Drag Coefficient Prediction of a Single Bubble Rising in Liquids

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Thermodynamics, Transport, and Fluid Mechanics

Drag coefficient prediction of a single bubble rising in liquids Xiaokang Yan, Kaixin Zheng, Yan Jia, Zhenyong Miao, lijun wang, Yijun Cao, and Jiongtian Liu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04743 • Publication Date (Web): 02 Apr 2018 Downloaded from http://pubs.acs.org on April 3, 2018

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Industrial & Engineering Chemistry Research

Drag coefficient prediction of a single bubble rising in liquids

Xiaokang Yan a,b , kaixin Zhengc, Yan Jiac,d, Zhenyong Miao a,b, Lijun Wang *c , Yijun Cao b Jiongtian liu b,e (aSchool of Chemical Engineering, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China; b

National Engineering Research Center of Coal Preparation and Purification, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China; c

School of Electric Power Engineering, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China) d

Department of Mechanical and Electrical Engineering, Beijing Jiaotong University Haibin College, Huanghua 061199, Hebei, China

e

School of Chemical Engineering and Energy, Zhengzhou University, 450001, Zhengzhou, China

Abstract The motion of single bubbles rising in 2-octanol solutions was investigated experimentally. By using a high-speed video system to follow the rising bubble, the sequences of the recorded frames were digitized and analyzed using image analysis software. The periodical fluctuation of the bubble terminal velocity was observed, which is indicative of a non-constant bubble drag coefficient. Then the measured drag coefficient was compared with correlations available in literature. The comparison shows that these correlations cannot give fully satisfactory results in predicting the fluctuated drag coefficient. Using the extensive experimental data, the authors proposed a new correlation to predict the drag coefficient calculated from the fluctuated motion with the added mass force and history force included. In virtue of non-linear curve fitting, the drag coefficient of single bubble is correlated as a function of the Re number, Eo number, We number and Mo number based on the 1 * Corresponding author. Tel: +86 15162110547; Email:[email protected]

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equivalent bubble diameter. This new model successfully predicts the periodical fluctuations of the drag coefficients. For the standard drag prediction (when averaged terminal velocity is used), this model agrees better with our experimental results and has a maximum relative error of 2.2% and an average relative error of 0.87%. This model is also verified by bubble rising in MIBC and OP10 aqueous solutions, and the errors are similar with that of 2-octanol solutions. Based on a comparison with measurements in other liquids from literature, this new correlation represents well the experimental data which covers 21 pure liquids and contaminated liquids in a very wide range of parameters as follows: 10-3≤Re≤105, 10-2≤Eo≤103, and 10-14≤Mo ≤107. This work is an extension of our previous one

1

in which the old correlation

predicts the fluctuated drag coefficient for bubble rising in pure water.

Keywords: Drag coefficient fluctuation; Bubble; Drag correlation; Solutions 1 Introduction The characterization of rising bubbles is important for the design of heat and mass transfer operations in chemical engineering 2-6. Drag force plays a critical role in controlling bubble motion, and a number of theoretical or empirical models are available for the prediction of bubble drag coefficients. Generally, theoretical predictions of the drag coefficients are appropriate for spherical bubbles under low Reynolds number (Re) flow, e.g.: the Stokes drag law7, the Cran and Chan correlation 8, and the Moore correlation 9. The progress in numeric algorithms has enabled attempts to solve the complex flow problems

10-12

. Dhole et

al.13, 14, Feng et al.15 and Kishore et al.16 developed drag models for spherical bubbles 2


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rising at intermediate Re flow based upon the drag forces calculated by numerical simulations. As Reynolds number increases, bubble deformation will happen when it is rising in liquids17, 18. For the convection terms which are required to be solved, theoretical solutions for the Navier-Stokes equation become difficult. Therefore, empirical or semi-empirical correlations based on experimental or numerical data are used to derive the drag coefficients. Different literature sources

18-20

experimentally proposed

drag models for deformed bubbles in terms of the Re, Eo, We, and Morton number (Mo). Kishore and Gu21, Tran-Cong et al.22 and Dijkhuizen et al.23 gave drag models for irregularly shaped particles based on the numerical simulation results. Furthermore, deformed bubbles were found to be moving in a zigzag or nearly zigzag path and undergoing a periodic oscillation in velocity as described in literature 1, 24-27

. This indicates that the drag coefficient during bubble deformation is not

constant. However, most available drag prediction models are developed in steady motion, and they cannot provide satisfactory predictions of drag coefficients in oscillating situations 1, 28. In a previous study1, we presented an improved correlation for the drag coefficient of a bubble undergoing periodic motion in pure water by an experimental approach. The problem is that this model is not valid for bubble rising in other liquids or solutions. However, due to the overwhelming applications in many chemical, environmental and mineral processing industries, such as froth flotation, waste treatment, bubble 3



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columns, gas-liquid reactors, etc., the motion of bubbles in different liquids and solutions has received considerable attention. As it is known, in a contaminated gas-fluid system, the properties of the gas/liquid interface and the solution of the system affect the bubble’s deformation, velocity, and drag force. Much work has been performed to study the motion of bubbles in contaminated liquids

29-36

. A bubble’s

deformation, terminal velocity, and diameter decreases rapidly as the concentration of surfactant in the liquid is increased, because an adsorption layer forms at the surfaces of rising bubbles, decreasing the fluidity of the gas/liquid interface and increasing viscous drag. Therefore, the drag coefficient correlations for contaminated system and pure system should be quite different. Some previous work has been done to predict the drag coefficient of a single bubble moving in various liquids and solutions. Table 1 lists four representative models. These models are chosen by imposing the following criteria: a) applicable for a variety of fluid properties and bubble diameters; b) the bubbles can have a range of shapes; c) applicable for flow with a wide range of Re number. Table 1 Correlations for bubble drag coefficient Investigator

Correlations

Peebles and

   24 18.7  CD = max max  , 0.68  , min 0.0275EoWe2 , 0.82Eo0.25 We0.5   (1) Re Re    

For contaminated

 24 8 2  CD = max  (1 + 0.1Re0.75 ) , min  , Eo   3 3   Re  48  8 Eo   24 CD = max min  (1 + 0.15 Re0.678 ) ,  ,  Re Re   3 Eo + 4  

and not applicable

Garber

17

Ishii and Chawla

37

Tomiyama et al.38

Remarks

(2)

system used only in pure system.

(3)

For both pure and contaminated system.

(for pure system)

The traveling

 72  8 Eo   24 C D = max  min  (1 + 0.15 Re0.678 ) , Re  , Re   3 Eo + 4   4


(4)

distance z to reach a terminal velocity is

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used to judge the

(for slightly contaminated system)

contaminated

8 Eo   24 CD = max  (1 + 0.15Re0.678 ) ,  3 Eo + 4   Re

degree. In literature24, he determines a full

(5)

contaminated system by z 4 mm and 62.5×50 mm for measuring 100 = 0.09765 mm/pixel or S =0.04883 mm/pixel. 1024 u ∆t (c) The shutter speed is set as ∆t 50 , then dP = is the maximum possible 50 ⋅ S

bubbles of deq ≤ 4 mm. S =

positional error due to the unsharp image result from the slow shutter speed. (d) The maximum calibration error was less than 1mm, such that dS =

1 mm/pixel 1024

Overall, the maximum uncertainty in local velocity ∆u/u was 3.5%. The uncertainty on the average terminal velocity, being obtained on a longer path, 9



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was less affected by the error in the position of the bubble’s center. The uncertainty in the average terminal velocity was much less than 3.5%.

3 Results and discussion 3.1 Bubble velocity Figure 2 shows an image sequence of bubbles rising in different concentrations of 2-octanol. The nozzle diameter is 2.7×10-3 m and the time interval between two consecutive images is 0.04 s. When the concentration of 2-octanol was increased, 1) bubble size, 2）bubble deformation, and 3)bubble velocity decreased. The change in bubble size is explained by Tate’s Law for bubble size, which is proportional to a solution’s surface tension. Deformations in bubble shape are counterbalanced by surface tension forces, which tend to restore a bubble’s spherical shape. The decrease in bubble velocity can be explained as follows. Frumkin and Levich’s theory41 states that the surfactant concentration is highest at the bubble’s stagnation point and lowest at its leading pole. A spherical stagnant cap will be formed at the rear end. Within the framework of the spherical stagnant cap model7, the degree of contamination can be expressed by the cap angle. When the concentration of contaminants is increased, the stagnant cap angle becomes larger cause drag force to rise7, 44; hence the velocity of bubble slows down.

10


42, 43

, which will

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S1

S2

S3

S4

S5

Figure 2 Image sequence of bubbles motion in solutions with different surface tensions (dn=2.7×10-3 m ∆t=0.04 s)

Generally, bubbles reach their terminal velocity after 0.1 m of being detached from a nozzle 1, 25. A typical plot of bubble terminal velocity local value vs. position is shown in Figure 3. Clearly, bubble terminal velocity undergoes a periodic fluctuation, which is indicative of a non-constant bubble drag coefficient.

Figure 3 Terminal velocities of bubbles as a function of height

11



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3.2 Drag coefficient When a single bubble is rising unsteadily in liquid, the main forces acting on it are buoyancy ( Fb ) and total drag force ( FDA = FD + FV + FH , The drag force, virtual mass force and Basset force are summed as a total drag force28, 45-47), and these forces are balanced. VG ρ G

dvT = Fb − ( FD + FV + FH ) dt

( FDA FD + FV + FH 4 = = d eq CD = 3 1 1 2  π 2  2 π 2   ρ LvT   d eq   ρ LvT   d eq  2 4 2 4      

(10)

ρ L − ρG ) g − ρ G ρ LvT2

dvT dt (11)

where VG is the equivalent volume of the bubble; ρL and ρG are the density of water and air, respectively; g is the acceleration due to gravity; A0 is the projected area of the bubble; and vT is the local velocity of the bubble and calculated by Eq.(7) after the bubble reaching the terminal velocity. Since the inertial force is much smaller than the buoyancy force. Therefore, Eq.(11) can be simplified to

CD =

4d eq ( ρL − ρG ) g 3ρ L vT2

(12)

where deq is the equivalent diameter of the bubble, which is given by

d eq =

4 A0

π

(13)

where A0 is projected bubble area measured at the nozzle orifice immediately before bubble departure. Since the terminal velocity of the bubble undergoes a periodic fluctuation (Figure 3), vT in Eq.(12) can be calculated using its local value or the average value. Drag coefficients predicted with Eqs.(1), (2), (5) and (6) were compared with 12


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experimental data (calculated with Eq.(12) and vT was the average value). Here Eq.(5) in Tomiyama’ s model is chosen due to the fact that the traveling distance to reach a terminal velocity was less than 100mm in experiment. Figure 4 presents the ratio of the predicted and experimental drag coefficients vs. Re. Values predicted with Eq.(1) either over- or underestimated the experimentally observed values over the entire range of bubble diameters tested. The Rodrigue correlation, Eq.(6), underestimated the data throughout the total range tested. This result is consistent with Rodrigue’s publication

20

in which his model may not give good prediction for Re≥100. Eq.(2)

and Eq.(5) fit the experimental data better than Eq.(1) and Eq.(5) with an average relative error within 9.1% and 12.2%, respectively.

Figure 4 Measured drag coefficient vs. calculated drag coefficient 13



However, as mentioned in section 3.1, the actual terminal velocity of the bubble underwent a periodical fluctuation. Since the Ishii and Chawla model (Eq.(2)) is likely not a valid predictor of the drag coefficients under fluctuating conditions, this model required further verification. A comparison of Eq.(2) to the experimental data (where the experimental drag coefficient was calculated with Eq.(12) and v T was the local value) is shown in Figure 5. Clearly, Eq.(2) did not reliably predict the drag coefficient of a bubble with a periodically fluctuating motion. A major reason for this is that Ishii and Chawla adopted a simple correlation (

2 Eo ) for non-spherical 3

bubbles at Eo

Drag Coefficient Prediction of a Single Bubble Rising in Liquids

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