One-dimensional drift-flux model of gas holdup in fine-bubble jet reactor

Chemical Engineering Journal xxx (xxxx) xxxx

Contents lists available at ScienceDirect

Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej

One-dimensional drift-flux model of gas holdup in fine-bubble jet reactor Hongzhou Tian1, Shaofeng Pi1, Yaocheng Feng, Zheng Zhou, Feng Zhang , Zhibing Zhang ⁎

⁎

Key Laboratory of Mesoscopic Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210046, China

HIGHLIGHTS

GRAPHICAL ABSTRACT

holdup in the column of a fine• Gas bubble jet reactor was modeled. dissipation rate, bubble size, • Energy and average rising velocity were modeled.

model can predict effects of de• The sign parameters on gas holdup.

ARTICLE INFO

ABSTRACT

Keywords: Gas holdup Energy dissipation rate Bubble Sauter mean diameter Bubble average rising velocity Model

A one-dimensional drift flux model of the gas holdup in the fine-bubble jet reactor (for short FJR) was developed and experimentally verified. The model consists of three key sub-models (bubble Sauter mean diameter, bubble average rising velocity and energy dissipation rate). Effects of downcomer length, liquid dynamic viscosity and surface tension on the bubble Sauter mean diameter, energy dissipation rate, and gas holdup were discussed theoretically. Furthermore, three kinds of systems (air-tap water, air-aqueous ethanol, and air-aqueous glycerol) were chosen to validate the model under different conditions including liquid flow rate, nozzle diameter, downcomer length, and temperature. Theoretical results indicated that the energy dissipation rate in the downcomer of FJR was the critical factor influencing gas holdup in the column of FJR. It is also found that gas holdups predicted by the model were roughly identical to experimental results and errors were within ± 15%.

1. Introduction Bubbles in liquid are popular in industrial processes including gasliquid reaction, water treatment, biological fermentation and the like. In many gas-liquid multiphase reaction systems, the global reaction rate is mainly controlled by the gas-liquid interfacial area a, which is determined by bubble Sauter mean diameter d32 and gas holdup ϕG. Relative to millimeter bubbles, fine bubbles (with diameter between 1 μm and 1 mm) can provide not only a larger interfacial area but also greater gas dissolubility due to higher internal pressure. Therefore, the

design and scale-up of reactors capable of producing a large number of fine bubbles are required to the intensification of mass transfer and reaction for gas-liquid systems [1]. Despite the influences of mass transfer or reaction, bubble size in turbulent dispersion is mainly determined by the competition between bubble breakup [2,3] and coalescence [4,5], both of which are too complicated to be modeled successfully. However, in isotropic turbulence, the diameter of the maximum stable bubble, dmax is related closely to d32 and can be predicted reasonably by the classical theory of Kolmogorov-Hinze [6]. This is strictly limited to the conditions with the

Corresponding authors. E-mail addresses: [email protected] (F. Zhang), [email protected] (Z. Zhang). 1 These Authors Equally contributed to the work. ⁎

https://doi.org/10.1016/j.cej.2019.03.098

1385-8947/ © 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Hongzhou Tian, et al., Chemical Engineering Journal, https://doi.org/10.1016/j.cej.2019.03.098

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

Nomenclature a D d32 dmax dmin E e F (∙) Fr g hf L P Q Re S u us Wecrit

wt% z

gas-liquid interfacial area, m2 m−3 downcomer diameter, m bubble Sauter mean diameter, m diameter of the maximum stable bubbles, m diameter of the minimum bubbles, m mechanical energy, W relative roughness probability density function Froude number, dimensionless gravitational acceleration, 9.81 m s−2 head loss in pipeline, h downcomer length, m pressure, Pa volumetric flow rate, m3 h−1 Reynolds number, dimensionless cross-sectional area, m2 velocity, m s−1 superficial velocity, m s−1 critical Weber numbers defined in the theories of

Kolmogorov-Hinze mass fraction of solute relative height, m

Greek letters ε η λ μ ρ σ φG

energy dissipation rate, W kg−1 Kolmogorov’s length scale, m friction coefficient, dimensionless dynamic viscosity, Pa s density, kg m−3 surface tension, N m−1 gas holdup

subscripts G L O, C, I m

following characteristics: (1) bubble breakup results from a single interaction with a velocity fluctuation over a length comparable to bubble diameter; (2) the energy dissipation rate in the flow field surrounding bubble is large enough; (3) the contact time between bubble and turbulent eddies is long enough to develop an Equilibrium between surface tension and turbulent dispersion force. Anyhow, it is widely accepted that the higher the energy dissipation rate, the smaller the bubble size. Practically, the energy dissipation rate reflects the energy intensity and efficiency of bubble generators. Traditional gas-liquid reactors, such as bubble column reactors (BCRs) and stirred tank reactor (STRs), usually generate bubbles with diameter of several millimeters or bigger [7,8]. In BCRs, bubbles generated through gas spargers could be hardly split into fine bubbles due to the limited energy dissipation rate. For STRs, fine bubbles can be generated in the region close to the agitators, while most of the input energy offered by mechanical stirring is consumed on the macroscopic movement in other regions where fine bubbles easily become larger due to bubble coalescence. For the generation of fine bubbles, specific methods basing on different mechanisms have been developed [9–12]. Among them, the liquid ejector can transport the entrained fluid with the driving force of motive fluid, following with the generation of fine bubbles and millibubbles [13]. Due to the prominent performances, such as fluid transportation, strengthen mixing and heat removal, gas-liquid ejectors have been widely applied in industries [14–16]. Generally, the ejector is the key component of liquid jet pumps [17] and composed of nozzle, throat, and diffuser [18]. In the case of fine bubble jet reactors (named as FJR, which is essentially the bubble column reactors with a specific liquid ejector as fine bubble generator), throat and diffuser of the ejector are usually installed as a whole called downcomer [19,20] so as to increase the depth of bubble penetration and force bubbles enter into the column [21]. It was shown recently that FJR has good mass transfer performances, such as high gas holdup, volumetric mass transfer coefficient and mass transfer efficiency [22]. To characterize the mass transfer performance of FJR, bubbles’ fundamental morphological characteristics (bubble shape, bubble size, and bubble surface contamination) play a crucial role, since the gas-liquid mass transfer depends mainly on bubble characteristics including Marangoni effect when using surfactants, bubble wakes, bubble clustering. Bubble shape is one critical factor determining the drag coefficient (CD) and mass transfer [23]. In industrial systems, the adsorption and accumulation of surfactants at the bubble surface may hinder the mobility of the bubble interface, and

gas in bubbles liquid location, see Figs.1 and 2 mass, or mixture

the CD is increased [24]. Bubble sizes in gas-liquid systems can be characterized by d32, which was widely used in studies of mass transfer between gas and liquid. Some measurement techniques for bubble size have been developed, but it is a great challenge to identify copious bubbles when gas holdup is high or in the opaque device. Therefore, the global gas holdup that can be measured rather accurately is usually used to indicate the gas-liquid mixing of a gas-liquid reactor [8,25,26]. The prevailing flow regime in the reactor column should be considered for gas holdup prediction since the gas-liquid hydrodynamics changes with flow regime [27]. Design parameters (i.e., bubble column diameter, aspect ratio, and gas sparger opening) are depended mainly on flow regime, which could be observed orderly with the increase of gas superficial velocity: dispersed bubble flow, vertical-spiral flow and turbulent flow [28]. Even in the simple case of dispersed bubble flow, the liquid velocity (i.e., co-current or counter-current bubble columns) influence the bubble size, shape and bubble motion (and, thus, the whole fluid dynamics in the bubble column). In the case of FJR, gas entrainment rate that has great influence on the gas holdup in the column of FJR is mainly determined by several factors including the nozzle geometry [29,30], gas-liquid mixture height [31], and downcomer configuration [21]. The effect of ε in the ejector's throat on the gas entrainment was developed firstly by Cunningham [32] and adopted by Evans et al. [33] in a confined plunging liquid jet bubble columns. Since the gas entrainment of the ejector has been carefully studied, the present work pays more attention to the relationship between energy dissipation rate ε in the downcomer and gas holdup in the column of the FJR. The bubble average rising velocity is affected by the flow regime surrounding the bubble. With the increase of uG and uL, flow regime in bubble columns changes orderly from discrete bubble flow to dispersed bubble flow, slug flow, churn flow, bridging flow and annular flow [27], and the bubble average rising velocity changes correspondingly [34]. The size of bubbles generated depends largely on ε in the downcomer. Generally, there are two regions with different hydrodynamics in the downcomer, i.e., mixing zone and pipe flow zone [33,35]. The mixing zone is the region of fine bubble generation, which result from the intense interaction between plunging liquid jet and entrained gas. Pipe flow region is usually characterized as homogeneous bubble flow, but it is shown that the existence of this particular region also depends on ε which is affected by the design parameters of downcomer [35]. Besides the importance of energy dissipation rate in the downcomer, 2

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

gas-liquid flow in the column of FJR should also be considered. Since the theory of one-dimensional drift flux has been used widely for modeling of two-phase flows [36,37], the present work will adopt the theory for the modeling of the gas holdup in FJR. The following sections were organized as: in Section 2, one-dimensional drift flux model of gas holdup in the FJR was developed, and the mathematical expressions of three critical parameters including bubble average rising velocity, bubble Sauter mean diameter and energy dissipation rate were established accordingly; in Section 3, experimental information was introduced details; in Section 4, effects of FJR design parameters on gas holdup were discussed, and the model developed was verified experimentally. Some conclusions were drawn in Section 5.

2.2. The average rising velocity of bubbles in the column In the case of the homogeneous bubble flow with one-dimensional motion, the mean rise velocity of bubbles can be calculated as: (4)

v32 = v0 + u

where, vo is the bubble rising velocity in the infinite quiescent liquid, and u stands for the interstitial velocity of liquid between bubbles.

QL

u=

(1

G ) S0

=

vL (1

(5)

G)

vL is the superficial liquid velocity in the column of FJR.

vL =

QL S0

(6)

Substituting Eqs. (5) and (6) into Eq. (4) gives:

2. Theoretical As mentioned above, modeling of the gas holdup for FJR relies on three critical parameters including (1) energy dissipation rate in the mixing zone of the downcomer; (2) bubble Sauter mean diameter d32; (3) bubble average rising velocity. Due to the complex mechanisms of bubble generation, some following assumptions have to be set for the simplification of gas holdup modeling under steady-state conditions.

v32 = v0 +

(1) Only homogeneous fine bubble flow exists in the downcomer. The assumption is reasonable since the minimum liquid superficial velocity in the downcomer is 1.54 m/s (downcomer diameter is a constant as 0.019 m, the minimum liquid flowrate is 1.575 m3/h), which is larger than 0.5 m/s, the transitional value of liquid superficial velocity in the downcomer [38]. Accordingly, it can be considered that only the mixing zone exists in the downcomer. (2) The Equilibrium of bubble sizes in downcomer can be developed since the L/D for downcomer in the present work varies between 37.5–56.25, which is beyond the range (8–24) for the development of Equilibrium bubble size. (3) Gas-liquid flow in the column of FJR is homogeneous bubble flow and can be described with a one-dimensional drift flux model [39,40]. To release the turbulent kinetic energy of outflow from the downcomer and thus disperse bubbles fully to decrease interactions between bubbles as far as possible, the volume of the column space below the downcomer exit should be large enough [41]. In the present work, the volumetric ratio ranges from 0.37 to 0.79 (correspondingly, the length of downcomer changes from 450 mm to 150 mm).

v32 =

v322

(1

vL vG/ v32 )

(7) (8)

(vL + vG + v0 ) v32 + vG v0 = 0

Ignoring the unreasonable result in Eq. (8), v32 can be obtained, accordingly.

(vG + v0 + vL ) +

(vG + v0 + vL ) 2 2

4vG v0

(9)

It can be seen from Eq. (9) that vG and vL are two factors that determine v32, especially when v0 is small. For fine spherical bubbles with diameter d32, v0 is usually predicted based on the famous Stokes Equation:

v0 =

L gd32

2

(10)

18µL

where ρL and μL stand for the density and dynamic viscosity of the liquid, respectively, g represents the gravitational acceleration. It should be pointed out that predictions by Eq. (10) for fine bubbles in unpurified systems only partly agree with the measurements [42,43]. The discrepancy may be determined by some factors including the potential and contamination on the surface of bubbles [44]. Accordingly, some correlations with practical applications need to be adopted in the present work. One typical relationship related is shown in Eq. (11).

v0 =

Lg

1/4

L

Mo 1/4 2 2c d de + + e Kb de 2

n /2

1/ n

(11)

where 2.1. One-dimensional drift flux model of gas holdup

Mo =

L L

In the case of homogeneous bubble flow, the superficial gas velocity vG can be calculated as follows:

vG =

QG S0

QG S0 G

= vG / v32

(12) (13)

The values of Kb, c and n in Eq. (11) for different systems can be determined experimentally.

(1)

2.3. Modeling of bubble Sauter mean diameter d32 Sauter mean diameter, d32 is an essential parameter for bubbles in the liquid [45,46]. Theoretically, modeling of d32 should be based on bubbles’ population balance Equation (PBE) [47], as well as the theories of bubble breakup [2] and coalescence [5,47]. d32 can be characterized by dmax and dmin, the minimum diameter of the stable bubbles. It has been reported that d32 is proportional to dmax for some gas-liquid systems, while dmin has rarely been concerned [48] and could be determined by different mechanisms [49,50]. For FJR, since bubbles are mainly generated by high energy dissipation rate in the downcomer, dmin may be determined by the theory of turbulent eddy collision. Accordingly, only eddies with about 11.4–31.4 times

(2)

where φG is the gas holdup in FJR. Substituting Eq. (2) into Eq. (1) gives: G

3

de = d32 ( L g / L )1/2

where QG and S0 are the volumetric flow rate of gas entering into the ejector and the cross-sectional area of the column FJR, respectively. The mean rising velocity of bubbles in the column can be expressed as:

v32 =

gµL 4

(3)

Since vG can be measured conveniently, the prediction of v32 is required for the calculation of the gas holdup. 3

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

Kolmogorov scale have enough energy and living time to breakup bubble with the same size of eddies. Considering the difficulty in choosing one specific proportionality constant (11.4–31.4) for dmin prediction and the fact that eddies with 11.4 times Kolmogorov scale are regarded as the minimum scale of eddies responsible for bubble breakup, the proportionality constant 11.4 is taken for dmin prediction [51], that is:

respectively. If gas and liquid flow rates, as well as the ejector structures, are all defined, only PO in Eq. (18) is needed. Basing on Bernoulli Equation, mechanical energy conservation between the nozzle outlet and section I in the circulation pipe can be developed as Eq. (20) and PO can be calculated.

(14)

(20)

dmin = 11.4(µL / L )3/4

1/4

uN 2 P u2 P + o + Z 0 = L + I + ZI 2g 2g Lg Lg

where ε is the energy dissipation rate in the downcomer. Since the region for bubble generation can be approximate to isotropic turbulence and turbulent dispersion is dominant, dmax can be predicted with Kolmogorov-Hinze theory. 2/5 (

dmax =

L Wecrit /2 L )

where PI is the pressure at section I, as shown in Fig. 1. hf1, the mechanical energy loss between the nozzle outlet and section I, is related to the liquid velocity and friction coefficient in the inner pipe wall and can be calculated as

(15)

3/5

hf 1 =

where Wecrit, the critical Weber number is defined as the ratio of liquid fluctuation pressure surrounding bubble to surface tension when bubble breakup. It is hard to predict Wecrit accurately since Wecrit is affected by many factors including flow regime, and viscosity ratio of dispersed phase to continuous phase. Considering the following facts: (1) the values of Wecrit reported are about 1.20 [52], which is close to the result from the resonance theory of bubble breakup; (2) bubble generation in the downcomer of FJR is similar to that in plunging liquid jet bubble column. The value of Wecrit in the present work was chosen as 1.2. Basing on the theory in reference and assumption that the size distribution of spherical bubbles in homogeneous bubbly flow is continuous and log-normal, calculation of d32 can be derived as:

d32 = dmax (dmax / dmin)

1/2 exp [3.5(

4 + 0.5ln(dmax / dmin)

uL2 = 2

Ki ×

Li u2 × L D 2

1

(21)

Li is the Eqsuivalent length between the nozzle outlet and where section I. Friction coefficient λ can be calculated with Harland's formula:

1

=

e/ d 3.7

1.8lg

1.11

+

6.9 Re

(22)

where e/d is the relative roughness of pipe, and the definition of Reynolds number Re is:

duL µL

Re =

(16)

2)2 ]

hf 1

L

(23)

As for the calculation of ε in the downcomer with Eq. (17), another parameter that needs to be determined is the mean density for the gasliquid system ρm, which is related to the average gas holdup G [53].

It can be seen from Eq. (16) that if bubble size distribution in homogeneous bubbly flow is continuous log-normal, d32 is just the function of dmax and dmin, which are related to the liquid physical properties (ρL, σL and μL) and energy dissipation rate ε. Generally, ε depends on the energy input, reactor structure and operating condition of the reactor. It should be noted that ε is different in the regions in FJR and the size of the bubbles in FJR is mainly determined by ε in the downcomer. Therefore, the modeling of ε in the downcomer is necessary and essential for the design and operation of FJR.

m

=

G

=

L (1

G)

usg usg + usl

+

(24)

G G

K

(25)

where usg and usl are the superficial velocity of the gas phase and liquid Eddy-current flowmeter

2.4. Energy dissipation rate ε in the downcomer The energy dissipation rate ε in downcomer is determined by the liquid ejector structure shown in Fig. 2. Since energy dissipated due to the inner wall friction is converted to heat, energy dissipation rate ε (dissipation energy per unit mass fluid) useful for bubble generation in the downcomer of the ejector (Fig. 2) can be expressed as:

=

E mV

=

E mV

=

Eo

Ec

Thermometer

Feed port

(17)

Lc Q L gZc

+ Q L Pc +

Lo Q L

2 Lc Q L

2

uN 2 +

uc2 +

go Q go gZ o

gc Q gc gZc

+ Q go Po +

+ Qgc Pc +

go Q go

2 gc Q gc

2

Lc=710mm

+ QL Po +

Heat exchanger

ugo 2

Refrigerating machine

(18)

ugc 2 ΔL

Ec =

Lo Q L gZo

Air inlet

m Qhf 2

mV

where ρm is the density of gas-liquid mixture in downcomer with volume V. hf2 stands for the mechanical energy loss in the downcomer. Eo and Ec are, respectively, the mechanical energy input and outflow from the ejector. According to the law of mechanical energy conservation, Eo and Ec can be calculated as follows, respectively:

Eo =

Location I Gas flowmeter

Pressure gage

(19)

Centrifugal pump

where Zo and Zc represent the relative height of the location o and c, respectively. uN and uc stand for the liquid velocity at the nozzle outlet and location c, respectively. ugo and ugc are the gas velocity at location o and c, respectively. Po and Pc represent the pressure at location o and c,

Fig. 1. Schematic diagram of experiments. 4

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

the pump and the ejector, a heat exchanger was installed to maintain the liquid temperature to a scheduled value (Fig. 1).

Ø55mm Ø33mm

Ls2= 25mm

3.2. Physical properties of the liquids Three kinds of liquids including tap water, aqueous ethanol solution (3.11–11.52 wt%) and aqueous glycerol solution (5.02–17.64 wt%) were used as motive liquid. The physical properties of the motive liquids are listed in Table 2. According to the public report by the Nanjing Ecological Environment Bureau of China, the tap water used in the present work (2018.5–2018.10) in Nanjing University meets the national standard for drinking water quality (GB5749-2006). In the standard, the limits of two salt (chloride and sulfates) are both the 250 mg/L.

Air inlet

Ls1= 33mm

38.29°

Suction chamber Nozzle Location O

Ø8mm

Downcomer L

3.3. Gas holdup measurement

Energy dissipation zone,ε

Since the liquid level difference before and after aeration is very important for gas holdup measurement [58], the initial liquid level H0 in the column should be determined accurately. Before the experiment, a certain amount of liquid was introduced into the column, and the air inlet was closed to avoid the effect of entrained gas. When the centrifugal pump worked, the liquid level was decreasing to a stable value. Then, a certain amount of water was injected into the column to maintain the liquid level at 430 mm, which was marked as the initial liquid level, H0. Under steady conditions, the flow rates of liquid (0.6 m3/h–3.0 m3/ h) and entrained air were measured, respectively. In the downcomer of the ejector, gas was sufficiently mixed with liquid to form fine bubbles, which entered the column and formed a bubble bed with a height of H1. Therefore, the gas holdup can be calculated.

Ø19mm

Location C

Fig. 2. Structure of the ejector.

phase, respectively. The coefficient K in Eq. (25) reflects the effect of gas-liquid relative speed on gas holdup and is expressed as a function of Z [54]. In case Z < 10,K = 0.16367 + 0.310372Z 0.03525Z2 + 0.001366Z3; In case Z ≥ 10, K = 0.75545 + 0.003585Z 0.1436 × 10 4Z2 . Z was defined by Hughmark [53] as follows:

Z = Rem1/5Frm1/8Cl

1/4

(26)

where Rem, Frm, and Cl are the Reynolds number, Froude number and liquid volume fraction in the mixed phase, respectively:

G

= (H1

(31)

H0 )/H1

(27)

To reduce the measurement error, each measurement was repeated three times to take the average value. The relevant uncertainties of instrumentation used are listed in Table 3.

gD

(28)

4. Results and discussion

usl Cl = usg + usl

(29)

Rem =

Frm =

D(

G usg G µG

+ +

L usL ) L µL

(usg + usl ) 2

As shown in Section 2, the model of gas holdup includes three key parameters: bubble energy dissipation rate, bubble Sauter mean diameter in the downcomer and bubble average rising velocity in the column. The effects of downcomer length, liquid viscosity and surface tension on gas holdup, as well as the energy dissipation rate and bubble Sauter mean diameter would be discussed in this section.

D in Eqs. (27) and (28) is the downcomer diameter. Since the mechanical energy loss is mainly caused by the inner wall friction in the downcomer, hf2 can be calculated as follows.

hf 2 =

2

Li u2 × c D 2

(30)

Table 1 Dimensions of the ejector.

Accordingly, with Eq. (17), ε can be calculated. Combination of Eqs (14)–(17) presents a relationship of d32 with the liquid physical properties and the operating parameters of FJR. Then, v32 can also be obtained. Finally, the gas holdup of FJR can be calculated with Eq. (3). The Equations above construct a mathematical relationship of the gas holdup with liquid physical properties, reactor structure and operating parameters of FJR. 3. Material and methods 3.1. Structure of FJR The dimension parameters of the ejector are listed in Table 1. The total height and effective volume of the column were 710 mm and 60 L, respectively. The inner diameter of the column was 200 mm. Between 5

Description

Value

Diameter of liquid inlet, d Diameter of downcomer, D Diameter of the nozzle, DN Length of the downcomer, L Length of the reactor chamber, Lc Height of location I, ZI Height of location O, ZO Height of location C, ZC Equivalent length from I to O, ΣLI-O Equivalent length from O to C, ΣLO-C Resistance coefficient from I to O, KI-O Resistance coefficient from O to C, KO-C Relative roughness, e

15.7 mm 19 mm 8 mm 150–450 mm 710 mm 1560 mm 1018 mm 518 mm(when L = 450 mm) 3270 mm 1170 mm 450 mm 11910 mm 0.04

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

increases faster with a rise in liquid viscosity when the liquid viscosity is less than 1.5 cp, while the gas holdup goes up slowly or even decrease when the liquid viscosity increases above 1.5 cp. The variation of the gas holdup with liquid viscosity is consistent with some studies [60–62]. Obviously, the effect of viscosity on gas holdup is limited in a specific range of liquid dynamic viscosity. The dual effect of liquid viscosity has been investigated comprehensively [63].

Table 2 Physical properties of three aqueous solutions. Liquid phases

wt %

T (°C)

Density* (g/ cm3)

Viscosity* (cp)

Surface tension** (dyn/cm)

Water

100 100 100 100 100 5.02 10.98 17.64 17.64 17.64 17.64 17.64 3.11 6.97 11.52 11.52 11.52 11.52 11.52

10 15 20 25 30 20 20 10 15 20 25 30 20 20 10 15 20 25 30

0.9997 0.9991 0.9982 0.9970 0.9956 0.9901 0.9803 0.9702 0.9696 0.9690 0.9670 0.9647 0.990131 0.9804 0.9702 0.9696 0.9690 0.9670 0.9647

1.8077 1.1404 1.0050 0.8937 0.8007 0.9968 0.9990 1.3041 1.1332 1.0016 0.9053 0.8099 0.9967 0.9990 1.3041 1.1332 1.0016 0.9053 0.8099

74.36 73.62 72.88 72.14 71.40 72.74 72.30 71.43 71.78 71.81 71.43 70.94 62.27 54.07 51.90 55.72 47.80 67.62 75.92

Glycerol aqueous solutions

Ethanol aqueous solutions

4.1.3. Effects of surface tension Since the liquid surface tension is not concerned in Eqs. (15)–(30), liquid surface tension seems to have little influence on the energy dissipation rate in downcomer (Fig. 5(a)). However, it should be noted that liquid surface tension is related to bubble surface energy, which is essentially the product of bubble surface area and surface tension. Strictly speaking, liquid surface tension is a parameter connecting with energy transfer in the downcomer. Fig. 5(b) indicates that with a rise in surface tension, d32 increases while the gas holdup decreases. The variation of d32 is determined by Eqs.(14)–(16), and the change of gas holdup with surface tension could be described by the correlation proposed by Akita et al. [64]. According to Fig. 5(b), it can be concluded that to increase gas holdup and enlarge the gas-liquid interfacial area, the small surface tension of the liquid is preferred.

* : Data of density and viscosity were adopted or interpolated from the handbook [55]. ** : For aqueous ethanol and aqueous glycerol, the data of surface tension were adopted or interpolated from [56] and [57], respectively.

4.2. Comparison of measurements with theoretical results 4.2.1. Air -tap water system It can be seen from Table 4 that with the increase of liquid flow rate, both the flow rate of entrained gas and energy dissipation rate in downcomer increase, therefore, d32 decrease and gas holdup increase accordingly. Two typical models describe the nonlinear relationship between liquid flow rate and gas entrainment. The first model proposed by Evans et al. [65] is based on the natural essence of gas entrainment and some assumptions. The other model suggested that gas entrainment rate was related closely to the pressure drop in the downcomer [66]. The first model is more reasonable since the minimum liquid flow rate for gas entrainment can be predicted. As shown in Table 5, with an increase of the nozzle diameter, the gas entrainment rate, energy dissipation rate, and gas holdup are all decreased, while d32 goes up gradually. Under the similar liquid flow rate, since the energy dissipation rate in the downcomer and the gasliquid interfacial area a (a = 6ϕG/d32) both decrease with a rise in nozzle diameter, it may be concluded that the energy efficiency for bubble generation is reduced at larger nozzle diameter. However, considering the stability of liquid jet [67] and the formation of the mixing zone in the downcomer, the nozzle diameter should be larger than a specific value. With the increase of downcomer length, the gas entrainment rate increases while the energy dissipation rate decreases (Table 6). The rise in gas entrainment rate may result from the increase of bubble penetration [21], and the decrease of the energy dissipation rate may come from the increase of fluid volume and the rise of energy loss due to friction.

4.1. Calculation of gas holdup for air-water system 4.1.1. Effects of downcomer length It can be found from Fig. 3(a) that energy dissipation rate ε in downcomer decreases with a rise in the downcomer length. This can be explained from two aspects: on the one hand, with the increase of the downcomer length, the ratio of air pressure near the plunging liquid jet to atmospheric pressure increases, which led to the decrease of liquid jet length. Accordingly, the plunging liquid jet kinetic energy at the impact point decreases, resulting in the decrease of momentum transferred [21]; on the other hand, an excessive downcomer length leads to an additional pressure drop [59], which means more energy is needed for overcoming the higher static pressure of the column to let gas-liquid mixture leave from the downcomer. Fig. 3(b) indicates that the shorter the downcomer length, the smaller the bubble size and thus the higher the gas holdup in the column of FJR. However, if the downcomer is too shorter or disappear, the contacting time and momentum between the two phases are limited to obtain small bubbles. Therefore, the downcomer length should be appropriate to enlarge the gas holdup and gasliquid interfacial area. To ensure sufficiently the contacting time and momentum transfer for the bubble generation, the downcomer length was taken as 450 mm in the following. 4.1.2. Effects of liquid viscosity It seems from Fig. 4 (a) and (b) that in a narrower range of liquid dynamic viscosity (0.5–2.5 cP), energy dissipation rate ε in the downcomer decreases linearly with an increase of the liquid dynamic viscosity, leading to a rise in d32. This may be caused by more friction consumption due to the higher viscous dissipation in liquid. However, unlike ε and d32, the variation of the gas holdup with liquid dynamic viscosity is relatively complex. As shown in Fig. 4(b), the gas holdup

4.2.2. Air - aqueous glycerol solutions system Besides water, aqueous glycerol and aqueous ethanol were also chosen as working liquids to show the effect of liquid physical

Table 3 List of instrumentation used and relevant uncertainties. Instrument

Liquid flowmeter

Gas flow meter

pressure gauge

Thermometer

Manufacturer & model Precision

LWGY ± 0.5%

MF5700 ± (2 + 0.5FS) %

Yichuan ± 2.5%

UNI-T UT321 ± (0.5% + 0.8)

6

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

Fig. 3. (a) Effect of downcomer length L on the energy dissipation rate ε in the downcomer, (b) Effects of downcomer length L on the bubble Sauter mean diameter d32 and gas holdup φG. (DN = 8 mm, D = 19 mm, QL = 2.052 m3/h, QG = 0.245 L/s, PI = 0.162 MPa, purified water, 20 °C).

Fig. 4. (a) Effect of liquid dynamic viscosity μ on the energy dissipation rate ε in the downcomer, (b) Effects of liquid dynamic viscosity μ on the bubble Sauter mean diameter d32, and gas holdup φG. (DN = 8 mm, D = 19 mm, L = 0.45 m, QL = 2.050 m3/h, QG = 0.224 L/s, PI = 0.171 MPa, 20 °C).

Fig. 5. (a) Effect of surface tension σ on the energy dissipation rate ε in the downcomer, (b) Effects of surface tension σ on the bubble Sauter mean diameter d32, and gas holdup φG. (DN = 8 mm, D = 19 mm, L = 0.45 m, QL = 2.052 m3/h, QG = 0.245 L/s, PI = 0.162 MPa, 20 °C).

properties on the bubble generation in the downcomer and the gas holdup in the column. As shown in Table 7, with an increase of glycerol content, d32 and the gas holdup are raised, while ε drops dramatically. The higher glycerol concentration leads to the larger viscosity and the lower surface tension decrease [68]. Noticeably, the higher gas holdup results from, the higher liquid viscosity (Fig. 4(b)) but, the lower surface tension decrease (Fig. 5(b)).Table 8.

With the increase of the aqueous glycerol temperature, d32 reduces gradually, while the other parameters change rather complicatedly. Both the liquid viscosity and surface tension decrease with the increasing temperature, and the decrease of d32 can be predicted from Figs. 4(b) and 5(b). It should be noted that the trends for variations of the experimental data and the calculated results with temperature are similar to each other.

7

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

Table 4 Effect of liquid flow rate QL. QL(m3/h)

QG(L/s)

1.575 2.003 2.409 2.799

PI(MPa)

0.255 0.346 0.417 0.554

d32(mm)

ε(W/kg)

0.134 0.159 0.188 0.212

4.09 9.22 17.06 25.66

Gas holdup (%)

0.69 0.52 0.42 0.37

δ (%)

Measured

Theoretical

8.51 15.35 20.96 27.36

9.54 16.93 23.05 29.72

12.25 9.93 9.75 8.45

Note: L = 0.45 m, DN = 0.008 m, D = 0.019 m. Table 5 Effect of nozzle diameter DN. DN (mm)

8 9 10 11 12

QL(m3/h)

2.799 2.802 2.813 2.796 2.814

QG(L/s)

0.554 0.478 0.389 0.247 0.186

PI(MPa)

ε(W/kg)

0.341 0.275 0.239 0.215 0.203

25.66 21.38 19.25 14.53 10.19

d32(mm)

0.37 0.39 0.40 0.44 0.50

Gas holdup (%)

δ (%)

Measured

Theoretical

27.36 23.76 18.87 11.70 8.12

29.72 25.68 20.93 12.80 8.66

8.45 7.88 10.73 9.39 6.86

Note: L = 0.45 m, L/D changes from 56.25 to 37.5. Table 6 Effect of downcomer length L. L(m)

0.15 0.30 0.45

QL(m3/h)

2.84 2.806 2.799

QG(L/s)

0.306 0.477 0.554

PI(MPa)

0.223 0.222 0.221

ε(W/kg)

37.01 35.73 25.66

d32(mm)

0.32 0.33 0.37

Gas holdup (%)

δ (%)

Measured

Theoretical

20.37 25.86 27.36

19.70 28.49 29.72

−3.44 9.99 8.45

Table 7 Effect of aqueous glycerol concentration at 20 °C. wt (%)

5.02 10.98 17.64

QL(m3/h)

2.015 2.010 1.988

QG(L/s)

0.220 0.282 0.213

PI(MPa)

ε(W/kg)

0.162 0.161 0.161

19.01 17.81 16.14

d32(mm)

0.42 0.45 0.49

Gas holdup (%)

δ (%)

Measured

Theoretical

13.33 13.53 14.34

14.94 14.86 14.93

12.06 9.88 4.12

Table 8 Variation of φG with temperature in aqueous glycerol solution (17.64 wt%). T (°C)

10 15 20 25 30

QL(m3/h)

2.050 1.967 1.988 2.090 2.010

QG(L/s)

0.224 0.227 0.213 0.254 0.235

PI(MPa)

ε(W/kg)

d32(mm)

Gas holdup (%)

δ (%)

Measured

Theoretical 16.34 16.14 14.93 17.26 16.39

0.166 0.160 0.161 0.161 0.161

16.63 15.73 16.14 17.82 19.58

0.52 0.51 0.49 0.45 0.42

17.31 15.69 14.34 15.69 14.85

PI(MPa)

ε(W/kg)

d32(mm)

Gas holdup (%)

−5.58 2.90 4.12 10.02 10.36

Table 9 Effect of aqueous ethanol concentration at 20 °C. wt(%)

3.11 6.97 11.52

QL(m3/h)

1.966 1.926 1.926

QG(L/s)

0.256 0.294 0.273

0.156 0.155 0.155

5.45 7.76 8.56

0.62 0.55 0.53

8

δ (%)

Measured

Theoretical

10.97 15.19 14.51

10.46 13.87 13.46

−4.70 −8.71 −7.23

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

Table 10 Variation of φG with temperature in aqueous ethanol solution (11.52 wt%). t (°C)

10 15 20 25 30

QL(m3/h)

1.948 1.938 1.926 1.945 1.918

QG(L/s)

0.290 0.282 0.273 0.267 0.277

PI(MPa)

0.156 0.155 0.155 0.154 0.154

ε(W/kg)

8.01 8.11 8.56 11.15 14.28

d32(mm)

0.58 0.56 0.53 0.47 0.42

Gas holdup (%)

δ (%)

Measured

Theoretical

14.17 14.34 14.51 14.68 14.85

14.92 14.05 13.46 14.20 14.60

5.28 −2.03 −7.23 −3.27 −1.71

diameter d32 and bubble average rising velocity. Comparison between measurement and prediction by the model indicates that the model can accurately predict the gas holdup for the FJRs. Moreover, for FJRs or jet reactors related, in addition to the model of bubble size and gas holdup, another critical task is to predict the flow rate of entrained gas based on the liquid flow rate and structure of the ejector, which will significantly benefit the proper design and operation of the jet reactor. Acknowledgments The authors are grateful to the financial support of National Natural Science Foundation of China (Nos. 91634104 and 21776122), National Key Research Program of China (No. 2018YFB0604605), Jiangsu Science and Technology Plan Project (No. BM2018007), the Fundamental Research Funds for the Central Universities. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.cej.2019.03.098.

Fig. 6. Comparison between measurement and prediction by the present model.

References

4.2.3. Air-aqueous ethanol solutions system As shown in Table 9, the energy dissipation rate rises while d32 drops with an increase in ethanol concentration. It seems that liquid surface tension may be the main factor that determines the bubble size in aqueous ethanol solution. With the increase of ethanol concentration, liquid viscosity increases while surface tension decreases, and the former makes bubble larger while the latter makes the bubble smaller. For the gas holdup in the column, when the concentration of aqueous ethanol is less than 6.97%, liquid viscosity may be the dominant factor. When the concentration increases further from 6.97%, the effect of liquid viscosity on the gas holdup becomes complex. Moreover, the foam was observed over the liquid when ethanol concentration was above 11.52%. Thus the liquid level needed to be determined carefully. It can be seen from Table 10 that with the increase of temperature, energy dissipation rate increases and d32 decreases accordingly, while the gas holdup changes weakly. The slight variation of gas holdup may be due to the contrary effect of both decreases in viscosity and surface tension, which has been discussed above. The experimental data were compared with the theoretical results as shown in Fig. 6. It can be found that the model can accurately predict the gas holdup in FJR, i.e., almost all the deviations are in the range of ± 15%.

[1] K.N. Yao, et al., The effect of microbubbles on gas-liquid mass transfer coefficient and degradation rate of COD in wastewater treatment, Water Sci. Technol. 73 (8) (2016) 1969–1977. [2] Y. Liao, D. Lucas, A literature review of theoretical models for drop and bubble breakup in turbulent dispersions, Chem. Eng. Sci. 64 (15) (2009) 3389–3406. [3] J. Solsvik, H.A. Jakobsen, A review of the statistical turbulence theory required extending the population balance closure models to the entire spectrum of turbulence, AIChE J. 62 (5) (2016) 1795–1820. [4] G. Montoya, et al., A review on mechanisms and models for the churn-turbulent flow regime, Chem. Eng. Sci. 141 (2016) 86–103. [5] Y.X. Liao, D. Lucas, A literature review on mechanisms and models for the coalescence process of fluid particles, Chem. Eng. Sci. 65 (10) (2010) 2851–2864. [6] J.O. Hinze, Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes, AIChE J. 1 (3) (1955) 289–295. [7] A.A. Kulkarni, Mass transfer in bubble column reactors: effect of bubble size distribution, Ind. Eng. Chem. Res. 46 (7) (2007) 2205–2211. [8] N. Kantarci, F. Borak, K.O. Ulgen, Bubble column reactors, Process Biochem. 40 (7) (2005) 2263–2283. [9] M. Ansari, H.H. Bokhari, D.E. Turney, Energy efficiency and performance of bubble generating systems, Chem. Eng. Process. 125 (2018) 44–55. [10] T. Moucha, V. Linek, K. Erokhin, J.F. Rejl, M. Fujasová, Improved power and mass transfer correlations for design and scale-up of multi-impeller gas-liquid contactors, Chem. Eng. Sci. 64 (3) (2009) 598–604. [11] F. Azizi, A.M. Al Taweel, Mass transfer in an energy-efficient high-intensity gasliquid contactor, Ind. Eng. Chem. Res. 54 (46) (2015) 11635–11652. [12] D.E. Turney, M. Ansari, D.V. Kalaga, R. Yakobov, S. Banerjee, J.B. Joshi, A micro-jet array for economic intensification of gas transfer in bioreactors, Biotechnol. Prog. (2018). [13] H. Seo, A.M. Aliyu, K.C. Kim, Enhancement of momentum transfer of bubble swarms using an ejector with water injection, Energy 162 (2018) 892–909. [14] S. Elbel, N. Lawrence, Review of recent developments in advanced ejector technology, Int. J. Refrig.-Revue Int. Du Froid 62 (2016) 1–18. [15] S.K. Majumder, G. Kundu, D. Mukherjee, Mixing mechanism in a modified cocurrent downflow bubble column, Chem. Eng. J. 112 (1–3) (2005) 45–55. [16] B.W. Atkinson, et al., Increasing gas-liquid contacting using a confined plunging liquid jet, J. Chem. Technol. Biotechnol. 78 (2–3) (2003) 269–275. [17] R.G. Cunningham, R.J. Dopkin, Jet breakup and mixing throat lengths for the liquid jet gas pump, J. Fluids Eng., Trans. ASME 96 (3) (1974) 216. [18] A.B. Little, S. Garimella, A critical review linking ejector flow phenomena with component- and system-level performance, Int. J. Refrig.-Revue Int. Du Froid 70

5. Conclusions The reactor structure, operating condition, and the motive fluid properties have significant influences on the bubble generation in the downcomer and the gas holdup in the FJR. For the intensification of the mass transfer and reaction in gas-liquid systems, the small bubbles and high gas holdup can be achieved by the appropriate combination of the parameters concerning reactor structure, fluid physical properties, and operation parameters. The mathematical model for the gas holdup of FJR consists of calculation of the energy dissipation rate, Sauter mean 9

Chemical Engineering Journal xxx (xxxx) xxxx

H. Tian, et al.

[44] R. Perez-Garibay, et al., Effect of surface electrical charge on microbubbles' terminal velocity and gas holdup, Miner. Eng. 119 (2018) 166–172. [45] S.S. Alves, et al., Bubble size in aerated stirred tanks, Chem. Eng. J. 89 (1–3) (2002) 109–117. [46] T. Hibiki, M. Ishii, Experimental study on interfacial area transport in bubbly twophase flows, Int. J. Heat Mass Transfer 42 (16) (1999) 3019–3035. [47] A. Buffo, D.L. Marchisio, Modeling and simulation of turbulent polydisperse gasliquid systems via the generalized population balance equation, Rev. Chem. Eng. 30 (1) (2014) 73–126. [48] A. Gordiychuk, et al., Size distribution and Sauter mean diameter of micro bubbles for a Venturi type bubble generator, Exp. Therm Fluid Sci. 70 (2016) 51–60. [49] H. Luo, H.F. Svendsen, Theoretical model for drop and bubble breakup in turbulent dispersions, AIChE J. 42 (5) (1996) 1225–1233. [50] R.M. Thomas, Bubble coalescence in turbulent flows, Int. J. Multiphase Flow 7 (6) (1981) 709–717. [51] W.B. Shi, et al., Modelling of breakage rate and bubble size distribution in bubble columns accounting for bubble shape variations, Chem. Eng. Sci. 187 (2018) 391–405. [52] G.M. Evans, G.J. Jameson, B.W. Atkinson, Prediction of the bubble size generated by a plunging liquid jet bubble column, Chem. Eng. Sci. 47 (13–14) (1992) 3265–3272. [53] G.A. Hughmark, Holdup in gas-liquid flow, Chem. Eng. Prog. 58 (1962) 62–65. [54] J. Xu, Boiling Heat Transfer and Gas-liquid Two-phase Flow, Atomic Energy Press, Beijing, 1993. [55] T.X. Yunbing Yao, Yingmin Gao (Eds.), Handbook of Chemistry and Physics, Shanghai science and technology press, Shanghai, 1985. [56] G. Vazquez, Estrella Alvarez, Jose M. Navaza, Surface tension of alcohol water+ water from 20 to 50. degree. C, J. Chem. Eng. Data 40 (3) (1995) 611–614. [57] K. Takamura, Herbert Fischer, Norman R. Morrow, Physical properties of aqueous glycerol solutions, J. Pet. Sci. Eng. 98 (2012) 50–660. [58] G. Besagni, et al., The effect of aspect ratio in counter-current gas-liquid bubble columns: experimental results and gas holdup correlations, Int. J. Multiphase Flow 94 (2017) 53–78. [59] M. Opletal, et al., Gas suction and mass transfer in gas-liquid up-flow ejector loop reactors. Effect of nozzle and ejector geometry, Chem. Eng. J. 353 (2018) 436–452. [60] S. Godbole, M. Honath, Y. Shah, Holdup structure in highly viscous Newtonian and non-Newtonian liquids in bubble columns, Chem. Eng. Commun. 16 (1–6) (1982) 119–134. [61] G. Olivieri, et al., Effects of viscosity and relaxation time on the hydrodynamics of gas-liquid systems, Chem. Eng. Sci. 66 (14) (2011) 3392–3399. [62] S. Rabha, M. Schubert, U. Hampel, regime transition in viscous and pseudo viscous systems: a comparative study, AIChE J. 60 (8) (2014) 3079–3090. [63] G. Besagni, et al., The dual effect of viscosity on bubble column hydrodynamics, Chem. Eng. Sci. 158 (2017) 516–545. [64] K. Akita, F. Yoshida, Gas holdup and volumetric mass transfer coefficient in bubble columns. Effects of liquid properties, Ind. Eng. Chem. Process Des. Dev. 12 (1) (1973) 76–80. [65] G.M. Evans, G.J. Jameson, C.D. Rielly, Free jet expansion and gas entrainment characteristics of a plunging liquid jet, Exp. Therm Fluid Sci. 12 (2) (1996) 142–149. [66] J. Sokolov, H.M. Jingel, Ejector, Science Press Beijing, PR China, 1977. [67] R.P. Grant, S. Middleman, Newtonian jet stability, AIChE J. 12 (4) (1966) 669–678. [68] G. Padmavathi, K.R. Rao, Effect of liquid viscosity on gas holdups in a reversed flow jet loop reactor, Can. J. Chem. Eng. 70 (4) (1992) 800–802.

(2016) 243–268. [19] A.K. Biń, Gas entrainment by plunging liquid jets, Chem. Eng. Sci. 48 (21) (1993) 3585–3630. [20] C.A. Jakubowski, et al., Ozone mass transfer in the mixing zone of a confined plunging liquid jet contactor, Ozone-Sci. Eng. 28 (3) (2006) 131–140. [21] A. Ohkawa, et al., Some flow characteristics of a vertical liquid jet system having downcomers, Chem. Eng. Sci. 41 (9) (1986) 2347–2361. [22] S.-K. Park, H.-C. Yang, An experimental investigation of the flow and mass transfer behavior in a vertical aeration process with orifice ejector, Energy 160 (2018) 954–964. [23] R. Clift, J.R. Grace, M.E. Weber, Bubbles, drops, and particles, Courier Corporation, 2005. [24] S.S. Alves, S.P. Orvalho, J.M.T. Vasconcelos, Effect of bubble contamination on rise velocity and mass transfer, Chem. Eng. Sci. 60 (1) (2005) 1–9. [25] A. Shaikh, M. Al-Dahhan, Scale-up of bubble column reactors: a review of current state-of-the-art, Ind. Eng. Chem. Res. 52 (24) (2013) 8091–8108. [26] P.M. Wilkinson, A.P. Spek, L.L. van Dierendonck, Design parameters estimation for scale-up of high-pressure bubble columns, AIChE J. 38 (4) (1992) 544–554. [27] A. Shaikh, M.H. Al-Dahhan, A review on flow regime transition in bubble columns, Int. J. Chem. Reactor Eng. (2007) 5(1). [28] R.C. Chen, J. Reese, L.-S. Fan, Flow structure in a three-dimensional bubble column and three-phase fluidized bed, AIChE J. 40 (7) (1994) 12. [29] H.N. Oguz, The role of surface disturbances in the entrainment of bubbles by a liquid jet, J. Fluid Mech. 372 (1998) 189–212. [30] D.D. Lima, I.E. Lima Neto, Effect of nozzle design on bubbly jet entrainment and oxygen transfer efficiency, J. Hydraul. Eng.-ASCE 144 (8) (2018) 06018010. [31] A. Mandal, G. Kundu, D. Mukherjee, Gas holdup and entrainment characteristics in a modified downflow bubble column with Newtonian and non-Newtonian liquid, Chem. Eng. Process. 42 (10) (2003) 777–787. [32] R.G. Cunningham, Gas compression with the liquid jet pump, J. Fluids Eng.-Trans. ASME 96 (3) (1974) 203–215. [33] G.M. Evans, A.K. Biń, P.M. Machniewski, Performance of confined plunging liquid jet bubble column as a gas-liquid reactor, Chem. Eng. Sci. 56 (3) (2001) 1151–1157. [34] D.J. Nicklin, Two-phase bubble flow, Chem. Eng. Sci. 17 (9) (1962) 693–702. [35] G.M. Evans, A study of a plunging jet bubble column, Ph. D thesis Univ. of Newcastle, 1990. [36] R. Kong, et al., Void fraction prediction and one-dimensional drift-flux analysis for horizontal two-phase flow in different pipe sizes, Exp. Therm Fluid Sci. 99 (2018) 433–445. [37] T. Hibiki, M. Ishii, One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes, Int. J. Heat Mass Transfer 46 (25) (2003) 4935–4948. [38] K. Yamagiwa, D. Kusabiraki, A. Ohkawa, Gas holdup and gas entrainment rate in downflow bubble column with gas entrainment by a liquid jet operating at high liquid throughput, J. Chem. Eng. Jpn. 23 (3) (1990) 343–348. [39] T.-Y. Sun, G.M. Faeth, Structure of turbulent bubbly jets—II. Phase property profiles, Int. J. Multiphase Flow 12 (1) (1986) 115–126. [40] T.Y. Sun, G.M. Faeth, Structure of turbulent bubbly jets—I. Methods and centerline properties, Int. J. Multiphase Flow 12 (1) (1986) 99–114. [41] M. Iguchi, et al., Structure of turbulent round bubbling jet generated by premixed gas and liquid injection, Int. J. Multiphase Flow 23 (2) (1997) 249–262. [42] A. Haapala, et al., Hydrodynamic drag and rise velocity of microbubbles in papermaking process waters, Chem. Eng. J. 162 (3) (2010) 956–964. [43] L. Parkinson, et al., The terminal rise velocity of 10–100 micron diameter bubbles in water, J. Colloid Interface Sci. 322 (1) (2008) 168–172.

10