FLUID FLOW AND TRANSPORT PHENOMENA Chinese Journal of Chemical Engineering, 18(6) 923—930 (2010)
Hydrodynamic Behavior of a Single Bubble Rising in Viscous Liquids* CAI Ziqi (蔡子琦), BAO Yuyun (包雨云) and GAO Zhengming (高正明)**
State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Abstract The rising behavior of single bubbles has been investigated in six systems with different viscosity and Morton number (Mo) from 3.21×10−11 to 163. Bubbles with maximum equivalent diameter of up to 16 mm were investigated. The bubble Reynolds number (Re) ranged from 0.02 to 1200 covering 3 regimes in which two functions are obtained relating the drag coefficient, CD, with Re and Mo. It has been found that in the high Reynolds number regime the drag coefficient increases until the Reynolds number of about 1200. The classic expression of Jamialahmadi (1994) is improved and extended to high viscosity liquids. A new relationship for the aspect ratio of deformed bubbles in terms of Re, the Eötvös number and Mo, applicable to a wide range of system properties, especially in high viscosity liquids, is also suggested. Keywords bubble, rising velocity, viscous liquid, drag coefficient, deformation
1
INTRODUCTION
Gas-liquid reactors are widely used in industrial processes such as food processing, microbe cultivation, wastewater treatment, the production of detergents, lubricants, cosmetics and many other fine chemicals, where the performance depends on mass and heat transfer between phases. The overall efficiency of a gas-liquid reactor is often determined by the characteristics of the dispersion in which the gas phase is usually in the form of bubbles. Gas holdup and bubble diameter are crucial parameters for reactor performance as these directly affect bubble velocity and residence time. Of all the factors mentioned above, the single bubble rising velocity, which is determined largely by shape and size, is the main factor determining bubble behavior, and thus the subject of the present investigation. There has been much research in the past on single bubble rising velocity, also termed as terminal velocity. Generally speaking, three approaches have been used to develop correlations of bubble rising velocity: force balance, dimensional analysis and wave theory. Davies and Taylor [1] proposed that bubble rising velocity, Ub, could be related to bubble radius of curvature, r, and Clift et al. [2] described bubble rising velocity in terms of equivalent diameter. The force balance approach was commonly applied to calculate the bubble rising velocity by researchers like Stokes [3]. In this approach, the gravity, buoyancy, surface tension and other forces can easily be obtained from known equations, but it is relatively difficult to determine the drag force. Most investigations have used a Reynolds number based drag coefficient (CD) to estimate the drag force and in this way developed corelations between bubble size and rise velocity. Levich [4] proposed a simple correlation between CD and Re for single clean bubble in water. Levich’s correlation was
derived from Stokes’s Law, 48 CD = Re = f ( Db ) (1) Re Peebles and Garber [5] used liquids with a wide range of physical properties deriving a series of correlating equations (2) for CD involving the system properties in terms of the Reynolds number:
CD = 48 Re−1
Re < 2
(2) CD = 29 Re Re > 2 Using a dimensional analysis approach, Rodrigue [6] obtained a correlation for CD in terms of Reynolds number Re and Morton number Mo, extending the correlation in terms of velocity number Vn and flow number Fn according to his definition (which were related to bubble conditions and liquid properties), and his correlation of drag coefficient could be simplified as CD ∝ Re −0.65 Mo0.05 . Also using the dimensional analysis, Tomiyama [7] introduced the concept of aspect ratio, E, and proposed a correlation for bubbles of all possible shapes in pure and contaminated Newtonian liquids. Sanada et al. [8] used especially purified water to investigate bubble behavior. They reported that drag coefficients at low Reynolds numbers are greatly affected by traces of contaminants and also examined the effects of bubble shape on the motion. Using wave theory as a starting point, Mendelson [9] has suggested that bubble surface dynamics are similar to the waves on the surface of an ideal liquid and so can be correlated in terms of the bubble size and fluid properties. To reach a general understanding of bubble motion in a quiescent Newtonian fluid the behavior of a freely rising single bubble is investigated in this work, in which new correlations of bubble drag coefficient are derived and the calculation of bubble rising velocity is extended to relatively higher viscosity liquids −0.68
Received 2010-04-27, accepted 2010-08-25. * Supported by the National Natural Science Foundation of China (20821004, 20990224) and by the National Basic Research Program of China (2007CB714300). ** To whom correspondence should be addressed. E-mail:
[email protected]
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Chin. J. Chem. Eng., Vol. 18, No. 6, December 2010 Table 1
Physical properties of aqueous glycerol solution at 18.5 °C
Solution
Volume fraction of glycerol
Mass fraction
Density/kg·m−3
Viscosity/Pa·s
Surface tension/N·m−1
Mo
1
0
0
995
0.00106
0.072
3.21×10−11
2
0.20
0.241
1054
0.00948
0.0564
4.19×10−7
3
0.816
0.849
1220
0.105
0.0607
4.35×10−3
4
0.914
0.931
1241
0.338
0.0610
0.456
5
0.98
0.984
1264
0.972
0.0601
31.9
6
1
1
1265
1.515
0.059
163
compared with the literature reports. The terminal velocity is determined from non-intrusive high speed camera visualization, and the relationship between the three dimensionless numbers: Re, Eötvös number Eo and the Morton number Mo with the drag coefficient CD is plotted over a wide range. Bubble shape and deformation are also investigated in terms of Re, Eo and Mo. 2
(a) In S2, 2.7 mm, 4.9 mm, 8.8 mm, 12 mm
(b) In S4, 2.7 mm, 5.5 mm, 9.2 mm, 11.5 mm
EXPERIMENTAL
In this experiment, a cuboid plexiglass tank is used as the test container (20 mm×200 mm×600 mm, wall thickness 2 mm). De-ionized water and aqueous solution of glycerol are used as the liquids with Mo ranging from 3.21×10−11 to 163. The physical properties of these liquids are shown in Table 1. The liquid viscosity is measured with a RS150 Rheometer (Haake, Germany), the surface tension is measured using the bubble-pressure method. An air bubble of less than 5 mm equivalent diameter is introduced using a mini-syringe. Larger bubbles (up to 20 mm diameter) are made by injection from glass tubes of between 3 and 9 mm diameter. The equivalent spherical diameter, De, is given in terms of bubble volume Vb, by 1/ 3
⎛ 6V ⎞ De = ⎜ b ⎟ ⎝ π ⎠
(3)
The bubble behavior is recorded by a high speed CMOS camera (FASTCAM-ultima APX, Photron Co., USA). In order to obtain clear pictures the frame rate can be up to 1000 fps, which provides a sharp edge to the bubble images. The image with a resolution of 1024×1024 pixels was produced by a Nikkor Micro 60 mm f 2.8D lens. The position of the bubble barycenter and measurements of the rising velocity and diameter were obtained using the MATLAB image processing programs. The image handling procedure involves background subtraction, noise reduction with a median filter, bit quantization by the automatic discriminator threshold selection method, edge shape check-up and fitting, bubble center spot fitting, and velocity calculation. Original images of bubbles of differing sizes in three different liquids are as shown in Fig. 1. The bubble barycenter is estimated by Eq. (4) as
(c) In S6, 2.2 mm, 5.6 mm, 8.9 mm, 12 mm Figure 1 Images of bubbles of various equivalent spherical diameters in solutions 2, 4 and 6
X=
1 N ∑ xi N i
and
Y=
1 N ∑ yi N i
(4)
where X and Y are the center of mass for a bubble, xi and yi are the pixel position of the ith pixel of the bubble spot and N is the total number of pixels. In the investigation of single bubble behavior, the camera position is set about 45 cm above the point of the syringe needle, where the bubble velocity has become stabilized. The recording area is a rectangle of 20 mm×20 mm or 40 mm×40 mm, depending on the bubble size, which means the precision of the bubble images are 0.019 mm per pixel and 0.039 mm per pixel, respectively. 3 3.1
RESULTS AND DISCUSSION Definitions
There are some important parameters in describing the bubble behavior. Ignoring the density and viscosity of the gas phase, the dynamic behavior of a single bubble rising in quiescent liquid can be described with three dependent dimensionless numbers. These are Reynolds number Re, Eötvös number Eo and Morton number Mo, defined by Re =
ρU b De μ
Eo =
g ρ De2
σ
Mo =
gμ4
ρσ 3
Re and Eo reflect the effects of liquid viscosity and
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surface tension on bubble rising velocity and deformation, Mo is a characterizing number dependent on the liquid properties only. The aspect ratio, E, which describes the bubble deformation, is defined as the ratio between the lengths of the minor and major axes of the bubble and always less than 1 when rising in stagnant liquid. For spherical and ellipsoidal bubbles, E is easy to be evaluated, but when bubble is large enough, it becomes sphericalcapped or ellipsoidal-capped, so here its cross section should be “repaired” to be a whole sphere or ellipse, then E equals to the lengths of minor axis and major axis of the “repaired” sphere or ellipse [15]. 3.2
Evaluation of measurement accuracy
It is inevitable that there is some uncertainty in experiments. Eq. (5) [10] gives the uncertainty of this experiment in calculating the bubble velocity: ΔU = 2
2
2
2
⎛ ∂U ⎞ ⎛ ∂U ⎞ ⎛ ∂U ⎞ ⎛ ∂U ⎞ ⎜ ∂P dP1 ⎟ + ⎜ ∂P dP2 ⎟ + ⎜⎝ ∂Δt dΔt ⎟⎠ + ⎜⎝ ∂S dS ⎟⎠ ⎝ 1 ⎠ ⎝ 2 ⎠ (5) In this P1 and P2 are the vertical positions of the barycenter of a bubble in two consecutive images, S is the precision of the bubble image (mm per pixel) and Δt is the time interval between two images. The velocity U = ( P2 S − P1S ) / Δt , so that ∂U P2 S − P1S S S ∂U ∂U =− , = , , = ∂Δt ∂P1 Δt ∂P2 Δt Δt 2 ∂U P2 − P1 = ∂S Δt When referring to instantaneous rising velocity, the relevant calculation is made on two consecutive frames. So, in that case, (1) Δt = (1000 fps) −1 = 0.001 s; dΔt = 10−6 . (2) dP is the positional error of the image taken by the camera, which is associated with bubble velocity. When the bubble is rising in the tank, if the shutter speed is slow and the bubble velocity high, the image will not be sharp and lead to an error in determining the bubble barycenter. At slow rise velocities the
(a) 8 ms, De = 5.2 mm, U = 225 mm·s−1
(b) 4 ms, De = 1.5 mm, U = 293 mm·s−1 Figure 3
barycenter positional calculation will be accurate. We assume that dP is a function of bubble velocity U. When U is as great as 400 mm·s−1, dP equals to ±1 pixel; when U is zero, dP is also zero. Meanwhile, the increase of dP is greater at greater U. It is therefore assumed that there is a relation of dP = kU 2 , where k = 6.25×10−6 mm2·(s2·pixel)−1. 21 mm 19 mm − = 1.96×10−3 mm (3) dS < 1024 pixel 1024 pixel per pixel. dS is the error of calibration. In an area of 1024×1024 pixels, if we recall that the real area is 19 mm×19 mm or 21 mm×21 mm, there might some artificial error of ±1 pixel. When ΔU is calculated, the relative error in measurement can be obtained as Er = (ΔU / U ) × 100% , and its relation with bubble velocity Ub is established. The resulting relative percentage error is shown in Fig. 2
Figure 2 The relation between Ub and relative error area 40 mm×40 mm; area 20 mm×20 mm
Figure 2 shows that the relative error increases from 5% to 8.5% of an area 20 mm×20 mm, and from 2.6% to 13.7% of an area 40 mm×40 mm, for rising velocities from 10 mm·s−1 to 400 mm·s−1. As we can see from Eq. (5), the major source of uncertainty in dP is precision of the edge of the processed image. It is necessary to decrease the effect of dP by using an appropriate shutter speed. Furthermore, any deviation of focusing plane for the bubble will produce an unsharp image and be another source of error in dP. Fig. 3 indicates the different circumstances of bubbles images with various shutter speeds, from left to right, 8 ms, 4 ms, 2 ms and 1 ms, respectively.
(c) 2 ms, De = 3.3 mm, U = 237 mm·s−1
(d) 1 ms, De = 7.4 mm, U = 241 mm·s−1
Original bubble images at various shutter speeds in water
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It is obvious that relatively slow shutter speed makes the edge of the image out of focus and it is too blurry for the computer to recognize bubble edge exactly. The appropriate shutter speed is required since the increasing of the shutter speed is helpful for getting an exact original bubble image and over-high shutter speed and photographing frame will increase the calculation quantities at the same time. 3.3
Single bubble behavior (a) High Re (Re>20) and CD in S1 fitting line; Peebles & Garber (1953); Levich (1949); Bhaga & Weber (1981); ■ S1
3.3.1 Bubble drag coefficient The drag coefficient CD for a spherical bubble was calculated from experimental data using Eq. (6) below, which is derived from the force balance between the buoyancy and drag forces on a bubble when it reaches the terminal velocity. In Eq. (6), A is the maximum cross-sectional area (perpendicular to the direction of bubble motion) corresponding to the bubble with the equivalent diameter of De. For spherical bubbles A = πDe2 / 4 is used, we have
CD =
πDe3 g 3 AU b2
=
4 gDe 3U b2
(6)
For deformed bubbles, A is calculated by other integral or image recognition ways. We derived experimental curves of bubble rise velocity plotted as a function of bubble equivalent diameter in Fig. 4. The drag coefficients are plotted as a function of bubble Reynolds number in Fig. 5.
(b) Low Re (Re
