Fractal Structure in Gas–Liquid–Solid Circulating Fluidized Beds with

Article pubs.acs.org/IECR

Fractal Structure in Gas−Liquid−Solid Circulating Fluidized Beds with Low Solid Holdups of Macroporous Resin Particles Jianhua Liu,† Mingyan Liu,*,†,‡ and Zongding Hu† †

School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China State Key Laboratory of Chemical Engineering, Tianjin 300072, China

‡

ABSTRACT: The fractal structures of flow images of gas−liquid−solid circulating fluidized beds (GLSCFBs) with low solid holdups of macroporous resin particles at different operating conditions and liquid physical properties were investigated by the fractal analysis method. The images were obtained with the high-speed image acquisition and treatment system of complementary metal oxide semiconductor (CMOS). The results show that the structure of the gas−liquid−solid circulating flow exhibits certain fractal behavior and the fractal dimension D can be used to describe the multiphase self-organization flow structure from the point of view of geometry. The fractal dimension changes with the phase holdup and the agglomeration degree of bubbles or/and particles. Generally, higher solid or gas holdup leads to larger fractal dimension and more extensive aggregation or self-organization degree results in lower entropy and thus lower fractal dimension. In order to avoid the appearance of the aggregation in the GLSCFB riser, higher superficial velocity of primary liquid flow, lower superficial velocity of auxiliary liquid flow, and lower liquid surface tension may be exerted.

1. INTRODUCTION A gas−liquid−solid circulating fluidized bed (GLSCFB) is an important multiphase flow operation mode or installation with relatively high superficial fluidized liquid velocity and certain circulation rate of solid particles and can be utilized in many industry processes such as chemical, petrochemical, and biochemical, etc.1−15 However, the GLSCFB is an open nonlinear dynamic system and the three-phase flow in the bed often exhibits itself in a self-organization agglomeration structure which is a complex image and is difficult to describe quantitatively.4 As mentioned above, GLSCFB is actually an open nonlinear dynamic system, in which, the flow structure is coherent or ordered under the continuous inputs of gas and liquid flows from the environment. Self-organization phenomena in the GLSCFB refer to the movement of fluidized solid particle in groups, the spirally ascending motion of bubbles and their wakes. The self-organization behavior can be characterized by analyzing clustering characters of particle or bubble phases.4 In order to understand the complex multiscale structure of gas−liquid−solid flow including gas−liquid flow, the energyminimization multi-scale (EMMS) method,16,17 computational fluid dynamics (CFD) simulations,18,19 and advanced measurement techniques such as a three-dimensional (3D) image reconstruction technique for electrical capacitance tomography (ECT) imaging based on a neural-network multicriterion optimization (NNMOIRT)20 were employed. It was shown that these tools are effective to a certain degree. However, gas−liquid−solid flow self-organization structure is also a kind of nonlinear phenomenon and may be studied by the nonlinear analysis method, including the fractal and chaotic tools. Self-organization and chaos in a gas−solid fluidized bed were investigated by analyzing the time series of pressure fluctuation, and experimental evidence that a complex system of particles © 2013 American Chemical Society

suspended by upward-moving gas can exhibit low-dimensional bulk behavior or a large-scale collective particle motion referred to as slugging in an industrial device was presented.21 Particle self-organization behaviors in a liquid−solid expanded bed and a liquid−solid circulating fluidized bed were studied with fractal analysis and reconstruction methods by An and Liu,22−24 respectively. The self-organization structure of gas bubbles in a gas−liquid−solid expended bed was studied with fractal methods by Duan.25 Pressure fluctuations in a gas−liquid−solid fluidized bed under different batch operating conditions were analyzed in terms of Hurst’s rescaled range (R/S) analysis, thus yielding the estimates for the so-called Hurst exponent, H. The time series of the pressure fluctuations has a local fractal dimension of dFL = 2 − H. An H value of 1/2 signifies that the time series follows Brownian motion; otherwise, it follows fractional Brownian motion (FBM).26 The turbulent motion of particles in gas− liquid−solid systems was characterized by the fractal analysis of particle trajectories obtained from direct visualization using a two-dimensional (2D) column. The particle motion depends strongly upon large-scale turbulent eddies in the bubble wake structure. The fractal dimension of particle trajectories was found to be a maximum at a gas flow rate corresponding to the transition of bubble flow from homogeneous bubbly to churnturbulent flow regimes.27 The deterministic chaotic motions of bubbles and particles in a 3D gas−liquid−solid fluidized bed were studied by the fractal analysis of time series data of bubble and particle frequencies measured with a novel optical Special Issue: Multiscale Structures and Systems in Process Engineering Received: Revised: Accepted: Published: 11404

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transmittance probe with a narrowed laser beam. The correlation dimensions of gas and solid phases were found to be significantly dependent on the flow regimes.28 The complex axial-mixing characteristics and flow behavior of fluidized particles in a gas−liquid−solid fluidized bed with an i.d. of 0.152 m were interpreted by means of the fractal analysis of pressure fluctuations generated in the bed. The Hurst exponent recovered from the Pox diagram of pressure fluctuations decreases with increasing gas flow rate, but it displays a local maximum with variation of the liquid flow rate in the bed containing a binary mixture of particles.29 Hydrodynamic studies were conducted in gas−liquid−solid systems of 3.0 mm glass beads and of 2.1 mm polypropylene low-density particles. Simultaneous measurement of differential pressure and bubble conductivity probe signals enabled the investigation of the change in flow structure in relation to the flow regime transitions. The measurements were extensively analyzed using fractals and chaos, power spectra frequency analysis, and wavelet decomposition in addition to the standard statistical analyses.30 Recently, Liu et al.4 experimentally investigated the clustering behaviors of solid particles and bubbles and solid particles in the gas−liquid−solid circulating fluidized bed at low solid holdups of resin particles with traditional linear analysis methods. They found that the clusters of solid particles largely exist as doublets and triplets, and the mixed groups of particles and bubbles mostly exist as one bubble carrying two to four particles. In the present paper, self-organization behaviors of solid particle, gas bubble, and bubble−particle mixed groups in the GLSCFB at different operation conditions and liquid properties will be investigated with the fractal analysis tools to further enhance the fundamental understanding on the multiphase flow in such a complex system. The multiphase flow images will be obtained with the complementary metal oxide semiconductor (CMOS) high-speed image acquisition and treatment system.

Figure 1. Experimental setup of the gas−liquid−solid circulating fluidized bed:4,5 1 riser; 2 downer; 3 separator; 4 primary liquid flow entrance; 5 auxiliary liquid flow entrance; 6 distributor of auxiliary liquid flow; 7 gas distributor of the riser; 8 square test section of the riser; 9 pressure test port of the riser; 10 upper circulating pipe; 11 butterfly valve; 12 pressure test port of the downer; 13 square test section of the downer; 14 gas distributor of the downer; 15 liquid distributor of the downer; 16 lower circulating pipe; 17 distributor of primary liquid flow; 18 personal computer; 19 high speed camera.

256 gray levels. Generally the size of the field of view shown in this work is 0.1105 m long and 0.05 or 0.06 m wide. Some images are cut from the original ones, which are about 0.05525 m long and 0.05 or 0.06 m wide. The depth of field of the images in the experiments is about 21 mm. Four 220-V 100-W incandescent lamps were arranged to the four corners of the square test section in the fluidized bed. When doing the experiments, the four lamps were turned on for lighting. The illumination system provides a volume light source. Datum and image processing was carried out by using the developed software Chemical 2.031 and common software Photoshop. The identification of particle, bubble, and mixed clusters was done by browsing a series of consecutive images by hand.4 A real cluster and a false group or pseudocluster cannot be determined only by a 2D image but should be judged by a series of consecutive images. If a cloud of particles, bubbles, or mixture of particles and bubbles in an image is a pseudocluster, there will be an obvious relative motion between them when the consecutive images are rapidly browsed. Those which are together with each other where there is not an obvious relative motion during the rapidly browsing of 20−30 images are considered as real clusters.4 The fundamental criterion is that the velocities of the cluster members are basically the same in a certain period of time. Bubble separation from a gas−liquid−solid flow image includes the removal of background noise, drawing of the bubbles, and binary conversion of gray levels or image threshold value split. The bubble was identified by the gray level of the image. Any point for which the gray level is higher than the global threshold is called an object point; otherwise, the point is called a background point. After this step, a binary image is obtained. The estimation of solid holdup of images consists of median filtering of the image after threshold value split, profile extraction, thinning, and solid holdup calculation.31 The relative error of calculation of solid holdup is ±0.01 and gas holdup ±0.05.

2. EXPERIMENTS, IMAGE ACQUISITION, AND TREATMENT Details of the experimental setup and reagents, high-speed CMOS imaging acquisition and datum processing systems are available in the literature,4,5 and only a brief description is given here. The calculation steps of the fractal dimension is also be shown in this section. As shown in Figure 1, the GLSCFB apparatus includes a riser with inner diameter of 0.05 m and height of 3 m, a downer with inner diameter of 0.06 m and height of 3.5 m, and a separator. In the visual measurements, the cross-section shape of the test sections both in the riser and in the downer are square and are made of plexiglass. The side length of the riser cross-section is 0.05 m and the height is 1.3 m. The side length of the downer cross-section is 0.06 m and the height is 1.0 m. The experiment was carried out at room temperature. Compressed air was adopted as gas phase; tap water and the solutions of sodium carboxymethyl cellulose (CMC-Na) and sodium dodecyl sulfate (SDS) were adopted as liquid phase; the AB-8 macroporous resin, D101 resin, and D101-I resin were adopted as solid particle phase. CMC-Na and SDS were used to adjust the viscosity and the surface tension of the liquid phase, respectively. The high-speed CMOS imaging system mainly contains a digital camera, a digital grabbing card, and the software. The image sampling rate was set as 500 fps. The images grabbed are 11405

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Figure 2. Box-counting dimensions of three mathematical fractal images: (a) Koch curve; (b) Sierpinski triangle; (c) Sierpinski rectangle.

Each image of the gas−liquid−solid flow can be broken down into a bubble image and a liquid−solid image after bubble separation treatment. On the basis of these images, the fractal dimension (exactly box dimension) of a bubble image, a liquid−solid image, or a gas−liquid−solid image can be calculated since the fractal dimension analysis is a powerful tool to quantitatively describe the nonlinear character of a image and to enhance the understanding of the selforganization behavior in the GLSCFB. The fractal dimension was estimated by box-counting method according to the distribution of image gray. The detailed principles and calculation method of the fractal dimension are available in the literature.32,33 The concrete calculation steps are as follows. (1) Cover the image with a box of size, δ, and count the number of boxes required to cover the target object N(δ). Repeat the procedure with a smaller box that is to reduce the box size by half every time. Record its box size and the number of boxes N(δ) required to cover the object. (2) A log−log plot is generated where the x-axis is the log (1/δ) and the y-axis is the log(N(δ)). Linear relationships between N(δ) and δ are obtained. The slope of the resulting plot gives the value of fractal dimension of the image, D. In order to verify the developed software of box-counting method for fractal dimension estimation, we take three mathematical fractal images: Koch curve, Sierpinski triangle, and Sierpinski rectangle as calculation examples because their values of fractal dimension are exactly known. The calculation results are shown in Figure 2 and Table 2. The largest relative error of calculation is 3.17%, which means that the developed software is accurate enough. The examples of the fractal dimension calculations of the gas bubble, liquid−solid, and gas−liquid−solid flow images are given as Figure 3a, b, and c. The values of the fractal dimension of the three images are 1.5545, 1.5512 and 1.4254, respectively.

3. RESULTS AND DISCUSSION 3.1. Confirmation of Self-Similarity Structure in a Multiphase Flow Image of GLSCFB. The similarity between the part and the whole is a key character of a fractal body.32 In order to confirm the self-similarity, a typical image of the gas− liquid−solid flow in the GLSCFB was enlarged step by step, as shown in Figure 4. It is found that there are good similarities between the images. Good similarities indicate that there is a reasonable basis to use the fractal dimension to describe the nonlinear flow structures in the GLSCFB. It should be noted that absolute or ideal self-similarity does not exist in nature, but general and limited self-similarity can be found in nature, as shown in Figure 4. A detailed description of the relationship of the fractal dimension and the image will be shown in the following. 3.2. Relationships between the Fractal Dimension, Operation Conditions, and Self-organization or Clustering Behavior in the GLSCFB. 3.2.1. Superficial Gas Velocity of the Riser. Typical original gas−liquid−solid flow pictures, three-phase flow images after binary conversion of image gray levels and gas bubble flow images extracted from the original gas−liquid−solid flow images in the GLSCFB riser at different superfical gas velocities of the riser are shown in Figure 5. The relationship between the superficial gas velocity of the riser and the fractal dimension of the image is shown in Figure 6. It can be seen from Figure 6 that the fractal dimension of gas bubble flow image D3 experiences an increase with the increase of the superficial gas velocity in the riser with a slight decrease at higher superficial gas velocity. The increase of superficial gas velocity ugr can increase the gas holdup and thus increase the turbulence of the system. Hence, the fractal dimension D3 climbs. However, when the bubble number increases to a relatively high, as shown in Figure 5 (d3), the appearance of bubble aggregation decreases the disorder degree of the system, leading a low fractal dimension value of the bubble flow image D3. The trend of the fractal dimension of the liquid−solid flow image D2 with superficial gas velocity of the riser is somewhat complex and a fluctuant rule is seen. There may be some competitive impacts of the solid holdups and the particle aggregation on the fractal dimension D2. Investigations on the solid holdup of the riser show that with the increase of ugr, the solid holdup increases generally,4 which is different from the rule of the fractal dimension D2. Thus, the aggregation behavior may affect the fractal dimension D2. The variation tendency of the fractal dimension D1 with the superficial gas velocity in Figure 6 is the result of the interactions between gas, liquid, and solid phases. As shown in Figure 5a2−d2, with the increase of ugr, solid and gas holdups

Table 2. Box-Counting Dimensions of Mathematical Fractal Images fractal images Koch curve Sierpinski triangle Sierpinski rectangle

theoretical fractal dimensions, D

computation fractal dimensions, D

relative errors

1.26 1.585

1.28165 1.59866

1.70% 0.86%

1.74

1.68480

3.17%

11406

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Figure 3. Fractal dimension calculations of the gas bubble, liquid−solid, and gas−liquid−solid flow images (AB-8 resin, um = 4.44 mm/s, ua = 4.44 mm/s, ugr = 2.00 mm/s, ugd = 0.00 mm: (a) gas bubble flow image; (b) liquid−solid flow image; (c) gas−liquid−solid flow image.

Figure 4. Amplified pictures of a typical gas−liquid−solid flow image in the GLS riser (AB-8 resin, dp = 0.6 mm, um = 5.56 mm/s, ua = 4.44 mm/s, ugr = 2.00 mm/s, ugd = 0 mm/s, εs,r = 0.000552, εg,r = 0.006134, Gs = 80.66 kg/(m2·h)).

In order to find the physical meaning that the fractal dimension conveys, the relationship between the fractal dimension of the image and the number of the clusters with superficial velocity of primary liquid flow is shown in Figure 10. The direction from left to right along the horizontal ordinate axis in Figure 10 stands for the increase of the superficial velocity of primary liquid flow. The fractal dimension of the gas−liquid−solid flow image D1 under the superficial velocity of primary liquid flow is figured out on the horizontal ordinate axis and the number of the each kind of cluster is shown in the vertical ordinate axis. In Figure 10, it can be seen that the particles are largely exist in a single particle manner in the GLSCFB, and the main clusters are the doublets and triplets, and the number of the doublets are far higher than other kinds of clusters. The aggregation behavior can be reflected by the fractal dimension in a certain degree. It has been shown that increasing primary liquid flow rate can inhibit the particle aggregation but enhance the bubble−particle multiplex aggregation to a certain degree.4 Meanwhile in Figure 9a, D1 and D3 decrease both, but D2 experiences a decrease first and then an increase. The degree of the system disorder decreases due to the appearance of the aggregation (i.e., from an irregular single particle distribution to certain particle-organized structure) and of course leads to the decrease of the system entropy and thus the fractal dimension. On the one hand, the bubble−particle aggregation degree is enhanced due to the increase of the main flow rate, which leads to the decrease of the fractal dimension of the bubble flow images D3. On the other hand, the decrease of the particle aggregation degree which is caused by the increase of the main flow rate results in the increase of D2 at the later stage. For the strong disturbance of the bubble on the three-phase flow, the influence of the bubbles on the fractal dimension is dominant

increase obviously. But the fractal dimension D1 decreases with the increase of ugr when ugr is smaller than 2.00 mm/s. When ugr is higher than 2.00 mm/s, the solid holdup increases obviously and so do the disorder degree and the fractal dimension. The relationship between D2 and the number of clusters in the GLSCFB riser on variable superficial gas velocity in the riser is shown in Figure 7. Combining the variation of the fractal dimension D2 and the change of the number of the particle clusters, it can be seen that D2 can reflect the combined effects of the particle aggregation and the solid holdup. When ugr is smaller than 2.00 mm/s, the influence of the solid holdup’s increase on the fractal dimension is dominant. When ugr is higher than 2.00 mm/s, the aggregation degree results in the decrease of the fractal dimension D2. 3.2.2. Superficial Velocity of Primary Liquid Flow. The original gas−liquid−solid circulating flow images of the GLS riser at different superficial velocities of primary liquid flow are shown in Figure 8. It can be found from Figure 8 that higher superficial velocities of primary liquid flow lead more dilute the gas bubble and particle flows. The values of fractal dimension of the original images of three-phase flows, liquid−solid flow images with gas bubble separated and separated gas bubble flow images with different superficial velocities of primary liquid flow are shown in Figure 9. Gas or solid holdup is also shown in Figure 9. It can be seen from Figure 9a that with the increase of superficial velocity of primary liquid flow, the fractal dimensions of gas−liquid−solid and gas bubble flow images, namely D1 and D3, decrease continuously. The fractal dimensions of liquid− solid flow images D2 first decrease and then increase. The solid or gas holdup goes down in general, which can be seen in Figure 9b. 11407

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Figure 5. Typical original gas−liquid−solid flow images, images after binary conversion of image gray levels, and gas bubble flow images extracted from the original gas−liquid−solid flow images in the GLSCFB riser at different superfical gas velocities of the riser: (a) ugra = 1.33 mm/s, (b) ugrb = 2.00 mm/s, (c) ugrc = 2.67 mm/s, (d) ugrd = 3.33 mm/s; 1 original three-phase flow images, 2 three-phase flow images after binary conversion of image gray levels, 3 gas bubble flow images extracted from the original three-phase flow images (AB-8 resin, dp = 0.6 mm, um = 5.56 mm/s, ua = 4.44 mm/s, ugd = 0 mm/s; εs,a = 0.001 625, εs,b = 0.001 233, εs,c = 0.001 560, εs,d = 0.002 019; εg,a = 0.003 224, εg,b = 0.004 832, εg,c = 0.006 104, εg,d = 0.009 502; Gsa = 172.79 kg/(m2·h), Gsb = 198.64 kg/(m2·h), Gsc = 112.23 kg/(m2·h), Gsd = 161.46 kg/(m2·h)).

superficial velocities of auxiliary liquid flow, more solid particles are circulated and fewer gas bubbles are found. The values of fractal dimension of the original three-phase flow images, liquid−solid flow images with gas bubble separated, and separated gas bubble flow images with different superficial velocities of auxiliary liquid flow are shown in Figure 12.

and the changing trends of D1 and D3 are the same to each other. 3.2.3. Superficial Velocity of Auxiliary Liquid Flow. Typical three-phase flow images in the GLSCFB riser at different superficial velocities of auxiliary liquid flow are shown in Figure 11. As can be found in Figure 11 that with the increase of 11408

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Figure 6. Relationship between superficial gas velocity of the riser and fractal dimension of multiphase flow images in the GLSCFB (AB-8 resin, dp = 0.6 mm, um = 5.56 mm/s, ua = 4.44 mm/s, ugd = 0.00 mm/ s): D1 fractal dimension of gas−liquid−solid flow image; D2 fractal dimension of liquid−solid flow image separated from the three-phase flow image; D3 fractal dimension of gas bubble flow image separated from the three-phase flow image.

Figure 8. Typical flow images in the GLSCFB riser at different superficial velocities of primary liquid flow: (a) uma = 5.56 mm/s; (b) umb = 6.67 mm/s; (c) umc = 7.78 mm/s; (d) umd = 8.89 mm/s (AB-8 resin, dp = 0.6 mm, ua = 4.44 mm/s, ugr = 2.00 mm/s, ugd = 0 mm/s; εs,a = 0.000 875, Gsa = 74.27 kg/(m2·h), Gsb = 72.97 kg/(m2·h), Gsc = 73.81 kg/(m2·h), Gsd = 68.57 kg/(m2·h)).

Figure 7. Relationship between D2 and the number of clusters in the GLSCFB riser on variable superficial gas velocity in the riser.

It can be seen in Figure 12 that the fractal dimension of the bubble flow images separated from the three-phase flow images D3 decreases continuously with the increase of the auxiliary liquid flow velocity, which is agreeable to the rule of gas holdup variation. This is because with the increase of the auxiliary liquid flow rate, the gas bubble number, and the gas holdup decrease. However, the bubble−particle aggregation degree increases. The variation rules of D1 and D2 are somewhat complex. With increase of the auxiliary liquid flow velocity, the fractal dimension of the liquid−solid flow images separated from the original three-phase flow images D2 increases first and decreases later. There are two sides of influence of the auxiliary liquid flow on the multiphase flows. On the one hand, with the increase of the auxiliary liquid flow velocity, the solid holdup increases and the gas holdup decreases; on the other hand, with the increase of the auxiliary liquid flow velocity, the particle and bubble−particle multiplex aggregation enhance. Both the average characteristics of the fluid dynamics and the aggregation behaviors influence the fractal dimensions, and the two sides have an opposite effect on the fractal dimensions. Namely the increase of the phase holdups increases the disorder degree of the system, and thus the entropy and fractal dimension increase. Meanwhile, the aggregation behaviors of particle and bubble−particle result in the decrease of the fractal

Figure 9. Fractal dimensions and phase holdups of gas−liquid−solid, liquid−solid, and gas flow images in GLS riser at different superficial liquid velocities of primary liquid flow: (a) fractal dimension; (b) phase holdup.

dimension. For the above two reasons, at the early stage of the increase of auxiliary liquid flow velocity, the increase of the solid holdup results in the increase of D2, but with the further increase of the aggregation behavior, the fractal dimension D2 decreases. The influence of the auxiliary flow on the fractal 11409

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Figure 12. Relationship between fractal dimension and superficial velocity of auxiliary liquid flow in GLSCFB riser (AB-8 resin, dp = 0.6 mm, um = 5.56 mm/s, ugr = 2.0 mm/s, ugd = 0.00 mm/s). Figure 10. Relationship between fractal dimension of the images of the riser and the number of the clusters when changing the superficial liquid velocity of primary liquid flow.

the downer were also investigated. There are so many data obtained for these contents. Detailed figures will not been given here to save space, and only the main results are described briefly as follows. Fractal dimension of gas−liquid−solid flow images D1 and gas bubble flow images D3 of the downers change little with superficial velocity of primary liquid flow, which indicates that superficial velocity of primary liquid flow hardly affects the flow and clustering behavior in the downers due to the obvious disturbance of the gas bubble in the downers. However, the fractal dimension of liquid−solid flow images D2 of the gas− liquid−solid downers, and that of liquid−solid flow images in the liquid−solid downers, fluctuate slightly with the superficial velocity of primary liquid flow. Fractal dimension of multiphase flow images in the downers generally decreases with the superficial velocity of auxiliary liquid flow, which indicates that superficial velocity of auxiliary liquid flow enhances the particle and bubble−particle clustering behaviors in the downers. The fractal dimension of multiphase flow images in the liquid−solid and gas−liquid−solid downers fluctuates with the increase of the superficial gas velocity of the riser, which is related to the complex variations of phase holdups and clustering behaviors. The fractal dimension of multiphase flow images in the liquid−solid and gas−liquid−solid downers increases overall with the increase of the superficial gas velocity of the downer due to the dominant effect of the enhancement of the gas bubble turbulence in the downer. These results indicate that the higher the fractal dimension, the weaker the clustering behavior and the higher the entropy. 3.3. Liquid Physical Properties. The fractal dimensions of multiphase flow images in the liquid−solid and gas−liquid− solid risers and downers at different liquid surface tension and viscosity are discussed here. 3.3.1. Liquid Surface Tension. Typical original multiphase flow images in the GLSCFB riser at different surface tension are shown in Figure 13, and the numbers of gas bubbles and particles fluctuate. Figure 14 shows the relationship between fractal dimension of the multiphase flow pictures in the GLSCFB riser and liquid surface tension. It can be found form Figure 14 that with the increase of liquid surface tension, the fractal dimension of the gas−liquid− solid flow images D1 in the GLSCFB riser decreases slightly. But the fractal dimensions of the liquid−solid flow images D2 and the gas bubble flow images D3 in the GLSCFB riser fluctuate when increasing the liquid surface tension, and the

Figure 11. Typical images in the GLSCFB riser at different superficial velocities of auxiliary liquid flow: (a) uaa = 3.33 mm/s; (b) uab = 3.89 mm/s; (c) uac = 4.44 mm/s; (d) uad = 5.00 mm/s (AB-8 resin, dp = 0.6 mm, um = 5.56 mm/s, ugr = 2.00 mm/s, ugd = 0 mm/s; εs,a = 0.000 444, εs,b = 0.000 925, εs,c = 0.001 112, εs,d = 0.001 599; εg,a = 0.005 025, εg,b = 0.004 267, εg,c = 0.003 338, εg,d = 0.002 874; Gsa = 36.51 kg/(m2·h), Gsb = 70.66 kg/(m2·h), Gsc = 117.90 kg/(m2·h), Gsd = 124.68 kg/ (m2·h)).

dimension of the three-phase flow images is complex. But under high auxiliary liquid flow rate, all the fractal dimension values of three-phase flow images D1, liquid−solid flow images D2, and gas bubble flow images decrease. This illustrates that under a relatively higher auxiliary flow rate, the aggregation behaviors become more prominent. 3.2.4. Other Operation Conditions. Fractal dimension values of multiphase flow images in the liquid−solid and gas−liquid−solid downers at varied superficial velocity of primary liquid flow, superficial velocity of auxiliary liquid flow, superficial gas velocity of the riser, and superficial gas velocity of 11410

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the multiphase flow pictures in the GLSCFB riser and liquid viscosity. It can be found from Figure 16 that with the increase of the liquid viscosity, the fractal dimensions of the gas−liquid− solid flow images D1 increase gradually, which agrees with the trend of solid holdup with the liquid viscosity. The fractal dimensions of the liquid−solid flow images D2 and gas bubble flow images D3 even though fluctuate but slightly decrease are found with the increase of the liquid viscosity. It has been shown that the increase of the liquid viscosity enhances the particle clustering behavior while weakens the mixed aggregation in the GLS riser,4 which may related to the high fractal dimension of three-phase flow, low fractal dimension of liquid−solid flow or gas bubble flow. The above investigations on the influences of operation conditions and liquid physical properties on the fractal dimensions indicate that the phase holdup and the aggregation degree are the main factors to result in different fractal structures. Increasing phase holdups results in a relatively high fractal dimension, while enhancing aggregation decreases the fractal dimension due to the enhancement of the order structure in the multiphase flows. The phase holdups and the aggregation behavior are determined by the physical properties of the gas bubbles, solid particles and the fluids, operation conditions, and the geometry structure of the fluidized beds. The fractal dimension can generally be used to characterize the self-organization structure or aggregation behavior in the GLSCFB system. The fractal dimension of three-phase flow images D1 appears to be a comprehensive quantitative parameter to evaluate the self-organization structure and can be affected by the physical properties of the gas bubbles, solid particles, and the fluids besides the operation conditions and the geometry structure of the fluidized beds. D1 is generally round about 1.35−1.65. The fractal dimension of liquid−solid flow images D2 and that of the bubble flow images D3 appear to be a local quantitative parameter to evaluate the particle or the bubble selforganization structure. Among the phase holdup, aggregation behavior, and fractal dimension, knowing the trends of two of the three quantities, the tendency of the other quantity may be predicted. However, the close and accurate relationships among them should be further studied in the future. Further work should focus on the reconstruction of the fractal images in the GLSCFB system and on the investigation of the fractal phenomena from the control equations of multiphase flows.

Figure 13. Images of three-phase flow in the GLSCFB riser with various values of liquid surface tension: (a) γa = 69.3 mN/m, (b) γb = 53.8 mN/m, (c) γc = 48.2 mN/m, (d) γd = 40.7 mN/m (AB-8 resin, um = 5.56 mm/s, ua = 4.44 mm/s, ugr = 2.00 mm/s, ugd = 0.00 mm/s; εs,a = 0.000 627, εs,b = 0.000 728, εs,c = 0.000 542, εs,d = 0.000 419; εg,a = 0.00 2495, εg,b = 0.003 432, εg,c = 0.004 329, εg,d = 0.003 096; Gsa = 67.78 kg/(m2·h), Gsb = 60.73 kg/(m2·h), Gsc = 33.52 kg/(m2·h), Gsd = 26.61 kg/(m2·h)).

Figure 14. Relationship between fractal dimension of the multiphase flow pictures in the GLSCFB riser and liquid surface tension.

concrete variation rules of D2 and D3 agree with those of the solid and gas holdups. It has been shown4 that when increasing the liquid surface tension, the particle clustering behavior in the GLSCFBs is generally enhanced which agrees with the decreasing trend of the fractal dimension of the gas−liquid−solid flow images D1. Meanwhile, it seems that there is an optimal liquid surface tension on the bubble−particle mixed clustering behavior, which can also be explained by the variations of the bubble− particle mixed clustering behavior. 3.3.2. Liquid Viscosity. Typical original multiphase flow images and gas bubble flow images extracted from the original three-phase flow images in GLSCFB riser at varied liquid viscosity are shown in Figure 15, and the rules are not obvious. Figure 16 shows the relationship between fractal dimension of

4. CONCLUDING REMARKS Even though the self-organization flow structures in the GLSCFBs are very complex and the present work is quite preliminary, some results were obtained. (1) The fractal dimension is able to evaluate the selforganization or aggregation degree in the GLSCFB to a certain degree. (2) The phase holdup and the aggregation behavior are the main factors that influence the fractal dimension D1, D2, and D3. Increasing solid or gas phase holdup can increase the fractal dimension, while obvious aggregation decreases the fractal dimension. The fractal dimension 11411

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Figure 15. Original gas−liquid−solid flow images and gas bubble flow images extracted from the original three-phase flow images in the GLSCFB riser at varied liquid viscosity: (a) η = 349.4 × 10−5 Pa·s; (b) η = 720.3 × 10−5 Pa·s; (c) η = 1169.8 × 10−5 Pa·s; (d) η = 1501.0 × 10−5 Pa·s; 1 original three-phase flow images, 2 gas bubble flow images extracted from the original three-phase flow images (AB-8 resin, um = 5.56 mm/s, ua = 4.44 mm/s, ugr =2.00 mm/s, ugd = 0 mm/s; εs,a = 0.000 455, εs,b = 0.000 414, εs,c = 0.000 576, εs,d = 0.001 005; εg,a = 0.004 653, εg,b= 0.004 343, εg,c = 0.005 741, εg,d = 0.004 673; Gsa = 62.80 kg/(m2·h), Gsb = 100.74 kg/(m2·h), Gsc = 113.81 kg/(m2·h), Gsd = 71.28 kg/(m2·h)).

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (No. 20576091), the Open Foundation of State Key Laboratory of Multiphase Complex System, Institute of Process Engineering, Chinese Academy of Sciences.

■ Figure 16. Fractal dimension of multiphase flow pictures of the GLSCFB riser vs liquid viscosity.

of three-phase flow images D1 is a comprehensive quantitative parameter to evaluate the self-organization structure. (3) Among the three physical quantities of phase holdup, aggregation behavior and fractal dimension, knowing any two of them the changing trend of the other can be predicted. These results may provide certain guidance for the design and operation of the GLSCFB. For example, if one wants to avoid the aggregation phenomena appearing in the GLSCFB riser, he could increase the superficial velocity of primary liquid flow and reduce the superficial velocity of auxiliary liquid flow and liquid surface tension according the investigations of this work. Of course, further extensive and systematical studies are expected.

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NOMENCLATURE dp = diameter of particle, m D = fractal dimension, dimensionless D1 = fractal dimension of gas−liquid−solid flow image, dimensionless D2 = fractal dimension of liquid−solid flow image after the gas bubble was extracted from the original gas−liquid−solid flow image, dimensionless D3 = fractal dimension of gas bubble flow image extracted from the original gas−liquid−solid flow image, dimensionless GS = mass circulation rates of solid particles per unit area, kg/(m2·h) N = number t = time, s u = velocity, m/s

Greek Letters

ε = phase holdup, dimensionless γ = surface tension, N/m ρ = density, kg/m3 η = viscosity, Pa s Subscripts

a = auxiliary liquid flow b = bubble d = downer g = gas phase l = liquid phase m = primary liquid flow p = particle r = riser

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 11412

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s = solid

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