Sieve Plate—d_{0} = 6.0 mm |

R11 | [112] | 0.0707 | 12–33 | Single Nozzle—d_{0} = 2.25–7 mm |

R12 | [112] | 0.30 | 4–6.3 | Single Nozzle—d_{0} = 1.48–3.00 mm |

R13 | [108] | 0.3 | 16.7 | Single Sparger—d_{0} = 25.4 mm |

R14 | [18] | 0.3 | 12 | Single Sparger—d_{0} = 1 mm |

**Table 3.**References codes for Figure 24 data.

Code | Description |
---|---|

R0 | c_{EtOH}_{,wt} = 0%—Air-water gas hold-up curve |

R1 | c_{EtOH}_{,wt} = 0.3%—Gas hold-up curve measured after waiting 30 s for every gas hold-up measurement point from low to high gas flow rate: run 1. |

R2 | c_{EtOH}_{,wt} = 0.3%—Gas hold-up curve measured after waiting 30 s for every gas hold-up measurement point from low to high gas flow rate: run 2. |

R3 | c_{EtOH}_{,wt} = 0.3%—Gas hold-up curve measured after waiting 120 s for every gas hold-up measurement point from high to low gas flow rate. |

R4 | c_{EtOH}_{,wt} = 0.3%—Gas holdup curve measured after waiting 120 s for every gas hold-up measurement point after each flow rate increase. |

R5 | c_{EtOH}_{,wt} = 0.3%—Gas hold-up curve when foaming phenomenon was observed: run 1 |

R6 | c_{EtOH}_{,wt} = 0.3%—Gas hold-up curve when foaming phenomenon was observed: run 2 |

R7 | c_{EtOH}_{,wt} = 0.3%—Gas hold-up curve when foaming phenomenon was observed: run 3 |

Reference | Bubble Column Design | Gas and Liquid Phases | Phase Properties |
---|---|---|---|

Hughmark, 1967 [133] | Multi-orifice gas sparger d _{c} > 0.1 m U _{G} = 0.004–0.450 m/s | Air-Water; Air-Na_{2}CO_{3} aq. Soln.; Air-Kerosene; Air-Light oil; Ai-Glycerol aq. soln.; Air-ZnCl_{2} aq. soln.; Air-Na_{2}SO_{3} aq. soln. | ρ_{L} = 0.78–1.7 [g/cm^{3}] μ _{L} = 0.0009–0.152 [Pa∙s] σ = 0.025–0.076 [N/m] |

Kato et al. 1972 [134] | d_{c} = 0.066–0.214 m H _{c} = 2.01–4.05 m U _{G} = 0–0.30 m/s U _{L} = 0–0.015 m/s | Air-Water | |

Kato and Nishiwaki, 1972 [193] | Single-hole gas sparger d _{0} = 0.005 m d _{c} = 0.15–0.60 m U _{G} = 0.005–0.42 m/s | Air-Water; Air-Glycol aq. soln.; Air-Methanol; Air-CCl_{4}; Air-Na_{2}SO_{3} aq. soln. (0.15 M); Air-NaCl aq. soln. (0.03 M, 0.07 M, 0.15 M, 0.6 M, 1 M): O_{2}-Water; He-Water; CO_{2}-Water | ρ_{L} = 0.79–1.59 [g/cm^{3}] μ _{L} = 0.00058–0.0211 [Pa∙s] σ = 0.022–0.0742 [N/m] |

Akita and Yoshida, 1973 [47] | d_{0} = 0.0013–0.00362 m d _{c} = 0.10 and 0.19 m H _{c} = 1.5 and 2.4 m U _{G} = 0.043–0.338 m/s | Air-Water; Air-8.0 wt% Methanol aq. soln.; Air-15.0 wt% Methanol aq. soln.; Air-53.0 wt% Methanol aq. soln.; Air-35.0 wt% Cane sugar aq. soln.; Air-50.0 wt% Cane sugar aq. soln. | ρ_{L} = 0.91–1.24 [g/cm^{3}] μ _{L} = 0.001–0.0192 [Pa∙s] σ = 0.0382–0.0755 [N/m] |

Hikita and Kikukawa, 1974 [194] (as reported by [24]). | Perforated plates and sintered plates d _{0} = 0.3 and 0.005 m d _{c} = 0.0756–0.61 m H _{c} = 0.02–3.5 m U _{G} = 0.01–0.08 m/s | Air used as gas. Liquids: water, methanol, iso- and n-propanol, iso- and n-butanol, carbon tetrachloride, dichloroethane, methyl ethyl ketone, ethyl acetate, ethylene glycol, benzene | ρ_{L} = 0.8–1.6 [g/cm^{3}] μ _{L} = 0.00043–0.02 [Pa∙s] σ = 0.0214–0.0728 [N/m] K = 8×10 ^{4}–5×10^{10} |

Gestrich and Rähse, 1975 [195] | Perforated plate d _{0} = 0.0087–0.0309 m d _{c} = 0.05–0.10 m U _{G} = 0.002–0.14 m/s | Air-Water; Air-Glycerol aq. soln. (40%); Air-Kerosene Reacting system: Air/CO _{2}-aq. NaOH (2M) | ρ_{L} = 0.78–1.11 [g/cm^{3}] μ _{L} = 0.00088–0.0115 [Pa∙s] σ = 0.0312–0.072 [N/m] |

Kumar et al., 1976 [196] | Perforated plates d _{0} = 0.0005 and 0.001 m d _{c} = 0.10 m H _{c} < 2 m U _{G} = 0.01–0.20 m/s | Air used as gas; Liquids: n-octanol; tetrabromomethane, glycol; 1,3-butanediol. | ρ_{L} = 0.8–2.98 [g/cm^{3}] μ _{L} = (20–100)∙10^{−3} [Pa∙s] σ = 21.7–72 [dyn/cm] |

Mersmann,1978 [199] | Multi-nozzles d _{0} = 0.030 m d _{c} = 5.50 m H _{0} = 7.0 m (liquid height) U _{G} = 0.024–0.128 m/s | Air-Water | |

Koide et al., 1979 [135] | Downflow bubble column | Gases: Air, Ar, H_{2}, CCl_{2}F_{2}. Liquids: Water, CCl _{4}, Glycerol aq. soln., CMC (Carboxymethyl cellulose). | I = 148–336 [kg/m^{2}∙s] ε* = 0.003–0.24 ρ _{L}/ρ_{G} = 184–5340 μ _{L}/μ_{G} = 37–2220 σ = 0.055–0.07 [N/m] |

Friedel et al., 1980 [197] | Single-nozzle gas sparger d _{0} = 0.011 m d _{c} = 0.10 m H _{c} = 1.5 m U _{G} = 0.042–0.38 m | Pure liquids or non-electrolyte solutions: Air-Water; Air-30 wt% Sucrose; Air-50 wt% Sucrose; Air-Methanol; Air-53 wt% Methanol; Air-nButanol; Air-Aniline; Air-7 wt% iButanol; H_{2}–Water; CO_{2}-Water; CH_{4}-Water; C_{3}H_{8}-Water; H_{2}+N_{2} (1:1)-Water; H_{2}+N_{2} (5:1)-Water Electrolyte solutions: Air-(0.1–5.0 M) NaCl; Air-(0.1–1.5 M) Na _{2}SO_{4}; Air-(0.1–2.0 M) CaCl_{2}; Air-0.4 M MgCl_{2}; Air-(0.1–1.0 M) AlCl_{3}; Air-(0.1–3.0 M) KCl; Air-0.5 M K_{2}SO_{4}; Air-(0.16 M, 0.5 M) K_{3}PO_{4}; Air-0.5 M KNO_{3} | Pure liquids or non-electrolyte solutions ρ _{L} = 0.794–1.24 [g/cm^{3}] ρ _{G} = (0.0837–1.84) 10^{−3} [g/cm^{3}] μ _{L} = (0.658–17.8)∙10^{−3} [Pa∙s] μ _{G} = (0.008–0.0181) ∙10^{−3} [Pa∙s] σ = 0.0229–0.0759 [N/m] Electrolyte solutions ρ _{L} = 1.01–1.17 [g/cm^{3}] μ _{L} = 0.009–0.00187 [Pa∙s] σ = 0.0719–0.0796 [N/m] |

Hikita et al., 1980 [80] | Porous plate d _{c} = 0.153 m H _{c} = 2.5 m | Air-Distilled water | |

Iordache and Muntean, 1981 [202] | Gas distributor: perforated plate (749 holes,) d _{0} = 0.00166 m d _{c} = 0.305 m H _{c} = 2.44 m | Glycerine systems CMC solutions | Glycerine systems ρ _{L} = 1.010–1.249 [g/cm^{3}] μ _{L} = 0.0013–0.246 [Pa∙s] CMC solutions K = 0.0018–2.570 n = 0.495–1.0 ρ _{L} = 0.996–1.008 [g/cm^{3}] |

Godbole et al., 1982 [152] | Single nozzle (d_{0} = 1.5, 2.7 and 5.7 [mm]) d _{c} = 7.3 [cm] H _{c} = 0.95 [m] | Water-N_{2}; Water-He; Water-CO_{2;}; Methanol-N_{2}; NaNO_{3}-N_{2}; NaNO_{3}-He; LiCl-KCl-N_{2}; LiCl-KCl-He | ρ_{L} = 0.788–1.888 [g/cm^{3}] μ _{L}=(0.45–3.65)×10^{−3} [Pa∙s] σ = 0.0215–0.13 [N/m] |

Sada et al., 1984 [107] | Sintered glass disc d _{0} = 0.10 m d _{c} = 0.10 m H _{c} = 1.5 m U _{G} = 0.001–0.045 m/s | Air-water; Air-NaCl-water (NaCl = 0.25 M); Air-CuCl_{2}-water (CuCl_{2} = 0.25 M) T = 303–353 [K] | |

Grover et al., 1986 [78] | d_{c} = 0.30 m | Air-Wwater; Air-Varsol^{(*)}; Air-trichloroethylene ^{(*)} Varsol DX 3641 is a light hydrocarbon oil available from ESSO Chemicals. | ρ_{L} = 0.788–1.450 [g/cm^{3}] μ _{L} = (0.552–1.452) × 10^{−3} [Pa∙s] σ = 0.0283–0.0720 [N/m] |

Reilly et al., 1986 [108] | 40-L bubble column Perforated plate (20 holes) d _{0} = 0.001 m d _{c} = 0.23 m H _{c} = 1.22 m 1000-L pilot plant fermenter Ring sparger (100 holes) d _{0} = 0.003 m d _{c} = 0.76 m H _{c} = 3.21 m | Newtonian liquids: water, glycerine; dextrose aqueous solution; three fermentation media (glucose+mineral salt, molasses+mineral salt, Alpha-floc+mineral salt) Non-Newtonian fluids: Carboxy-methyl cellulose (CMC7H4, Hercules Inc.),; carboxypolymethylene (Carbopol 941, Goodrich Chemical Co.); and polyacrylamide (Separan NP10, Dow Chemical Co.) | Newtonian liquids ρ _{L} = 0.991–1.009 [g/cm^{3}] Non-Newtonian fluids ρ _{L} = 0.991–0.993 [g/cm^{3}] |

Kawase and Moo-Young, 1987 [207] | Single tube (d_{0} = 0.003 m); ring gas sparger (29 holes, d_{0} = 0.002 m; ring gas sparger (56 holes, d_{0} = 0.002 m) d _{c} = 0.06; 0.14; 0.30 m H _{c} = 1.8; 2.2; 2.0 m | Liquids: glycerol, CMC, PAA and Xanthan. | ρ_{L} = 0.999–1.248 [g/cm^{3}] σ = 0.0495–0.0720 [N/m] K = 3.2–9780 n = 0.180–1 |

Schumpe and Deckwer, 1987 [203] | Co-current and counter-current flow Single nozzle gas sparger d _{0} = 0.010 m d _{0} = 0.10 m H _{c} = 1.05 m U _{G} = 0.01–0.16 m/s U _{L} = 0.07 m/s | Air-Water; Air-Alcohol; Air-5%NaCl soln.T = 25–96.56 [°C] | ρ_{L} = 0.7483–1.0268 [g/cm^{3}] μ _{L} = (0.2946–0.8937) × 10^{−3} [Pa∙s] σ = 0.01877–0.07197 [N/m] |

Zou et al., 1988 [209] | Ejector gas sparger d _{c} = 0.30 m H _{c} = 1.46 m | Coalescent batches: distilled water; distilled water & OCENOL; 3% SOKRAT & OCENOL (SOKRAT is a commercial thickener consisting of water and a soluble liquid polymer based on acrylonitrile and acrylic acid in a ratio of 2:1. OCEANOL is a foam breaker consisting of a mixture of saturated and unsaturated alcohol from the fraction C_{16}–C_{18}.) (*); 6% SOKRAT & OCENOL; 10% SOKRAT & OCENOL; 58% Sucrose Non-coalescent batches: 0.5% SOKRAT; 3% SOKRAT ; 6% SOKRAT; 10% SOKRAT | Coalescent batches ρ _{L} = 0.995–1.270 [g/cm^{3}] μ _{L} = (0.7–25.6) × 10^{−3} [Pa∙s] σ = 0.0335–0.0654 [N/m] Non-coalescent batches ρ _{L} = 0.998–1.011 [g/cm^{3}] μ _{L} = (1.5–24.4) × 10^{−3} [Pa∙s] σ = 0.0487–0.0645 [N/m] |

Elgozali et al., 2002 [198] | Sieve plate gas sparger (25 holes, d_{0} = 0.005 m and 75 holes, d_{0} = 0.003 m) d _{c} = 0.09 m H _{c} = 0.61 m | Air used as gas. Pure liquids: 2-propanol Binary mixtures: 2-propanol/methanol (20 mol% 2-propanol); 2-propanol/water (15 mol% 2-propanol); ethanol/water; ethylene glycol/water |

Variable | All Batches | Coalescent Batches | Non-Coalescent Batches |
---|---|---|---|

K | (4.5 ± 5.9) × 10^{9} | (1.92 ± 0.74) × 10^{6} | (5.30 ± 11.70) × 10^{6} |

a | 0.67 ± 0.04 | 0.72 ± 0.01 | 0.44 ± 0.01 |

b | 0.22 ± 0.02 | 0.14 ± 0.01 | 0.13 ± 0.02 |

c | 1.95 ± 0.12 | 1.26 ± 0.04 | 1.41 ± 0.20 |

T [°C] | 40 | 60 | 70 | 80 | 85 | 90 | 95 |
---|---|---|---|---|---|---|---|

f | 1.210 | 1.2468 | 1.1804 | 0.1607 | 0.1215 | 0.1505 | 0.2634 |

c_{wt} [%] | z_{1} | z_{2} | z_{3} | k_{1} | k_{2} | z_{4} | z_{5} |
---|---|---|---|---|---|---|---|

Air-Water | |||||||

0 | −0.657 | 0.001 | 1.00 | 0.690 | −0.251 | 0.750 | −0.300 |

Air-Water-MEG | |||||||

0.05 | −0.570 | 0.001 | 1.00 | 0.691 | −0.300 | 0.670 | −0.280 |

0.1 | −0.550 | 0.001 | 1.00 | 0.701 | −0.294 | 0.692 | −0.280 |

0.5 | −0.666 | 0.001 | 1.00 | 0.674 | −0.308 | 0.666 | −0.300 |

1 | −0.560 | 0.001 | 1.00 | 0.689 | −0.321 | 0.667 | −0.300 |

5 | −0.420 | 0.001 | 1.00 | 0.730 | −0.295 | 0.735 | −0.300 |

8 | −0.340 | 0.001 | 1.00 | 0.744 | −0.300 | 0.745 | −0.300 |

10 | −0.625 | 0.001 | 1.00 | 0.693 | −0.385 | 0.710 | −0.300 |

80 | −0.340 | 0.001 | 1.00 | 0.730 | −0.237 | 0.703 | −0.300 |

Air-Water-EtOH | |||||||

0.05 | −0.282 | 0.001 | 1.00 | 0.844 | −0.140 | 0.808 | −0.100 |

Air-Water-NaCl | |||||||

0.4 (c/c_{t} = 0.48) | −0.605 | 0.001 | 1.00 | 0.697 | −0.286 | 0.713 | −0.300 |

1 (c/c_{t} = 1.17) | −0.485 | 0.001 | 1.00 | 0.722 | −0.284 | 0.738 | −0.300 |

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