Particle-laden bubble size and its distribution in microstructured

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Particle-laden bubble size and its distribution in microstructured bubbling bed in presence and absence of a surface active agent Ritesh Prakash, Subrata Majumder, and Anugrah Singh Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05625 • Publication Date (Web): 21 Jan 2019 Downloaded from http://pubs.acs.org on January 24, 2019

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Industrial & Engineering Chemistry Research

Particle-laden bubble size and its distribution in microstructured bubbling bed in presence and absence of a surface active agent Ritesh Prakash*, Subrata Kumar Majumder* and Anugrah Singh Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati781039, Assam, India *Corresponding author Prof. S. K. Majumder/ Mr. Ritesh Prakash Email: [email protected] or [email protected] Abstract This work reports experimental measurements of the bubble size and its distribution in the threephase (air-water-coal, air-water-coal-surfactant) system. A thorough investigation of the effect of particle concentration, particle size, viscosity, and axial positions on bubble size and its distribution in the presence and absence of the surfactant is reported. Bubble size distribution (BSD) in three different experimental conditions is studied which follows the beta, Weibull, and log-logistic nature of the distribution function. Correlations are developed for estimation of parameters of each distribution function. A generalized correlation model is also developed for the estimation of the Sauter mean bubble diameter, aspect ratio, and interfacial area in the threephase system by considering operating, geometric variables, and physical properties of the system. The current work will be useful for process intensification of chemical and biochemical processes based on the interfacial phenomena. Keywords: bubble size; bubble distribution function; particle concentration; particle size, surfactant

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1. Introduction Slurry columns are widely used three-phase contactors due to their simple construction, better transport efficiency, and low operating cost

1, 2.

The bubble size and its distribution in a non-

Newtonian systems are of great importance in many areas of chemical 3, mineral 4, and biological industries 5. Typical applications of the slurry columns are flotation, petroleum, wastewater treatment, chemical, and metallurgical industries

6.

Over the decades, the

hydrodynamic characteristics of slurry systems gathered considerable research and attracted the attention of multiphase flow reactor investigation 4, 7. The flow behaviour can significantly affect the performance of the slurry column. Accurate prediction of operating variables that govern the flow characteristics in multiphase flow is a necessary step for efficiency and design evaluation. The presence of particles substantially impacts the hydrodynamics of the multiphase system, but their effects are still not completely understood 8. Hydrodynamic characteristics of a slurry column are highly dependent on the operating variables (superficial gas velocity, superficial slurry velocity), physical properties (slurry density, slurry viscosity, slurry surface tension) and geometric variable (column length, diameter). Moreover, the efficiency of the slurry column is crucially dependent on the bubble size, rising velocity of the bubble and bubble wake phenomena 9. The comprehensive understanding of bubble generation technique, bubble size & its distribution is of fundamental significance, and they essentially affect the transport efficiency of the multiphase system. Though, in comparison with the understanding of bubbles size and its distribution in the Newtonian system, essential information about the effect of particle concentration, particle size, viscosity, axial height, and superficial gas velocity in the presence and absence of surfactant still needs to be understood in three-phase systems. As a result of the complicated flow behaviour of bubbles and the effect of the physical properties of the system, a thorough investigation of the bubble size and its distribution is a major challenge. Bubble size and its distribution significantly affect the hydrodynamics

10

and mass transfer characteristics of a slurry column. For scale-up,

modelling, and accurate design of the column it is essentially important to understand the flow and transport characteristics between different phases by studying the bubble size and its distribution

9, 11.

Numerous techniques for the measurement of bubble size and its distribution

have been used by many researchers. Some of the techniques as Mie scattering technique 2 ACS Paragon Plus Environment

12,

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pressure fluctuations 13, Bayesian magnetic resonance technique 14, resistivity probes 15, Coulter count method or pore electrical resistance method method

18,

photographic technique

19, 20,

16,

optical probe

and x-ray tomography

21

17,

two-point conductivity

are used for estimation of

bubble size. Intrusive techniques (e.g., optical probe) disturb the flow hydrodynamics, therefore non-intrusive methods (e.g., photographic technique) are favoured over intrusive technique

22.

The simplest technique to obtain the bubble size from an image is to manually draw lines on bubble interface 23. Interfacial area available for a process is a crucial variable for the design and scale-up purpose. The lower the bubble size is, the higher the interfacial area is, which improves the process performance. The interfacial area depends on the bubble size, gas holdup, sparger design, property of phases, and column geometry. Bubble size in the three-phase system depends on the physico-chemical property of the system and the type of gas distributor used 24. Passos et al.,

25

reported the measurement of the bubble size by an optical technique in a non-Newtonian solution. Addition of non-ionic surfactant (Triton X-100) results in a reduction in bubble size as well as the transition of homogenous to the heterogeneous regime is shifted to a higher gas flow rate. When smaller bubbles coalesce to form a bigger bubble, surface area for bubble-particle interaction decreases, thus process efficiency reduces

26.

Grau and Heiskanen

27

studied the

bubble size distribution and observed that reagents control the bubble size by decreasing the coalescence behaviour. The mean bubble size was found to increase with quartz particle addition and an increase in air flow rate. Yianatos et al.,

28

observed increased slurry density by an

increase in the particle concentration, which results in an increase in the bubble rise velocity 29. The rising velocity of the larger bubble is higher than the smaller bubble

30, 31,

hence the

residence time of the bubble decreases which impact the system performance 32. Since the past few years, several efforts have been carried out to investigate the bubble size and its distributions in the slurry system. The influence of slurry concentration and particle diameter on the bubble size has been studied by many investigators

30, 31, 33.

Numerous researchers

deduced that the addition of particles causes generation of bigger bubbles 11, 30-34. The increase in bubble size was because of the rise in the viscosity and mixture density of the slurry after particle addition

31.

The increase in liquid viscosity generally increases the bubble coalescence rate,

consequently enlarge the gas bubble size

34-36.

In contrast, Götz et al., 3

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reported that the small


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concentration of fine particles ( 0. The shape parameter and scale 36 ACS Paragon Plus Environment

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parameters are obtained from the experimental bubble size. The log-logistic distributions provide the closest match to the size distribution of the bubbles in (Air-water + SDS + particle) which can be represented as

 F d b   

 db     

 1

  d   1   b         

2

(13)

where  is the bubble shape parameter and  is the scale parameter. The shift parameter and scale parameter are obtained from the experiment. The experimental BSD in (Air-water + Tween 20 + particle) are well represented by a beta distribution function in agreement with the findings of many other researchers

F (d b ) 

1 (d b  a ) p 1 (b  d b ) q 1 B ( p, q ) (b  a ) p  q 1

(14)

where a is the minimum value of db, b is the maximum value of db, p is the lower shape parameter > 0, q is the upper shape parameter > 0, and B (p, q) is the beta function. The parameters b, p, and q are obtained from the experimental conditions. Generalized correlations based on the experimental conditions are developed for all the parameters of the log-logistic, beta and Weibull distribution function. Developed correlations for distribution function parameters are reported in Table 5. Table 5 Developed correlations for distribution parameters for different system. System

Correlation for distributions parameters

Air-water+ SDS+ particle (log-logistics distribution)

c    775.11 s   m 

0.477

 u sg d p  m       eff  

1.363

 u sg2     gd   p

0.092

 u sg2 d p  m       l  

Region of measure ment A

0.887

,

R2= 0.95

c    24.49 s   m 

0.176

 u sg d p  m       eff  

0.593

 u sg2     gd   p

0.419

0.98


 u sg2 d p  m       l  

0.260

, R2= (15a)


c    1.192 s   m 

0.473

1.367

 u sg d p  m       eff  

 u sg2     gd  p  

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 u sg2 d p  m       l  

1.085

 u sg2 d p  m       l  

0.0002

0.410

B R2=

,

0.94

c    0.32 s   m 

0.079

 u sg d p  m       eff  

0.067

 u sg2     gd  p  

0.354

, R2=

(15b)

0.98

c    1.192 s   m 

0.187

 u sg d p  m       eff  

0.373

 u sg2     gd   p

0.313

 u sg2 d p  m       l  

C

0.158

, R2=

0.94

c    0.32 s   m 

0.343

 u sg d p  m       eff  

0.907

 u sg2     gd   p

0.183

 u sg2 d p  m       l  

0.599

, R2= (15c)

0.92 For Airwater+Tween20 +particle (beta distribution)

c  b  1.03  10  s   m 

0.569

4

 u sg d p  m       eff  

2.179

1.066

 u sg2     gd   p

 u sg2 d p  m       l  

A

1.963

,

R2 = 0.98

c  p  2.12  10  s   m 

0.1074

3

1.063

 u sg d p  m       eff  

 u sg2     gd   p

0.426

 u sg2 d p  m       l  

0.916

,

R2 = 0.94

c  q  1.99 s   m 

0.10

 u sg d p  m       eff  

c  b  2.78  103  s   m 

0.534

0.051

 u sg2     gd   p

 u sg d p  m       eff  

0.238

1.407

 u sg2 d p  m       l  

u     gd  p   2 sg

0.133

0.319

, R2 = 0.98 (16a) B

1.204

 u d p m       l   2 sg

,

R2= 0.97

c  p  19.23 s   m 

0.138

 u sg d p  m       eff  

0.530

 u sg2     gd  p  

0.123

 u sg2 d p  m       l  

0.142

, R2=

0.96

c  q  4.18 s   m 

0.006

 u sg d p  m       eff  

0.055

 u sg2     gd  p  

0.268

 u sg2 d p  m       l  

0.169

, R2= (16b)

0.99

c  b  7.73  10  s   m  4

0.80

 u sg d p  m       eff  

2.260

 u sg2     gd  p  

R2= 0.95


0.469

C

1.844

 u sg2 d p  m       l  

,

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c  p  36.87 s   m 

0.272

 u sg d p  m       eff  

0.672

 u sg2     gd  p  

0.192

 u sg2 d p  m       l  

0.434

, R2=

0.94

c  q  1.21  10  s   m 

0.562

4

 u sg d p  m       eff  

1.872

 u sg2     gd  p  

0.471

1.405

 u sg2 d p  m       l  

, (16c)

R2= 0.97 Air-water + particle (Weibull distribution)

c    408.844 s   m 

0.406

 u sg d p  m       eff  

1.106

0.981

 u sg2 d p  m       l  

 u sg2     gd   p

A

0.418

,

R2= 0.91

c    2.37  10  s   m 

1.895

9

 u sg d p  m       eff  

5.283

 u sg2 d p  m       l  

3.929

 u sg2     gd   p

0.936

(17a)

, R2= 0.99

c    8.65  103  s   m 

0.604

 u sg d p  m       eff  

1.776

1.484

 u sg2 d p  m       l  

 u sg2     gd   p

B

0.518

,

R2= 0.89

c    2.54  105  s   m 

1.038

 u sg d p  m       eff  

2.854

 u sg d p  m       eff  

3.050

 u sg2 d p  m       l  

2.198

 u sg2 d p  m       l  

2.342

 u sg2     gd   p

0.405

 u sg2     gd  p  

0.797

(17b)

, R2= 0.96

c    6.90  10  s   m 

1.072

5

C

, R2= 0.93

c    1403.12 s   m 

0.399

 u sg d p  m       eff  

1.361

1.071

 u sg2 d p  m       l  

R2= 0.95

 u sg2     gd   p

0.046

, (17c)

Typical results of a comparison of experimental and predicted cumulative distribution function (CDF) are reported in Figure 17. Experimental values were calculated from the Eq. 12, Eq. 13, and Eq. 14 and predicted values were obtained from the developed correlations provided in the Table 5. The Figure 17a, reports the log-logistic distribution function where comparison between experimentally measured quantities (Eq. 13) and predicted values (Eq. 15a) are shown in system 1 (Table 4) with the AARE of 5.60% (SDS, 1 ppm, dp: 242.72 m) and 22.16% (SDS, 6 ppm, dp: 408.31 m). In Figure 17b, experimental (Eq. 14) and predicted quantities (Eq. 16a and Eq. 16b)



are obtained for system 3, in which AARE is found to be 25.28% (Tween 20, 1 ppm, dp: 408.31 m) and 9.46% (Tween 20, 6 ppm, dp: 242.72 m). Similarly, in Figure 17c, Weibull distribution function for the slurry system is reported in which experimental quantities are calculated using Eq. 12 whereas predicted values are obtained from Eq. 17a and Eq. 17c. In region A, AARE was found to be 6.60%, whereas in region C it was 14.92%. It can be concluded that the developed empirical correlations in Table 4 are suitable to predict the bubble size distribution function well within AARE of 30%. 1.05 SDS (1 ppm) dp: 242.72 m

Loglogistic distribution function

Beta distribution function

Experimentally obtained from Eq. 14 Predicted from Eq. 16a Experimentally obtained from Eq. 14 Predicted from Eq. 16b

1.0

Qg: 1 l/min

Region A AARE: 5.60% 0.75 ws: 0.2 wt.%

0.8

SDS (6 ppm) dp: 408.31 m

0.60

Qg: 5 l/min

CDF (-)

0.90

CDF (-)

Region A AARE: 22.16% ws: 0.8 wt.%

0.45

0.6

Tween 20 (6 ppm) dp: 242.72 m

0.4

Ql: 4 l/min

0.35

0.70

1.05

1.40

1.75

2.10

2.45

2.80

3.15

Ql: 2 l/min Region A AARE: 25.28%

0.2

Experimentally obtained from Eq. 13 Predicted from (Eq. 15a) Experimentally obtained from Eq. 13 Predicted from (Eq. 15a)

0.00 0.00

ws: 0.2 wt.%

Region B AARE: 9.46%

0.15

(a)

Tween 20 (1 ppm) dp: 408.31 m

ws: 0.8 wt.%

0.30

0.0

(b) 0.35

3.50

0.70

1.05

1.40

1.75

2.10

2.45

2.80

3.15

3.50

db (mm)

db (mm)

1.0

0.8

Weibull distribution Air-water-coal ws: 0.8 wt.%

Region A AARE: 6.60%

Ql: 3 l/min

0.6

CDF (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 59

Region C AARE: 14.92%

0.4

Experimentally obtained from Eq. 12 Predicted from Eq. 17a Experimentallly obtained from Eq. 12 Predicted from Eq. 17c

0.2

0.0

(c)

0.69

1.38

2.07

2.76

3.45

4.14

4.83

5.52

6.21

6.90

db (mm)

Figure 17. Comparison between experimental and predicted quantities (a) log logistic, (b) beta distribution, and (c) Weibull distribution. 6.4. Approach for prediction of Sauter mean bubble diameter, aspect ratio, and interfacial area 40 ACS Paragon Plus Environment

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Bubble formation in the three-phase system is a very complex phenomena and there is no availability of reliable correlations or model which can accurately predict the bubble diameter. Hence, a novel bubble size measurement correlation in three-phase is required and can be highly useful in the slurry system. Predicting the mean bubble size is essential for increasing the understanding of the formation of bubbles in the three-phase system. In the current investigation, an effort has been made to estimate the Sauter mean bubble diameter by developing a generalized empirical correlation. Empirical correlations are developed for the three-phase system. In the three-phase system, it was observed that Sauter mean bubble diameter depends on the superficial gas velocity, particle diameter, mixture density, slurry viscosity, slurry surface tension, axial height, particle concentration, and acceleration due to gravity. The general functional form considering the operating variables and physical properties of the system is expressed as

d 32 f u sg , d p ,  m ,  sl ,  sl , z, cs , g 

(18)

Using Buckingham’s pi theorem of dimensional analysis, the following relation is obtained

u d   d 32  A  sg p m  Z   sl 

b

 u sg2 d p  m       sl  

c

 u sg2     gd   p

d

 z    d   p

e

 cs      m

f

(19)

The values of A, b, c, d, e, and f were obtained by multiple linear regression analysis. An empirical correlation for the three-phase system can be expressed as

u d   d 32  1.79  10 11  sg p m  Z   sl 

4.033

 u sg2 d p  m       sl  

3.408

1.786

 u sg2     gd   p

 z    d   p

0.936

 cs      m

0.006

(20)

The Eq. (20) is valid for a definite range of the following parameters:

2.74  (u sg d p  m /  sl )  18.20 ;

141.18  ( z / d p )  1100 ;

2.49  10 4  (u sg2 d p  m /  sl )  9.02  10 3 ;

1.50  10 2  (u sg2 / gd p )  5.34  10 1 ;

1.90  10 3  cs /  m   1.13  10 2 .



The developed correlation (Eq. 20) fits the experimental data well with the correlation coefficient (R2 = 0.99) and standard error (0.070) and suitable for prediction of Sauter mean bubble diameter in three-phase with AARE of 5.59%. Developed correlation is suitable to predict the experimentally measured Sauter mean bubble diameter within the error range of 16.93%. The experimental values of Sauter mean bubble diameter were also compared with the correlations of Pohorecki et al., 97, Wilkinson et al., 98, Zhang et al., 99, and Lemoine et al.,

100

to evaluate their

range of validity. The comparison of the present correlation (Eq. 20) with those correlations is shown in Figure 18. It is noticed that correlations suggested by Pohorecki et al., 97 and Wilkinson et al., 98 overpredicts the Sauter mean bubble diameter. As per correlation given by Zhang et al., 99,

some values are in the range of present experimental data while the other data points are

underpredcited. The overprediction and underprediction of the values are due to variation in experimental conditions. The important relevant correlations suggested by different investigators, which were compared with our present developed correlation for Sauter mean bubble diameter is given in the Table S2 as supporting information.

5

34 6. +1

%

er

r ro

4

d32- predicted (mm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 59

%

.92 -16

3

e rr

or

Air-water (dp = 242.72 m) Air-water (dp = 408.31 m) SDS (dp = 242.72 m) SDS (dp = 408.31 m)

2

Tween 20 (dp = 242.72 m) Tween 20 (dp = 408.31 m) 97

Pohorecki et al. 98 Wilkinson et al. 99 Zhanag et al. 100 Lemoine et al.

1

AARE: 5.59% (Eq. 20)

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

d32- experimental (mm)


4.0

4.5

5.0

5.5

Page 43 of 59

Figure 18. Parity of Sauter mean bubble diameter in the three-phase system. The aspect ratio of a gas bubble can be expressed as the ratio of its minor and major axis. Estimation of aspect ratio is a matter of concern for assessment of bubble size and its distribution, flow regime transition101, and deformation of bubbles during the rising process102. Many researchers have attempted to make a correlation that relates the aspect ratio () with a dimensionless number. Some researchers have established a correlation based on Eotvos number (Eo) 101-104, Weber number (We)104-108, whereas some of the other researchers used dimensionless group which is the function of Reynolds number (Re) and Morton number (Mo)109 and Eotvos number110. A typical plot of the aspect ratio of gas bubbles for a discrete bubbly flow regime at different experimental condition is shown in Figure 19a  19d. It is observed that aspect ratio values of bubble shape are distributed widely. Based on the aspect ratio, the size discretizes to different equivalent bubble size classes. Each bubble size class represents the average aspect ratio of the gas bubbles belonging to that class. AR = 6 Qg = 3 l/min

1.0

AR: 7.8 Qg: 3 l/min

1.2

0.9 1.0

 (-)

0.8

 (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.7

0.6

0.6

0.5

0.4 (a)

0.8

0.4 (b)

1.26

1.68

2.10

2.52

2.94

3.36

3.78

4.20

4.62

1.44

1.80

2.16

2.52

2.88

deq (mm)

deq (mm)


3.24

3.60

3.96

4.32


1.1

1.1

AR: 6 Qg: 5 l/min

1.0

AR: 7.8 Qg: 5 l/min

1.0

0.9

0.9

0.8

0.8

0.7

 (-)

 (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 59

0.7

0.6

0.6 0.5

0.5 0.4

0.4 (c)

0.3 1.56

1.95

2.34

2.73

3.12

3.51

3.90

4.29

4.68

1.98

2.31

2.64

2.97

deq (mm)

3.30

3.63

3.96

4.29

4.62

deq (mm)

Figure 19. Distribution of aspect ratio of bubbles at different experimental conditions. As per report of Besagni and Inzoli

101

the aspect ratio can be well predicted statistically by

single dimensionless number only instead of considering various dimensionless numbers simultaneously. Therefore, based on the experimental data and numerical fitting, a correlation for prediction of aspect ratio is proposed in terms of Weber number, which is expressed as 

1

(21)

1.0012

1.2  0.0099We

Experimentally calculated aspect ratio is also compared with the correlations of Moore Taylor and Acrivos Maeda

109,

Li et al.,

106,Wellek 102,

et al.,

104,Fan

and Tsuchiya

and Besagni and Inzoli

101

correlations except the correlation of Li et al.,

103,Okawa

et al.,

111,

105,

Tadaki and

as shown in Figure 20. It is observed that all 102

either over predict or underpredict the

experimental gas bubbles aspect ratio. The correlation of Li et al.,

102

predicts the aspect ratio

with the AARE of 6.79%, whereas the present correlation (Eq. 21) predicts within the AARE of 4.48%. The correlations for the bubble aspect ratio against which the present developed correlation is compared is provided in the Supporting Information as a Table S3.


Page 45 of 59

1.1 1.0 0.9 0.8 0.7 0.6

 (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Present experiemntal data Present correlation (Eq. 21) 105 Moore et al. 106 Taylor and Acrivos 104 Wellek et al. 103 Fan and Tsuchiya 111 Okawa et al. 109 Tadaki and Maeda 102 Li et al. 101 Besagni and Inzoli

0.5 0.4 0.3 0.2 0.1 0.0 1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

deq (mm) Figure 20. Comparison of the experimental aspect ratio with predicted and others available aspect ratio correlation. Similarly, a generalized correlation for prediction of interfacial area of gas bubble is also developed by applying Buckingham’s Pi theorem. The generalized correlation for the interfacial area is represented as

u d   ai  1.61  1015  sg p m    sl 

4.783

 u sg2 d p  m       sl  

3.762

 u sg2     gd   p

1.126

 cs     m 

0.146

 z    d   p

0.080

(22)

The correlation coefficient and standard error for developed correlation (Eq. 22) are 0.92 and 0.174 respectively. The Eq. (22) is suitable to estimate the interfacial area within the error range of  29.90% and AARE of 13.90%. The developed correlation (Eq. 22) is also compared with the correlations present in the literature

47, 112, 113

to analyse their range of validity as shown in

Figure 21. All the literature correlations for the prediction of bubble interfacial area against which the current developed correlation is compared is provided in the Supporting Information 45 ACS Paragon Plus Environment


(Table S4). The Eq. (22) is valid for a definite range of the following parameters:

2.90  (u sg d p  m /  sl )  18.21 ;

141.18  ( z / d p )  1100 ;

2.50  10 4  (u sg2 d p  m /  sl )  9.02  10 3 ;

1.50  10 2  (u sg2 / gd p )  5.34  10 1 ;

1.99  10 3  cs /  m   1.17  10 2 . 900 Air-water (dp = 242.72 m) Air-water (dp = 408.31 m)

800

er ro r

SDS (dp = 242.72 m)

700

%

SDS (dp = 408.31 m)

.7 2

Tween 20 (dp = 242.72 m)

29

Tween 20 (dp = 408.31 m) 47

2

+

Besagni and Inzoli 112 Akita and Yoshida 113 Gestrich and Krauss

600

3

ai- predicted (m /m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 59

500

9 -2

400 300

.90

%

or err

AARE: 13.90% (Eq. 22)

200 100 0 0

100

200

300

400

500

600 2

700

800

900

3

ai- experimental (m /m ) Figure 21. Comparison of experimental and predicted interfacial area in the three-phase system.

Conclusion A photographic technique was used to measure the bubble size and its distribution in three axial regions of the microstructured slurry column. A detailed study was conducted to enunciate the effect of particle concentration, particle size, surfactant type, and surfactant doses on bubble size, bubble size distribution, and the type of the distribution function in a discrete bubbly flow 46 ACS Paragon Plus Environment

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regime. A linear relationship was observed between superficial gas velocity and bubble size. Addition of surfactant inhibits the rate of coalescence of bubbles, which result in smaller bubble size and narrower bubble size distribution compared to three-phase system. A slight difference in bubble size distribution observed when solid loading increases from 0.2 to 0.8 wt.% in the presence of the surfactant. The bubble diameter increased with increasing distance from the base of the column due to the coalescence of bubbles. The size distributions of bubbles in the presence of solid particles were wider than the two-phase system. It was also observed that as axial height increases the bubble size distribution curve shifted from smaller bubble diameter to the larger bubble diameter. In all systems, the obtained bubble size distributions were not symmetrical as normal distributions. The obtained bubble size distribution observed to follow distribution function, namely, beta, log-logistic, and Weibull in different experimental conditions. Correlations were developed for the estimation of each distribution parameters of all the distribution functions for the three axial regions of the column. A generalized correlation model is developed to predict the Sauter mean bubble diameter, bubble aspect ratio, and bubble interfacial area in the three-phase system and also compared with the published correlation in the literature. All the developed correlations are suitable to predict the experimental data well within the error range of  29.90% and AARE of 13.90%. The present experiment will be helpful for process intensification of a multiphase system. Supporting Information: 

Table S1 shows discretization of bubble classes in the axial region B in the presence of Tween 20 surfactant at 0.008 m/s superficial gas velocity is shown in Table S1



Table S2 shows the different correlations proposed by various investigators for Sauter mean bubble diameter against which the present proposed correlation is compared



Table S3 shows the different correlations proposed by various investigators for aspect ratio against which the present proposed correlation is compared



Table S4 shows the different correlations for the bubble interfacial area against which the present proposed correlation is compared

Nomenclature ai

Interfacial area (m2/m3) 47 ACS Paragon Plus Environment


a

Minimum value of bubble size in Eq. (14) (-)

AC

Cross section area of the column (m2)

AARE

Absolute average relative error (%)

AR

Aspect ratio (-)

b

Maximum value of bubble size in Eq. (14) (-)

B (p, q)

Beta Function in Eq. (14) (-)

BSD

Bubble size distribution (-)

CDF

Cumulative distribution function (-)

cs

Particle concentration (kg/m3)

cf

Surfactant concentration (ppm)

cv

Volumetric particle concentration (vol.%)

dp

Particle diameter (m)

db

Bubble diameter (m)

dc

Column diameter (m)

d32

Sauter mean bubble diameter (m)

deq

Equivalent bubble diameter (m)

d0

Sparger diameter (m)

F(db)

Bubble distribution function (-)

g

Acceleration due to gravity (m/s2)

hm

Gas-liquid-solid mixture height (m)

L

Column length (m) 48 ACS Paragon Plus Environment

Page 48 of 59

Page 49 of 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


lmajor

Maximum axial length of the bubble (m)

lminor

Minimum axial length of the bubble (m)

ms

Mass of solid (kg)

Mw-gas

Molecular weight of gas (kg/kmol)

n

Flow behaviour index (-)

ni

Number of bubbles (-)

N

Total number of variables (-)

p

Lower shape parameter in Eq. (14) (-)

q

Upper shape parameter in Eq. (14) (-)

R2

Correlation coefficient (-)

RF

Relative frequency (-)

RPM

Rotation per minute (-)

STDEV

Standard deviation (-)

T

Temperature (k)

usg

Superficial gas velocity (m/s)

U

Standard uncertainty (-)

Ur

Relative uncertainty (-)

ws

Particle concentration (wt.%)

Xw

Concentration of purest component in the liquid mixture (wt.%)

xi

ith component of the variables (-) 49 ACS Paragon Plus Environment


x

Average of data points (-)

z

Axial height (m)

Non-dimensional number Eo

2 Eotvos number g (  l   g )d eq /  l (-)

Mo

4 2 3 Morton number g (  l   g )  l /  l  l (-)

Re

Reynolds number  sl u sg d eq / l ) (-)

Fr

2 Froude number (u sg / d eq g ) (-)

We

2 Weber number (  l u sg d eq /  l ) (-)

Greek Letter 

Aspect ratio of bubble (-)

g

Gas holdup (-)



Bubble shape parameter in Eq. (12) (-)

l

Liquid holdup (-)

s

Solid holdup (-)



Energy dissipation rate per unit mass (m2/s3)

β

Scale parameter of bubble in Eq. (12) (-)

l

Surface tension of liquid (kg/m.s)

sl

Surface tension of slurry (kg/m.s)

m

Mixture (gas-liquid-solid) density (kg/m3)


Page 50 of 59

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l

Density of liquid (kg/m3)

s

Density of particle (kg/m3)

sl

Density of slurry (kg/m3)

g

Density of gas (kg/m3)

eff

Effective viscosity (Pa.s)

l

Viscosity of liquid (Pa.s)

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