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Numerical simulation of flow involving a fractal aggregate
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JTICE-874; No. of Pages 8 Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
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Numerical simulation of flow involving a fractal aggregate Nong Xu a, Long Fan b,* a b
Department of Chemical & Biological Engineering, University of British Columbia, Canada Department of Chemical & Materials Engineering, University of Alberta, Canada
A R T I C L E I N F O
A B S T R A C T
Article history: Received 25 September 2013 Received in revised form 20 March 2014 Accepted 23 March 2014 Available online xxx
Hydrodynamic properties of aggregate are different from those of impermeable particles and have not been well established, owing to their complex structural feature. Therefore, this paper focuses on the study of hydrodynamic properties of an aggregate by computational fluid dynamics method. An aggregate with self-similar structure composed of four primary spheres is constructed to account for its fractal geometry. Navier–Stokes equations are applied for the whole flow field in relation to the aggregate. Both exterior and interior flow through the aggregate could be seen clearly. Fluid collection efficiency, an indication of permeability, is also obtained. Reynolds number has complicated influence on fluid collection efficiency. Furthermore, the larger the space between four primary particles is, the larger the fluid collection efficiency. The smaller the fractal dimension is, the larger the fluid collection efficiency. These findings will enable the development of more realistic models for the flocculation and material transfer dynamics of fractal aggregate. All these results contribute to further research about mass transfer between the aggregate interior and the bulk liquid solution, as well as those researches related to irregular porous structures. ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Aggregate Computational fluid dynamics Fluid collection efficiency Self-similar structure
1. Introduction Suspended particles are often aggregated to improve their performance. This phenomenon has been encountered in many different industrial and scientific applications, such as water and wastewater treatment, atmospheric studies [1], public health research. For example, man-made aerosols in the atmosphere can occur in the form of aggregate. Primary or secondary sedimentation tanks make it possible for flocs to aggregate and settle in water and wastewater treatment, which improves the water quality greatly. The hydrodynamic behavior determines other important properties, such as settling velocity, coagulation rate. Hence, understanding the hydrodynamic behavior of aggregate is important to improve the water/wastewater treatment efficiency, atmospheric studies and other related research. Substantial research indicates that aggregate formed by particle coagulation are fractals [2–5]. The porous structure of fractal aggregate would permit significant interior flow into the aggregate, which would enhance particle collision and material transport. Accordingly, the hydrodynamic properties of the aggregate will be different from those of impermeable particles. Therefore, it is necessary to research on the hydrodynamic properties of the aggregate. The
* Corresponding author. Tel.: +1 6048275921. E-mail address: [email protected] (L. Fan).
relationship between the hydrodynamic properties, the fractal structure of aggregate and its internal permeation can be obtained. They will be helpful to the understanding of aggregation phenomenon, the improvement of water and wastewater treatment efficiency, as well as other research related to fractal aggregate. Nevertheless, two problems make the research of aggregate more difficult. One is the fractal structure. The complexity of fractal structure makes it difficult to simulate the aggregate with real structure. People have tried different methods to numerically construct the real structure for colloidal aggregate, for example, diffusion-limited cluster aggregate (DLCA) [1,6,7], chemically limited aggregate (CLA) [8,9], Tunable-Dimensional Method (TDM) [10]. More usually, people simplified the aggregate as a spherical particle or spheroid [11–15]. Though this makes the research much easier, it brings inevitable errors into the results. To solve the problem, people investigated the structure of aggregate in detail. It is found that aggregate occurring naturally may have multilevel structure [16–18]. Gorczyca and Ganczarczyk [19] proposed an activated sludge model that includes the following stages: (1) primary particles form floccule, (2) floccule group together to form microflocs, and (3) microflocs form the flocs. Floccule, microflocs, and flocs can exhibit different structures and perhaps different fractal dimensions. Thus the concept of selfsimilar structure is presented. It means that an aggregate of any size can be formed directly from its principal clusters with the
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Please cite this article in press as: Xu N, Fan L. Numerical simulation of flow involving a fractal aggregate. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.013
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Nomenclature A0 Ah d da g P p q R Re u u0 V x, y, z
m r h
the projected area of the aggregate [m2] the total area of the holes through a cross section [m2] diameter of primary particles [m] diameter of aggregate [m] acceleration of gravity [m/s2] pressure [Pa] number of units in a cell side length of a cell [m] radius of flow field [m] Reynolds number velocity [m/s] the fluid velocity (V) approaching the aggregate without the influence of the aggregate [m/s] fluid velocity [m/s] coordinates [m] viscosity of fluid [kg/m s] density [kg/m3] fluid collection efficiency
same general shape as the aggregate. The self-similar structure has been proven to reduce the computational time greatly and has been used in many researches successfully. For example, Rogak and Flagan [20] assumed clusters to be self-similar in their research about Stokes drag on cluster. In the work of Li and Logan [21], the particles in an aggregate were distributed according to a self-similar structure. Woodfield and Bickert [22] also treated the individual floc as a nested structure of settling swarms of permeable sub-spheres. Then an aggregate could be split into any number of levels, with any relative ratio between successive levels. Therefore, self-similar structure is a very good solution to the fractal structure of aggregate. If we could propose an appropriate self-similar structure, the aggregate could be represented quite well. The other problem comes from the porous structure of aggregate. Since the aggregate is permeable, fluid will flow through the interior of the aggregate. How to simulate the interior flow through the aggregate is another problem encountered in the simulation. Flow through permeable aggregate has been the subject of many theoretical and experimental investigations [11–15,23–31]. Different methods are used to account for the hydrodynamic behavior of aggregate. Kirkwook and Riseman [23], Riseman and Kirkwood [24], Hess et al. [25] employed Kirkwook–Riseman theory to calculate the translational diffusion coefficient, intrinsic viscosity for flexible or rigid chain-like polymers. Geller et al. [26] and Mondy et al. [27] used boundary element method to predict the rotation or translation motion of isolated, nonspherical particles in creeping Newtonian fluid, including isolated chain agglomerate particles, branched chains and flakes. Gastaldi and Vanni [28], Ka¨tzel et al. [10] used method of reflections and Filippov approach [29] to calculate the drag force, respectively. Babick et al. [30] employed Filippov approach to study the impact of particle size and fractal dimension on aggregate. Another promising method for solving the fluid flow through aggregate is the Lattice Boltzmann method (LBM). It allows the calculation of hydrodynamic behavior of aggregate. LBM was also combined with accelerated Stokesian-dynamics method to calculate the drag force of fractal aggregate [31,32]. However most of these methods are focused on aggregate in creeping flow, which limited their application somehow. A method is needed to account for the hydrodynamic behavior of aggregate with both low and high Reynolds number.
Different approaches are also used for the flow fields outside and inside the aggregate. The approach typically used for the flow exterior to the aggregate (under creeping or laminar flow conditions) is to solve the Navier–Stokes equations. As to the interior, there are two primary approaches [22], those that require knowledge of the positions of all particles within the aggregate and those that do not. The latter is paid more attention and it includes three different methods: Darcy’s law, Brinkman equation and Darcy–Brinkman model. For example, Adler [11] described the flow inside a spherical aggregate with the Brinkman equation of motion during the study on porous particles. Mountain et al. [12] used Darcy’s law for the flow inside a cluster in the research about the formation of high temperature aerosol agglomerates. Hsu and Hsieh [13] and Hsu et al. [14] treated the floc as a two-layer structure to investigate the boundary effect on the drag force acting on a spherical floc. Darcy–Brinkman model and the continuity equation were applied to the description of the flow field inside the floc. In the research of Veerapaneni and Wiesner [15], Vanni [33], they both described the flow exterior to the aggregate by the Stokes and continuity equations, while the flow inside the aggregate was assumed to be governed by Brinkman’s equation. This paper focuses on the above-mentioned two problems in the simulation of an aggregate settling in fluid by computational fluid dynamics (CFD) method. Self-similar structure is employed to represent the fractal geometry of the aggregate, instead of simplifying it as a spherical particle or using the above-mentioned methods. It will be more close to the aggregate in nature. The simulation of the flow through the interior of aggregate would become less difficult with the self-similar structure. Contrast to those simulating the inner and outer of the aggregate with different methods, the same method is used for the simulation of the flow fields outside and inside the aggregate since no obvious boundary between outside and inside, which makes the simulation much easier. Furthermore, the flow involving aggregate with different size and space is presented. Permeability is calculated and compared among different aggregate. This research tries to approach the real aggregate in nature. The results will benefit further research about aggregate, such as improvement of aggregation in wastewater treatment. 2. Simulation methods 2.1. Structure of the aggregate Self-similar structure can not only express the fractal geometry of aggregate in some degree, but also save the computation time remarkably. Different self-similar structures can be conceived according to the focus of different research. A fractal aggregate with self-similar structure is constructed in the present study. Four solid spherical particles compose the basic structure with three at the bottom and one at the top in a form of triangular cone, as shown in Fig. 1. For the sake of simplification, these four particles do not have special chemical composition. The aggregate is permeable for the four adjacent particles contact with each other at one point and a hole is present among the three bottom spheres. It could represent the porous fractal geometry of aggregate well. Based on the self-similar structure, every sphere itself is also a small aggregate composed of four small primary spheres. In other words, each aggregate has different levels, as given on the right of Fig. 1: (Level 1) Four primary particles form a small aggregate. We name the small aggregate as aggregate unit. (Level 2) Aggregate unit is the big primary particle. Four aggregate units form a big aggregate (aggregate cell). (Level 3) Aggregate cell is the bigger primary particle. Four aggregate cells form a bigger aggregate. The circle encompassing the four spheres gives the real shape and boundary of the aggregate. Red and black
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Fig. 1. The self-similar structure of the simulated aggregate.
circles in Fig. 1 are both aggregate composed of four primary particles. It would be very complicated if we consider every sphere as an aggregate in the simulation. Aggregate at each level are all composed of four primary ‘‘particles’’. The primary ‘‘particles’’ are either real spherical particles or small aggregate. Therefore aggregate will have the same structure, as well as the same flow type. Based on this, the flow type at any level will be similar to the four-sphere structure. Hence, we use the four-sphere structure to represent the aggregate, regardless of its level. The fractal structure of aggregate can be characterized by fractal dimension. Fractal dimension is an index for characterizing fractal patterns by quantifying their complexity as a ratio of the change in detail to the change in scale. It is a measurement of how particles convey information in duplication of a pattern of behavior at different size scales [34]. According to Rogak and Flagan [20], the fractal dimension is defined as log p/log q, where p units are arranged on a cell with side length q. Usually, fractal dimension varies from 0.26 to 3 [15]. The aggregate with a large fractal dimension will have a small permeability. We get the fractal dimension of 2 for the present aggregate. 2.2. Simulation methods The structure of the aggregate, even the interior, can be seen clearly with the present self-similar structure. There is no obvious interface between the outside and inside of the aggregate. Thus, it is unnecessary to use different methods for the flow fields outside and inside the aggregate, like the above-cited literature. Instead, we employ the same three-dimensional Navier–Stokes equations [35], Eqs. (1)–(4), for both the exterior and interior flow of the aggregate.
@ @ @ ðrux Þ þ ðruy Þ þ ðruz Þ ¼ 0 @x @y @z
(1)
! @ux @ux @ux @P @2 ux @2 ux @2 ux þ uy þ uz þm r ux þ þ ¼ @x @y @z @x @x2 @y2 @z2 þ rg x
r ux
(2)
@uy @uy @uy @P þ uy þ uz ¼ @x @y @z @y @2 uy @2 uy @2 uy þm þ þ @x2 @y2 @z2 þ rg y
!
(3)
r ux
@uz @uz @uz @P @2 uz @2 uz @2 uz þ uy þ uz þm þ þ ¼ @x @y @z @z @x2 @y2 @z2 þ rg z
!
(4)
where u and P denote the local flow velocity and pressure of the fluid, and m is the fluid viscosity. The general-purpose CFD software FLUENT is used to solve these equations. The simulated fluid is water. It is supposed to be incompressible Newtonian fluid with density and viscosity of 1000 kg/m3, 0.001 kg/m s, respectively. It is assumed that the spheres forming the aggregate are impermeable and there are no boundary effects on the surfaces of the aggregate. Considering that the aggregate settles at a steady speed V through an unbounded quiescent fluid, two simplifications are made. First, it is supposed that the domain can be used to represent an unbounded flow field if the ratio R/da (radius of flow field/diameter of aggregate) exceeds 7.5 [36]. Second, for the sake of easy definition of boundary conditions in the computational domain, the aggregate is assumed to be fixed, while the surrounding fluid is flowing at a uniform speed of V from infinity toward the fixed aggregate. No slip conditions are imposed at the surfaces of the primary spheres. The velocity is supposed to be uniform at the place far from the aggregate, as shown in Fig. 2. 2.3. Solution methods GAMBIT [37] is used for grid generation. Since the flow fields within and around the porous aggregate are of major concern, more grids are allocated near the aggregate for better accuracy. In order to resolve the contact points among the primary particles, tetrahedral mesh is employed at these regions. The discretization of equations is performed using the first-order upwind scheme and the pressure is solved by the SIMPLE algorithm [38]. Different discretization methods have been tried. First-order upwind scheme is found to be the best for the present research. The non-linearity in the phase momentum equations is dealt with the under-relaxation technique. When the residuals of the equations being solved meet the prescribed tolerance of a change of <0.1% from the previous iteration, a converged solution is considered to have been obtained. The choice of grids can influence the simulation significantly. Three grids are tested in our simulations with 310,383, 644,885, and 895,098 intersection points. It is found that the results for grids
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Fig. 2. Boundary conditions and the coordinates.
of 644,885 points are very similar to those for 895,098 grid points. Hence, 644,885 grid points are chosen for the simulations presented here. 3. Results and discussion The aggregate is located in the center of a cubic column in simulation. The side length of the column is 50 mm. The four primary spheres forming the aggregate in Fig. 1 are 1 mm in diameter. It is assumed that water at infinity flows upward with a velocity of 0.001 m/s. The origin of the coordinates is set at the center of the three bottom spheres shown in Fig. 2. Several planes are defined at y = 1.0, 0.5, 0, 0.5 and 1.0 mm for presentation of the simulation results, as plotted in Fig. 3. We only show the front views of the results in the following parts for the sake of simplification. 3.1. Determination of flow type To determine the flow type, a suitable Reynolds number is an index. For a settling particle, the Reynolds number is defined by Re ¼
dur
(5)
m
where d is the diameter of the particle. It has been found that the flow is laminar when the Reynolds number is below 1 [39,40].
However, the aggregate’s Reynolds number is not always much less than unity. In contrast, for aggregate with diameter greater than 100 mm, the aggregate’s Reynolds number can markedly exceed unity. In some cases the aggregate’s Reynolds number is found to be up to several tens. For example, most of the aggregate in secondary sedimentation tank was around 10–250 mm in diameter [41]. In the research of Lee and Hsu [42], the size of some aggregate was above 100 mm, even up to 160 mm. Correspondingly, the settling velocity would be relatively high. Regardless of the aggregate’s Reynolds number, some researchers directly employed some Stokes law-like correlations for their work [43,44]. In other words, the data for aggregate with a Reynolds number up to 100 are analyzed in the same way as those under creeping conditions. Although such an approach is apparently erroneous, it can give a reasonable result [45]. The reason may be the followings. For a permeable aggregate moving at an elevated Reynolds number, the boundary layer separation and the after-sphere wake might not occur as in the nonporous cases owing to the advection flow through the interior of the aggregate. Therefore the Stokes’ law-like correlation is still applicable at the elevated Reynolds number region. In this paper, the fluid velocity varies from 0.001 m/s to 0.1 m/s, corresponding to the Reynolds number of 2.28 to 227.8. On the basis of the analyses above-mentioned, the laminar flow model is employed for the whole simulation. 3.2. Flow field of the aggregate Fig. 4(a) and (b) shows the simulated velocity vectors and streamlines in the planes at y = 0.5, 0 and 0.5 mm with fluid velocity of 0.001 m/s, respectively. The fluid flows upward in a stable and uniform pattern at the bottom of the setting column. The velocity decreases approaching the aggregate. Most of the fluid flows around the aggregate, while a small portion of the fluid enters the aggregate through the hole among the spheres at the bottom. In the center plane at y = 0 mm, the fluid is mostly blocked by the two spheres at the bottom and the sphere at the top. However the fluid only changed the flow direction slightly to pass the aggregate. No circulation loops appear. The possible reason for this is that the Reynolds number is too low to produce circulation loops. The flow structure in the planes with y of 0.5 mm and 0.5 mm are similar to this. Fluid flows toward the centerline at the place just above the spheres. At the plane y = 0.5 mm, a small portion of the fluid flows through the gap between the two bottom spheres. Then the fluid flows around the top sphere, which can be seen clearly in Fig. 4(a) and (b). In the planes at y = 1.0 and 1.0 mm, the flow fields (not shown in the present paper) are simpler compared to the previous analyses. The streamlines are similar to velocity vectors, as plotted in Fig. 4(b). At the place far below the aggregate, all streamlines are uniformly distributed parallel to the centerline. The streamlines become curved approaching the aggregate. Most of the streamlines pass around from both sides of the aggregate, while a small number of streamlines enter the aggregate through the hole between the three bottom spheres. After passing the aggregate, the streamlines restore to uniform parallel distribution gradually with the increasing distance to the aggregate. 3.3. Fluid collection efficiency
Fig. 3. The selected planes (planform).
Fluid collection efficiency, h, is a direct indication of internal permeation of aggregate. It is defined as the ratio of the interior flow passing through the aggregate to the flow approaching it [11,46]. The fluid collection efficiency for a porous sphere ranges between zero and unity. By denoting the off-center distance of the limiting streamline from the axis of symmetry, the fluid collection
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Fig. 4. (a) Velocity vectors at different planes (front view). (b) Streamlines at different planes (front view).
efficiency is given as the square of the ratio of the off-center distance to the radius of the aggregate. Veerapaneni and Wiesner [15] presented the relationship between fractal dimension and fluid collection efficiency with this definition. Tsou et al. [47] utilized the definition of Adler [11] about fluid collection efficiency, but they named it as capture ratio. The fluid collection efficiency in the present simulation is calculated using the following equation. R Ah
h¼
u dA u0 A0
0
(6)
where u0 is the far upstream fluid velocity (V) through the projected area of the aggregate. A0 is the projected area of the aggregate in the horizontal plane (x–y plane). Ah is the total area of the hole through the cross section of the aggregate, and u is the velocity of the internal flow through the hole. The fluid collection efficiency of the aggregate is determined as 0.0117 with the flow velocity of 0.001 m/s. It means that only 1.17% of the fluid approaching the aggregate flows through the interior of the aggregate. Veerapaneni and Wiesner [15] found that the fluid
collection efficiency h is about 0.009 for aggregate with da/d = 100 and fractal dimension of 2. h increases with the decrease of da/d. When da/d changes from 100 to 500, h decreases from 0.009 to 0.002. Fluid collection efficiencies for aggregate with different da/d are very close, when fraction dimension is above 1.95. Hence different extrapolation methods will not have significant influence on fluid collection efficiency. Suppose h changes with da/d linearly. Then h of the present aggregate with da/d about 2 would be around 0.0107. It is close to our simulation value 0.0117. This gives a good indication that our simulations are reliable. This internal permeation may not be essential to the hydrodynamic behavior of the aggregate. However, it could make an important contribution to the mass transfer between the aggregate interior and the bulk liquid solution. It is helpful to understand the relationship between the hydrodynamic properties and the fractal structure of aggregate. 3.4. Effects of fluid velocity The fluid velocity varies from 0.001 m/s to 0.005, 0.01, 0.05 and 0.1 m/s in simulation. The velocity vectors through the aggregate are similar at different fluid velocities.
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Fig. 5. Evolution of fluid collection efficiency with Reynolds number. Fig. 7. Evolution of drag force with Reynolds number.
Different from velocity vectors, the fluid collection efficiency varies with the fluid velocity. Correspondingly, it varies with Reynolds number. The fluid collection efficiency increases with the increase of the Reynolds number, as plotted in Fig. 5. In other words, fluid is easier to flow through the interior of the aggregate with a larger Reynolds number. 3.5. Effects of primary particles’ diameter The simulation is performed for aggregate with diameter of 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50 and 100 mm to investigate the effect of primary particles’ diameter. The boundary of the flow domain are changed accordingly to satisfy the need of R/da > 7.5. The evolution of fluid collection efficiency with Reynolds number is shown in Fig. 6. When Reynolds number is larger than about 1000, correspondingly the diameter is larger than 5 mm, the fluid collection efficiency decreases with Reynolds number and the diameter. Veerapaneni and Wiesner [15] observed the similar phenomenon: The fluid collection efficiency would decrease with the increasing aggregate size for a given fractal dimension. The reason may come from drag force. The drag force comes up with the increase of Reynolds number, as plotted in Fig. 7. The markedly enhancing drag force inhibits the increase of the fluid collection efficiency. Hence, the fluid collection efficiency falls down with the
increment of Reynolds number and diameter. When Reynolds number is smaller, the influence of Reynolds number and diameter on fluid collection efficiency is not obvious. Since the results are based on several complicated equations, we still do not come up with a good reason for this phenomenon. The possible reason may come from the very low Reynolds number and the small diameter. The flow through the aggregate will be very slow with a small Reynolds number. It may behave like viscous fluid. Meantime, the space among the primary spheres will be much smaller when the diameter is small. Different forces may take into effect in this narrow space. Therefore the forces acting on the fluid should be considered differently. The involvement of these forces contributes to the slight effect on fluid collection efficiency. More research could be done toward the satisfactory explanation. Based on Figs. 5 and 6, it is concluded that Reynolds number has a non-trivial influence on the fluid collection efficiency. Fluid collection efficiency decreases with Reynolds number and diameter when Reynolds number is above 1000. For primary particles with diameter of 1 mm, fluid collection efficiency increases with increasing Reynolds number. However mostly the effect of Reynolds number is not obvious with a small Reynolds number. 3.6. Effects of space between primary spheres
Fig. 6. Evolution of fluid collection efficiency with Reynolds number.
To investigate the characteristic of the aggregate further, the relative locations of the four primary spheres are changed. The contact four primary spheres are detached so that there are spaces between any two of them. The space is represented by small spheres. Six small spheres representing space appear in the aggregate. The diameter of the small spheres is 10%, 20% of the primary spheres’ diameter, respectively. Fig. 8 illustrates the sketch of an aggregate with 10% space, where the big spheres with shadow are the primary particles. Since the small spheres are only used to illustrate the presence of space, they are shown as open circles. Fig. 9 compares the streamlines through different aggregate. The flow through the interior of the aggregate becomes easier because of the presence of the space between primary spheres. The fluid is apt to flow into the aggregate with a larger space. It could be anticipated that the fluid collection efficiency would be improved with the increase of the space. It is obvious that the space between the primary particles favors the improvement of the permeability, as shown in Fig. 10. With the inclusion of the space between primary spheres, not only the fluid
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collection efficiency decreases greatly with the increment of fractal dimension, as illustrated in Fig. 11. 4. Conclusions A porous fractal aggregate is simulated by introducing selfsimilar structure. Flow through and around the aggregate is obtained with CFD method. Permeability of the aggregate represented by fluid collection efficiency is also given. The effects of primary spheres’ diameter, fluid velocity, as well as the space between primary spheres, fractal dimension, are investigated. The following conclusions are obtained.
Fig. 8. Sketch of an aggregate with 10% space between primary spheres (planform).
collection efficiency but also the fractal dimension change. The fractal dimension for aggregate with space of 10% and 20% reduces to 1.75 and 1.58, respectively. Hence we could include ‘‘effective fractal aggregate’’ to describe the aggregate. The larger the space between the primary spheres is, the smaller the fractal dimension is, and the smaller the effective fractal aggregate is. The fluid
(1) A self-similar structure composed of four primary spheres is presented. It could represent the irregular porous structure of aggregate well. The flow through and around this aggregate is obtained. It provides a simple method for further research about fractal aggregate. (2) Reynolds number has a non-trivial influence on the fluid collection efficiency. Fluid collection efficiency decreases with fractal dimension. The more space between the primary spheres, the smaller the fractal dimension is, the smaller the effective fractal aggregate is. (3) The simulated internal permeation would be beneficial to further research about mass transfer between the aggregate
Fig. 9. Effects of space between primary spheres on streamlines.
Fig. 10. Effects of the space between primary spheres on fluid collection efficiency.
Fig. 11. Relationship between fluid collection efficiency and fractal dimension.
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interior and the bulk liquid solution. They will enable the development of more realistic models for the flocculation and material transfer dynamics of fractal aggregate. The method also provides basis for other research related to irregular porous structures.
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[23] Kirkwood JG, Riseman J. The intrinsic viscosities and diffusion constants of flexible macromolecules in solution. J Chem Phys 1948;16:565–73. [24] Riseman J, Kirkwood JG. The intrinsic viscosity, translational and rotatory diffusion constants of rod-like macromolecules in solution. J Chem Phys 1950;18:512–6. [25] Hess W, Frisch HL, Klein R. On the hydrodynamic behavior of colloidal aggregates. Z Phys B: Condens Matter 1986;64:65–7. [26] Geller AS, Mondy LA, Rader DJ, Ingber MS. Boundary element method calculations of the mobility of nonspherical particles – 1. Linear chains. J Aerosol Sci 1993;24:597–609. [27] Mondy LA, Geller AS, Rader DJ, Ingber M. Boundary element method calculations of the mobility of nonspherical particles – II. Branched chains and flakes. J Aerosol Sci 1996;27:537–46. [28] Gastaldi A, Vanni M. The distribution of stresses in rigid fractal-like aggregates in a uniform flow field. J Colloid Interface Sci 2011;357:18–30. [29] Filippov AV. Drag and torque on clusters of N arbitrary spheres at low Reynolds number. J Colloid Interface Sci 2000;229:184–95. [30] Babick F, Schiessl K, Stintz M. Characterization of pyrogenic powders with conventional particle sizing technique: I. Prediction of measured size distributions. Part Part Syst Char 2012;29:104–15. [31] Binder C, Feichtinger C, Schmid HJ, Thu¨rey N, Peukert W, Ru¨de U. Simulation of the hydrodynamic drag of aggregated particles. J Colloid Interface Sci 2006;301:155–67. [32] Binder C, Hartig AJM, Peukert W. Structural dependent drag force and orientation prediction for small fractal aggregates. J Colloid Interface Sci 2009;331:243–50. [33] Vanni M. Creeping flow over spherical permeable aggregates. Chem Eng Sci 2000;55:685–98. [34] Mandelbrot BB. The fractal geometry of nature. New York: Henry Holt and Company; 1983. [35] Zhou LX. Theory and numerical simulation of turbulent gas-particle flows and combustion. Beijing: Science Publication Press; 1993 [in Chinese]. [36] Wu RM, Chung HY, Lee DJ. Hydrodynamic drag forces on two porous spheres moving along their centerline. Chem Eng Sci 2003;59:943–50. [37] FLUENT Inc.. Fluent, FLUENT 6.2 user’s guide. Lebanon: FLUENT Inc.; 2003. [38] Patankar SV. Numerical heat transfer and fluid flow. New York: McGraw Hill; 1980. [39] Wu RM, Lee DJ. Hydrodynamic drag force exerted on a highly porous sphere moving towards an impermeable plate. Chem Eng Sci 1998;53:3571–8. [40] Wang L, Marchisio DL, Vigil RD, Fox RO. CFD simulation of aggregation and breakage processes in laminar Taylor–Couette flow. J Colloid Interface Sci 2005;282:380–96. [41] Xiao Y, Shi HC, Fan L. Numerical simulation of radial flow secondary sedimentation tank Based on CFD. China Water Wastewater 2006;22:100–4. [42] Lee DJ, Hsu YH. Fast freeze/thaw treatment on activated sludge: floc structure and sludge dewaterability. Environ Sci Technol 1994;28:1444–9. [43] Mitani T, Unno H, Akekata T. A structure model for activated sludge flocs. Jpn Water Res 1983;6:65–79. [44] Huang WT, Chang IL, Chen GW, Lee DJ. Uni-directional freezing of waste activated sludge: the effects of freezing speed. Environ Sci Technol 1996;30:2391–6. [45] Lee DJ, Chen GW, Liao YC, Hsieh CC. Use of free-settling test to estimate activated sludge flocs density. Water Res 1996;30:541–50. [46] Chellam S, Wiesner MR. Fluid mechanics and fractal aggregates. Water Res 1993;27:1493–6. [47] Tsou GW, Wu RM, Yen PS, Lee DJ, Peng XF. Advective flow and floc permeability. J Colloid Interface Sci 2002;250:400–8.
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JTICE-874; No. of Pages 8 Journal of the Taiwan Institute of Chemical Engineers xxx (2014) xxx–xxx
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Numerical simulation of flow involving a fractal aggregate Nong Xu a, Long Fan b,* a b
Department of Chemical & Biological Engineering, University of British Columbia, Canada Department of Chemical & Materials Engineering, University of Alberta, Canada
A R T I C L E I N F O
A B S T R A C T
Article history: Received 25 September 2013 Received in revised form 20 March 2014 Accepted 23 March 2014 Available online xxx
Hydrodynamic properties of aggregate are different from those of impermeable particles and have not been well established, owing to their complex structural feature. Therefore, this paper focuses on the study of hydrodynamic properties of an aggregate by computational fluid dynamics method. An aggregate with self-similar structure composed of four primary spheres is constructed to account for its fractal geometry. Navier–Stokes equations are applied for the whole flow field in relation to the aggregate. Both exterior and interior flow through the aggregate could be seen clearly. Fluid collection efficiency, an indication of permeability, is also obtained. Reynolds number has complicated influence on fluid collection efficiency. Furthermore, the larger the space between four primary particles is, the larger the fluid collection efficiency. The smaller the fractal dimension is, the larger the fluid collection efficiency. These findings will enable the development of more realistic models for the flocculation and material transfer dynamics of fractal aggregate. All these results contribute to further research about mass transfer between the aggregate interior and the bulk liquid solution, as well as those researches related to irregular porous structures. ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Aggregate Computational fluid dynamics Fluid collection efficiency Self-similar structure
1. Introduction Suspended particles are often aggregated to improve their performance. This phenomenon has been encountered in many different industrial and scientific applications, such as water and wastewater treatment, atmospheric studies [1], public health research. For example, man-made aerosols in the atmosphere can occur in the form of aggregate. Primary or secondary sedimentation tanks make it possible for flocs to aggregate and settle in water and wastewater treatment, which improves the water quality greatly. The hydrodynamic behavior determines other important properties, such as settling velocity, coagulation rate. Hence, understanding the hydrodynamic behavior of aggregate is important to improve the water/wastewater treatment efficiency, atmospheric studies and other related research. Substantial research indicates that aggregate formed by particle coagulation are fractals [2–5]. The porous structure of fractal aggregate would permit significant interior flow into the aggregate, which would enhance particle collision and material transport. Accordingly, the hydrodynamic properties of the aggregate will be different from those of impermeable particles. Therefore, it is necessary to research on the hydrodynamic properties of the aggregate. The
* Corresponding author. Tel.: +1 6048275921. E-mail address: [email protected] (L. Fan).
relationship between the hydrodynamic properties, the fractal structure of aggregate and its internal permeation can be obtained. They will be helpful to the understanding of aggregation phenomenon, the improvement of water and wastewater treatment efficiency, as well as other research related to fractal aggregate. Nevertheless, two problems make the research of aggregate more difficult. One is the fractal structure. The complexity of fractal structure makes it difficult to simulate the aggregate with real structure. People have tried different methods to numerically construct the real structure for colloidal aggregate, for example, diffusion-limited cluster aggregate (DLCA) [1,6,7], chemically limited aggregate (CLA) [8,9], Tunable-Dimensional Method (TDM) [10]. More usually, people simplified the aggregate as a spherical particle or spheroid [11–15]. Though this makes the research much easier, it brings inevitable errors into the results. To solve the problem, people investigated the structure of aggregate in detail. It is found that aggregate occurring naturally may have multilevel structure [16–18]. Gorczyca and Ganczarczyk [19] proposed an activated sludge model that includes the following stages: (1) primary particles form floccule, (2) floccule group together to form microflocs, and (3) microflocs form the flocs. Floccule, microflocs, and flocs can exhibit different structures and perhaps different fractal dimensions. Thus the concept of selfsimilar structure is presented. It means that an aggregate of any size can be formed directly from its principal clusters with the
http://dx.doi.org/10.1016/j.jtice.2014.03.013 1876-1070/ß 2014 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
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Nomenclature A0 Ah d da g P p q R Re u u0 V x, y, z
m r h
the projected area of the aggregate [m2] the total area of the holes through a cross section [m2] diameter of primary particles [m] diameter of aggregate [m] acceleration of gravity [m/s2] pressure [Pa] number of units in a cell side length of a cell [m] radius of flow field [m] Reynolds number velocity [m/s] the fluid velocity (V) approaching the aggregate without the influence of the aggregate [m/s] fluid velocity [m/s] coordinates [m] viscosity of fluid [kg/m s] density [kg/m3] fluid collection efficiency
same general shape as the aggregate. The self-similar structure has been proven to reduce the computational time greatly and has been used in many researches successfully. For example, Rogak and Flagan [20] assumed clusters to be self-similar in their research about Stokes drag on cluster. In the work of Li and Logan [21], the particles in an aggregate were distributed according to a self-similar structure. Woodfield and Bickert [22] also treated the individual floc as a nested structure of settling swarms of permeable sub-spheres. Then an aggregate could be split into any number of levels, with any relative ratio between successive levels. Therefore, self-similar structure is a very good solution to the fractal structure of aggregate. If we could propose an appropriate self-similar structure, the aggregate could be represented quite well. The other problem comes from the porous structure of aggregate. Since the aggregate is permeable, fluid will flow through the interior of the aggregate. How to simulate the interior flow through the aggregate is another problem encountered in the simulation. Flow through permeable aggregate has been the subject of many theoretical and experimental investigations [11–15,23–31]. Different methods are used to account for the hydrodynamic behavior of aggregate. Kirkwook and Riseman [23], Riseman and Kirkwood [24], Hess et al. [25] employed Kirkwook–Riseman theory to calculate the translational diffusion coefficient, intrinsic viscosity for flexible or rigid chain-like polymers. Geller et al. [26] and Mondy et al. [27] used boundary element method to predict the rotation or translation motion of isolated, nonspherical particles in creeping Newtonian fluid, including isolated chain agglomerate particles, branched chains and flakes. Gastaldi and Vanni [28], Ka¨tzel et al. [10] used method of reflections and Filippov approach [29] to calculate the drag force, respectively. Babick et al. [30] employed Filippov approach to study the impact of particle size and fractal dimension on aggregate. Another promising method for solving the fluid flow through aggregate is the Lattice Boltzmann method (LBM). It allows the calculation of hydrodynamic behavior of aggregate. LBM was also combined with accelerated Stokesian-dynamics method to calculate the drag force of fractal aggregate [31,32]. However most of these methods are focused on aggregate in creeping flow, which limited their application somehow. A method is needed to account for the hydrodynamic behavior of aggregate with both low and high Reynolds number.
Different approaches are also used for the flow fields outside and inside the aggregate. The approach typically used for the flow exterior to the aggregate (under creeping or laminar flow conditions) is to solve the Navier–Stokes equations. As to the interior, there are two primary approaches [22], those that require knowledge of the positions of all particles within the aggregate and those that do not. The latter is paid more attention and it includes three different methods: Darcy’s law, Brinkman equation and Darcy–Brinkman model. For example, Adler [11] described the flow inside a spherical aggregate with the Brinkman equation of motion during the study on porous particles. Mountain et al. [12] used Darcy’s law for the flow inside a cluster in the research about the formation of high temperature aerosol agglomerates. Hsu and Hsieh [13] and Hsu et al. [14] treated the floc as a two-layer structure to investigate the boundary effect on the drag force acting on a spherical floc. Darcy–Brinkman model and the continuity equation were applied to the description of the flow field inside the floc. In the research of Veerapaneni and Wiesner [15], Vanni [33], they both described the flow exterior to the aggregate by the Stokes and continuity equations, while the flow inside the aggregate was assumed to be governed by Brinkman’s equation. This paper focuses on the above-mentioned two problems in the simulation of an aggregate settling in fluid by computational fluid dynamics (CFD) method. Self-similar structure is employed to represent the fractal geometry of the aggregate, instead of simplifying it as a spherical particle or using the above-mentioned methods. It will be more close to the aggregate in nature. The simulation of the flow through the interior of aggregate would become less difficult with the self-similar structure. Contrast to those simulating the inner and outer of the aggregate with different methods, the same method is used for the simulation of the flow fields outside and inside the aggregate since no obvious boundary between outside and inside, which makes the simulation much easier. Furthermore, the flow involving aggregate with different size and space is presented. Permeability is calculated and compared among different aggregate. This research tries to approach the real aggregate in nature. The results will benefit further research about aggregate, such as improvement of aggregation in wastewater treatment. 2. Simulation methods 2.1. Structure of the aggregate Self-similar structure can not only express the fractal geometry of aggregate in some degree, but also save the computation time remarkably. Different self-similar structures can be conceived according to the focus of different research. A fractal aggregate with self-similar structure is constructed in the present study. Four solid spherical particles compose the basic structure with three at the bottom and one at the top in a form of triangular cone, as shown in Fig. 1. For the sake of simplification, these four particles do not have special chemical composition. The aggregate is permeable for the four adjacent particles contact with each other at one point and a hole is present among the three bottom spheres. It could represent the porous fractal geometry of aggregate well. Based on the self-similar structure, every sphere itself is also a small aggregate composed of four small primary spheres. In other words, each aggregate has different levels, as given on the right of Fig. 1: (Level 1) Four primary particles form a small aggregate. We name the small aggregate as aggregate unit. (Level 2) Aggregate unit is the big primary particle. Four aggregate units form a big aggregate (aggregate cell). (Level 3) Aggregate cell is the bigger primary particle. Four aggregate cells form a bigger aggregate. The circle encompassing the four spheres gives the real shape and boundary of the aggregate. Red and black
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Fig. 1. The self-similar structure of the simulated aggregate.
circles in Fig. 1 are both aggregate composed of four primary particles. It would be very complicated if we consider every sphere as an aggregate in the simulation. Aggregate at each level are all composed of four primary ‘‘particles’’. The primary ‘‘particles’’ are either real spherical particles or small aggregate. Therefore aggregate will have the same structure, as well as the same flow type. Based on this, the flow type at any level will be similar to the four-sphere structure. Hence, we use the four-sphere structure to represent the aggregate, regardless of its level. The fractal structure of aggregate can be characterized by fractal dimension. Fractal dimension is an index for characterizing fractal patterns by quantifying their complexity as a ratio of the change in detail to the change in scale. It is a measurement of how particles convey information in duplication of a pattern of behavior at different size scales [34]. According to Rogak and Flagan [20], the fractal dimension is defined as log p/log q, where p units are arranged on a cell with side length q. Usually, fractal dimension varies from 0.26 to 3 [15]. The aggregate with a large fractal dimension will have a small permeability. We get the fractal dimension of 2 for the present aggregate. 2.2. Simulation methods The structure of the aggregate, even the interior, can be seen clearly with the present self-similar structure. There is no obvious interface between the outside and inside of the aggregate. Thus, it is unnecessary to use different methods for the flow fields outside and inside the aggregate, like the above-cited literature. Instead, we employ the same three-dimensional Navier–Stokes equations [35], Eqs. (1)–(4), for both the exterior and interior flow of the aggregate.
@ @ @ ðrux Þ þ ðruy Þ þ ðruz Þ ¼ 0 @x @y @z
(1)
! @ux @ux @ux @P @2 ux @2 ux @2 ux þ uy þ uz þm r ux þ þ ¼ @x @y @z @x @x2 @y2 @z2 þ rg x
r ux
(2)
@uy @uy @uy @P þ uy þ uz ¼ @x @y @z @y @2 uy @2 uy @2 uy þm þ þ @x2 @y2 @z2 þ rg y
!
(3)
r ux
@uz @uz @uz @P @2 uz @2 uz @2 uz þ uy þ uz þm þ þ ¼ @x @y @z @z @x2 @y2 @z2 þ rg z
!
(4)
where u and P denote the local flow velocity and pressure of the fluid, and m is the fluid viscosity. The general-purpose CFD software FLUENT is used to solve these equations. The simulated fluid is water. It is supposed to be incompressible Newtonian fluid with density and viscosity of 1000 kg/m3, 0.001 kg/m s, respectively. It is assumed that the spheres forming the aggregate are impermeable and there are no boundary effects on the surfaces of the aggregate. Considering that the aggregate settles at a steady speed V through an unbounded quiescent fluid, two simplifications are made. First, it is supposed that the domain can be used to represent an unbounded flow field if the ratio R/da (radius of flow field/diameter of aggregate) exceeds 7.5 [36]. Second, for the sake of easy definition of boundary conditions in the computational domain, the aggregate is assumed to be fixed, while the surrounding fluid is flowing at a uniform speed of V from infinity toward the fixed aggregate. No slip conditions are imposed at the surfaces of the primary spheres. The velocity is supposed to be uniform at the place far from the aggregate, as shown in Fig. 2. 2.3. Solution methods GAMBIT [37] is used for grid generation. Since the flow fields within and around the porous aggregate are of major concern, more grids are allocated near the aggregate for better accuracy. In order to resolve the contact points among the primary particles, tetrahedral mesh is employed at these regions. The discretization of equations is performed using the first-order upwind scheme and the pressure is solved by the SIMPLE algorithm [38]. Different discretization methods have been tried. First-order upwind scheme is found to be the best for the present research. The non-linearity in the phase momentum equations is dealt with the under-relaxation technique. When the residuals of the equations being solved meet the prescribed tolerance of a change of <0.1% from the previous iteration, a converged solution is considered to have been obtained. The choice of grids can influence the simulation significantly. Three grids are tested in our simulations with 310,383, 644,885, and 895,098 intersection points. It is found that the results for grids
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Fig. 2. Boundary conditions and the coordinates.
of 644,885 points are very similar to those for 895,098 grid points. Hence, 644,885 grid points are chosen for the simulations presented here. 3. Results and discussion The aggregate is located in the center of a cubic column in simulation. The side length of the column is 50 mm. The four primary spheres forming the aggregate in Fig. 1 are 1 mm in diameter. It is assumed that water at infinity flows upward with a velocity of 0.001 m/s. The origin of the coordinates is set at the center of the three bottom spheres shown in Fig. 2. Several planes are defined at y = 1.0, 0.5, 0, 0.5 and 1.0 mm for presentation of the simulation results, as plotted in Fig. 3. We only show the front views of the results in the following parts for the sake of simplification. 3.1. Determination of flow type To determine the flow type, a suitable Reynolds number is an index. For a settling particle, the Reynolds number is defined by Re ¼
dur
(5)
m
where d is the diameter of the particle. It has been found that the flow is laminar when the Reynolds number is below 1 [39,40].
However, the aggregate’s Reynolds number is not always much less than unity. In contrast, for aggregate with diameter greater than 100 mm, the aggregate’s Reynolds number can markedly exceed unity. In some cases the aggregate’s Reynolds number is found to be up to several tens. For example, most of the aggregate in secondary sedimentation tank was around 10–250 mm in diameter [41]. In the research of Lee and Hsu [42], the size of some aggregate was above 100 mm, even up to 160 mm. Correspondingly, the settling velocity would be relatively high. Regardless of the aggregate’s Reynolds number, some researchers directly employed some Stokes law-like correlations for their work [43,44]. In other words, the data for aggregate with a Reynolds number up to 100 are analyzed in the same way as those under creeping conditions. Although such an approach is apparently erroneous, it can give a reasonable result [45]. The reason may be the followings. For a permeable aggregate moving at an elevated Reynolds number, the boundary layer separation and the after-sphere wake might not occur as in the nonporous cases owing to the advection flow through the interior of the aggregate. Therefore the Stokes’ law-like correlation is still applicable at the elevated Reynolds number region. In this paper, the fluid velocity varies from 0.001 m/s to 0.1 m/s, corresponding to the Reynolds number of 2.28 to 227.8. On the basis of the analyses above-mentioned, the laminar flow model is employed for the whole simulation. 3.2. Flow field of the aggregate Fig. 4(a) and (b) shows the simulated velocity vectors and streamlines in the planes at y = 0.5, 0 and 0.5 mm with fluid velocity of 0.001 m/s, respectively. The fluid flows upward in a stable and uniform pattern at the bottom of the setting column. The velocity decreases approaching the aggregate. Most of the fluid flows around the aggregate, while a small portion of the fluid enters the aggregate through the hole among the spheres at the bottom. In the center plane at y = 0 mm, the fluid is mostly blocked by the two spheres at the bottom and the sphere at the top. However the fluid only changed the flow direction slightly to pass the aggregate. No circulation loops appear. The possible reason for this is that the Reynolds number is too low to produce circulation loops. The flow structure in the planes with y of 0.5 mm and 0.5 mm are similar to this. Fluid flows toward the centerline at the place just above the spheres. At the plane y = 0.5 mm, a small portion of the fluid flows through the gap between the two bottom spheres. Then the fluid flows around the top sphere, which can be seen clearly in Fig. 4(a) and (b). In the planes at y = 1.0 and 1.0 mm, the flow fields (not shown in the present paper) are simpler compared to the previous analyses. The streamlines are similar to velocity vectors, as plotted in Fig. 4(b). At the place far below the aggregate, all streamlines are uniformly distributed parallel to the centerline. The streamlines become curved approaching the aggregate. Most of the streamlines pass around from both sides of the aggregate, while a small number of streamlines enter the aggregate through the hole between the three bottom spheres. After passing the aggregate, the streamlines restore to uniform parallel distribution gradually with the increasing distance to the aggregate. 3.3. Fluid collection efficiency
Fig. 3. The selected planes (planform).
Fluid collection efficiency, h, is a direct indication of internal permeation of aggregate. It is defined as the ratio of the interior flow passing through the aggregate to the flow approaching it [11,46]. The fluid collection efficiency for a porous sphere ranges between zero and unity. By denoting the off-center distance of the limiting streamline from the axis of symmetry, the fluid collection
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Fig. 4. (a) Velocity vectors at different planes (front view). (b) Streamlines at different planes (front view).
efficiency is given as the square of the ratio of the off-center distance to the radius of the aggregate. Veerapaneni and Wiesner [15] presented the relationship between fractal dimension and fluid collection efficiency with this definition. Tsou et al. [47] utilized the definition of Adler [11] about fluid collection efficiency, but they named it as capture ratio. The fluid collection efficiency in the present simulation is calculated using the following equation. R Ah
h¼
u dA u0 A0
0
(6)
where u0 is the far upstream fluid velocity (V) through the projected area of the aggregate. A0 is the projected area of the aggregate in the horizontal plane (x–y plane). Ah is the total area of the hole through the cross section of the aggregate, and u is the velocity of the internal flow through the hole. The fluid collection efficiency of the aggregate is determined as 0.0117 with the flow velocity of 0.001 m/s. It means that only 1.17% of the fluid approaching the aggregate flows through the interior of the aggregate. Veerapaneni and Wiesner [15] found that the fluid
collection efficiency h is about 0.009 for aggregate with da/d = 100 and fractal dimension of 2. h increases with the decrease of da/d. When da/d changes from 100 to 500, h decreases from 0.009 to 0.002. Fluid collection efficiencies for aggregate with different da/d are very close, when fraction dimension is above 1.95. Hence different extrapolation methods will not have significant influence on fluid collection efficiency. Suppose h changes with da/d linearly. Then h of the present aggregate with da/d about 2 would be around 0.0107. It is close to our simulation value 0.0117. This gives a good indication that our simulations are reliable. This internal permeation may not be essential to the hydrodynamic behavior of the aggregate. However, it could make an important contribution to the mass transfer between the aggregate interior and the bulk liquid solution. It is helpful to understand the relationship between the hydrodynamic properties and the fractal structure of aggregate. 3.4. Effects of fluid velocity The fluid velocity varies from 0.001 m/s to 0.005, 0.01, 0.05 and 0.1 m/s in simulation. The velocity vectors through the aggregate are similar at different fluid velocities.
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Fig. 5. Evolution of fluid collection efficiency with Reynolds number. Fig. 7. Evolution of drag force with Reynolds number.
Different from velocity vectors, the fluid collection efficiency varies with the fluid velocity. Correspondingly, it varies with Reynolds number. The fluid collection efficiency increases with the increase of the Reynolds number, as plotted in Fig. 5. In other words, fluid is easier to flow through the interior of the aggregate with a larger Reynolds number. 3.5. Effects of primary particles’ diameter The simulation is performed for aggregate with diameter of 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50 and 100 mm to investigate the effect of primary particles’ diameter. The boundary of the flow domain are changed accordingly to satisfy the need of R/da > 7.5. The evolution of fluid collection efficiency with Reynolds number is shown in Fig. 6. When Reynolds number is larger than about 1000, correspondingly the diameter is larger than 5 mm, the fluid collection efficiency decreases with Reynolds number and the diameter. Veerapaneni and Wiesner [15] observed the similar phenomenon: The fluid collection efficiency would decrease with the increasing aggregate size for a given fractal dimension. The reason may come from drag force. The drag force comes up with the increase of Reynolds number, as plotted in Fig. 7. The markedly enhancing drag force inhibits the increase of the fluid collection efficiency. Hence, the fluid collection efficiency falls down with the
increment of Reynolds number and diameter. When Reynolds number is smaller, the influence of Reynolds number and diameter on fluid collection efficiency is not obvious. Since the results are based on several complicated equations, we still do not come up with a good reason for this phenomenon. The possible reason may come from the very low Reynolds number and the small diameter. The flow through the aggregate will be very slow with a small Reynolds number. It may behave like viscous fluid. Meantime, the space among the primary spheres will be much smaller when the diameter is small. Different forces may take into effect in this narrow space. Therefore the forces acting on the fluid should be considered differently. The involvement of these forces contributes to the slight effect on fluid collection efficiency. More research could be done toward the satisfactory explanation. Based on Figs. 5 and 6, it is concluded that Reynolds number has a non-trivial influence on the fluid collection efficiency. Fluid collection efficiency decreases with Reynolds number and diameter when Reynolds number is above 1000. For primary particles with diameter of 1 mm, fluid collection efficiency increases with increasing Reynolds number. However mostly the effect of Reynolds number is not obvious with a small Reynolds number. 3.6. Effects of space between primary spheres
Fig. 6. Evolution of fluid collection efficiency with Reynolds number.
To investigate the characteristic of the aggregate further, the relative locations of the four primary spheres are changed. The contact four primary spheres are detached so that there are spaces between any two of them. The space is represented by small spheres. Six small spheres representing space appear in the aggregate. The diameter of the small spheres is 10%, 20% of the primary spheres’ diameter, respectively. Fig. 8 illustrates the sketch of an aggregate with 10% space, where the big spheres with shadow are the primary particles. Since the small spheres are only used to illustrate the presence of space, they are shown as open circles. Fig. 9 compares the streamlines through different aggregate. The flow through the interior of the aggregate becomes easier because of the presence of the space between primary spheres. The fluid is apt to flow into the aggregate with a larger space. It could be anticipated that the fluid collection efficiency would be improved with the increase of the space. It is obvious that the space between the primary particles favors the improvement of the permeability, as shown in Fig. 10. With the inclusion of the space between primary spheres, not only the fluid
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collection efficiency decreases greatly with the increment of fractal dimension, as illustrated in Fig. 11. 4. Conclusions A porous fractal aggregate is simulated by introducing selfsimilar structure. Flow through and around the aggregate is obtained with CFD method. Permeability of the aggregate represented by fluid collection efficiency is also given. The effects of primary spheres’ diameter, fluid velocity, as well as the space between primary spheres, fractal dimension, are investigated. The following conclusions are obtained.
Fig. 8. Sketch of an aggregate with 10% space between primary spheres (planform).
collection efficiency but also the fractal dimension change. The fractal dimension for aggregate with space of 10% and 20% reduces to 1.75 and 1.58, respectively. Hence we could include ‘‘effective fractal aggregate’’ to describe the aggregate. The larger the space between the primary spheres is, the smaller the fractal dimension is, and the smaller the effective fractal aggregate is. The fluid
(1) A self-similar structure composed of four primary spheres is presented. It could represent the irregular porous structure of aggregate well. The flow through and around this aggregate is obtained. It provides a simple method for further research about fractal aggregate. (2) Reynolds number has a non-trivial influence on the fluid collection efficiency. Fluid collection efficiency decreases with fractal dimension. The more space between the primary spheres, the smaller the fractal dimension is, the smaller the effective fractal aggregate is. (3) The simulated internal permeation would be beneficial to further research about mass transfer between the aggregate
Fig. 9. Effects of space between primary spheres on streamlines.
Fig. 10. Effects of the space between primary spheres on fluid collection efficiency.
Fig. 11. Relationship between fluid collection efficiency and fractal dimension.
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interior and the bulk liquid solution. They will enable the development of more realistic models for the flocculation and material transfer dynamics of fractal aggregate. The method also provides basis for other research related to irregular porous structures.
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Please cite this article in press as: Xu N, Fan L. Numerical simulation of flow involving a fractal aggregate. J Taiwan Inst Chem Eng (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.013