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Direct simulations of particle deposition and filtration in dual-scale porous media
Composites: Part A 42 (2011) 1344–1352
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Direct simulations of particle deposition and filtration in dual-scale porous media Wook Ryol Hwang a, Suresh G. Advani b,⇑, Shawn Walsh c a
School of Mechanical Engineering, Research Center for Aircraft Parts Technology (ReCAPT), Gyeongsang National University, Jinju 660-701, Republic of Korea Department of Mechanical Engineering, Center for Composite Materials, University of Delaware, Newark, DE 19716, USA c Army Research Laboratory, Aberdeen Proving Grounds, Aberdeen, MD 21005, USA b
a r t i c l e
i n f o
Article history: Received 3 March 2011 Received in revised form 17 May 2011 Accepted 21 May 2011 Available online 27 May 2011 Keywords: A. Particle-reinforcement B. Rheological properties C. Computational modeling E. Resin transfer molding (RTM)
a b s t r a c t A two dimensional direct numerical simulation technique is developed to describe particulate flows in dual-scale porous media to predict particle deposition on the permeable porous surface in liquid composite molding processes. This individual particle level simulation accounts for hydrodynamic interaction between particles and the fluid, especially near a porous wall (fiber tow), and can predict the deposition of the particles on solid or porous surfaces. A Stokes–Brinkman coupling is employed to describe the flow in dual-scale porous media and a fictitious domain approach is used to deal with freely suspended particles in the fluid stream. A single particle deposition process is investigated extensively along with effects of the permeability of porous media, the particle size and the pressure drop. Mechanisms leading to accelerated or delayed deposition of particles are analyzed by investigating the velocity fields around the particle in close proximity of the porous surface. Finally, particle filtration simulation are performed with a large number of particles to demonstrate the feasibility of this scheme to address particle deposition and filtration during manufacturing of composites using liquid composite molding processes in which the particles are mixed with the resin and the suspension is injected into a stationary dual scale preform. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Particle deposition and filtration within suspensions through porous media is of great significance in several natural and industrial processes: for example, formation of river beds, ground water contamination, deep bed filtration and manufacturing advanced composite materials. Among them, we are particularly interested in liquid composite manufacturing (LCM) processes, such as the resin transfer molding (RTM) or vacuum assisted resin transfer molding (VARTM), where particles (or fillers) are often added to achieve a specific property like fire resistance, abrasion resistance, electromagnetic shielding, or even for reduction in material costs. An interesting example is in manufacturing functionally graded composites (FGC) with the VARTM process in which particles are added to introduce multifunctional properties [1,2]. Depending on the specific application, the goal of infiltration in this case may be to entrap all particles in one layer, to achieve uniform distribution of particles, or to create a particle concentration gradient, usually in the thickness direction. The preforms are usually dual scale fabrics in which yarns or tows (bundles of thousands of micron size fibers) are woven, stitched or knitted together. Understanding particle deposition and filtration phenomena in
⇑ Corresponding author. Tel.: +1 302 831 8975; fax: +1 302 831 3619. E-mail address: [email protected] (S.G. Advani). 1359-835X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2011.05.017
such a heterogeneous dual-scale fibrous porous media will provide guidance to obtain a desired gradient of particle deposition in this application. Another example is a recently proposed manufacturing process for thermoplastic matrix fiber-reinforced composites by introducing thermoplastic powder as an aqueous suspension [3]. This novel process combines the VARTM process to impregnate the preform with the thermoplastic powder suspension together with the compression molding process to consolidate the thermoplastic matrix with the fabrics under pressure and heat. Again physical understanding of particle deposition and filtration within dual-scale fabrics will play a key role in designing proper material and process parameters such as geometrical architectures and the permeability of the fabrics, the particle size and concentration, the pressure or flow rate during impregnation. Few researchers have addressed particle filtration in liquid molding processes, in most cases intended to propose macroscopic constitutive models for the particle concentration and particle retention during flow. For example, Erdal et al. [4] formulated an evolution equation for the particle concentration during particlefilled resin infusion and filtration kinetics, considering resuspension of particles. Using these models for single scale porous media, they performed impregnation flow simulations for complex geometries and reported that irregular geometry of the mold greatly influences the local rate of filtration. Lefevre et al. [5,6] argued that a proper retention function needs to be introduced to account for their experimental observation in typical particle
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concentration profiles. For dual-scale porous media, Chohra et al. [1,2] modeled the filtration phenomena to predict the particle concentration distribution through single- and multi-layer fibrous tow geometries, employing a flow resistance network model with the viscosity variation due to the presence of particles. However, as was indicated in Nordlund et al. [7], such process models based on simple empirical relationship do not describe the phenomena causing particle depositions itself. Indeed micro-scale optical measurements and PIV experiments by the same group showed that particle filtration patterns are highly non-homogeneous and irregular and are closely related to the local flow fields [7,8]. In this regard, in-depth knowledge about a specific mechanism is necessary that leads to particle deposition and collectively to filtration within dual-scale porous media. Hence, it is important to examine individual particle behavior considering hydrodynamic interactions between particles and resin and specific flow characteristics in dual-scale porous media. In this work, we develop a direct solid/liquid meso-scale flow simulation to understand particle deposition and filtration in dual-scale porous media. The term ‘direct’ has been used, since hydrodynamic, inter-particle and particle-porous wall interactions are considered at the individual particle level. In addition, it is a meso-scale simulation because the flow in dual scale porous media is described by the Stokes–Brinkman coupling with the bi-periodic boundary condition that has been verified to successfully capture flow phenomena in such a system, especially near the interface between fluid and porous medium which invokes the particle deposition criteria. We first model the system by a set of governing equations, identifying important forces that govern the flow and particle deposition in this problem. Then we present our numerical method and implementation techniques. The numerical scheme is examined for the single-particle deposition on the porous surface case to understand the interplay between the particle deposition and flow in dual-scale porous media. This scheme is then extended to simulate the filtration and deposition process by introducing many particles in the flow that are randomly generated.
2. Modeling There are two distinct pore scales in dual scale porous media of liquid composite molding: one is the inter-tow space (macro-pore) between fiber bundles and the other is the intra-tow space (micropore) between micron-sized fibers within a tow. For a typical woven fabric, the inter-tow spacing is of the order of several hundreds of microns or more and the intra-tow spacing is of the order of one micron. The particle size of interest in the present study is in between, i.e. of the order of tens of microns, considering applications to manufacturing of functionally graded composites or thermoplastic matrix composites [1–3], as mentioned earlier. In this case, no surface filtration is expected in the macro-pores, but it takes places around the fiber tow boundary, as particles cannot penetrate into the micro-pores. We note that such particles in a fluid are nonBrownian: i.e., random displacement of particles due to the thermal fluctuation of fluid particles can be safely neglected. Fig. 1 describes the system of interest. The computational domain X is composed of the fixed region occupied by the porous material (Xp), the region occupied by the surrounding fluid (Xf) and the region occupied by particles. The interface between the porous and fluid regions is denoted by Cpf and the boundary of the computational domain is denoted by Ci(i = 1, 2, 3, 4). The horizontal (and vertical) periodicity in x (and y) is applied between C1 and C3 (and C2 and C4) to introduce the unit cell flow problem in X. The pressure difference Dp is assigned in the vertical direction. We consider only circular particles in this study and particles
Fig. 1. A schematic description of solid/liquid flow in dual-scale porous media in a bi-periodic unit cell. The subscripts ‘p’ and ‘f’ denote the porous media and the surrounding fluid, respectively. Particles are indicated by filled black circle and the i particle is denoted by Pi(i = 1, , N). The pressure drop in the vertical direction is taken into account.
S are denoted by Pi(t)(i = 1, , N) with P(t) for Ni¼1 Pi ðtÞ and N is the total number of particles. For a particle Pi, Xi = (Xi, Yi), Ri, Ui = (Ui, Vi) and xi = xik are used for the coordinates of the particle center, the particle radius, the translational velocity and the angular velocity respectively; and k is the unit normal vector out of the plane. 2.1. Governing equations for flows in dual-scale porous media Assuming that the inertia of the fluid is often neglected at this scale, the fluid domain can be modeled by the Stokes flow. For the porous media, we use the Brinkman equation as a momentum equation, considering the porous media as a continuum body that can be characterized by the permeability. This meso-scale description should be sufficient for our specific purpose, as the particle penetration through the micro-pore is excluded. Recently, we developed a single momentum equation called equivalent momentum equation, for the Stokes–Brinkman coupling with both the continuous stress and stress jump conditions that is valid everywhere in the domain, including the interface between the two media (for details, see Hwang and Advani [9]). In the present work, we employ this approach to model the coupled flow in dual-scale porous media. The single set of equations for both the regions (except the region occupied by particles, i.e. X n P) with the interface condition can be written as follows:
r u ¼ 0; r r a
gf Kp
in X n P;
ð1Þ
u ¼ 0;
ð2Þ
r ¼ p I þ 2g D;
in X n P;
ð3Þ
where u, r, p, g, Kp, I and D are the velocity, the stress, the pressure, the viscosity, the permeability of the porous medium, the identity tensor and the rate-of-deformation tensor, respectively. The superscript ‘’ defines the domain to which all physical quantities belongs to: For example in Xp, u = up, r = rp, p = pp and g = gp; in Xf, u = uf, r = rf, p = pf and g = gf. The function a is a material index which takes a value of ‘1’ in Xp (porous media) and ‘0’ in Xf (fluid domain). Eq. (2) is a special case of the Stokes– Brinkman coupling as described in Hwang and Advani [9] with the continuous stress condition on the interface Cpf. For the boundaries of the computational domain, we assign the bi-periodic boundary condition such that flow defined in a single unit cell could represent much larger periodic arrays of the cell structures. The periodicity in the horizontal and vertical directions can be expressed by the velocity continuity over the periodic boundary respectively:
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ujC2 ¼ ujC4
and ujC1 ¼ ujC3 :
ð4Þ
In addition to the periodicity, a uniform pressure difference Dp is applied in the vertical direction which is the driving force of the flow. 2.2. Particle domain by the rigid-ring description and hydrodynamic interaction Following the work by Hwang and co-authors [10–12], we consider the circular particle as a rigid ring, filled with the same fluid as in the fluid domain, and the rigid body condition is imposed on the particle boundary only. From the rigid-ring description, we use exactly the same set of equations for the particle domain as those of the fluid domain and it allows the implicit treatment of hydrodynamic interaction between particles and the fluid. Indeed, one can use a single set of governing equations (Eqs. (1)–(3)) for the entire domain Xð¼ Xf [ Xp [ PÞ and the discretization of the particle needs to be done only along its boundary. The rigid-body condition on the particle boundary states that the velocity of a fluid particle attached on the particle boundary is the same as that of the particle:
u ¼ U i þ xi k ðx X i Þ on @Pi ðtÞ:
ð5Þ
In addition, the movement of particle is given by kinematics
dX i ¼ Ui; dt
X i jt¼0 ¼ X i;0 :
ð6Þ
To determine the unknown particle rigid-body motion (Ui, xi), one needs balance conditions for the drag force and the torque on the particle boundary. In the absence of inertia and external forces or torques, the particle is subjected to the following force and torque balances:
F Hi þ F Pi ¼ 0;
Ti ¼
Z
F Hi ¼
Z
r n ds;
ð7Þ
@Pi ðtÞ
ðx X i Þ r n ds ¼ 0;
ð8Þ
@P i ðtÞ
where Ti = Tik and n is a unit normal vector on the particle boundary pointing out of the particles. Eq. (7) describes the balance between the hydrodynamic drag force F Hi and the repulsive particle–particle collision force F Pi , which is an artificial force to avoid (numerical) particle overlap. (The particle-porous wall interaction will be discussed in the next section separately.) We choose the following expression of Glowinski et al. [13] for this short-range particle– particle repulsive force: F Pi
¼
N X j¼1;i–j
( F Pi;j ;
F Pi;j
¼
0;
di;j > Ri þ Rj þ q;
C p ðX i X j ÞðRi þ Rj þ q di;j Þ2 ; di;j 6 Ri þ Rj þ q; ð9Þ
where di,j = |Xi Xj| is the distance between the centers of the ith and jth particle and q is the force range and Cp is a positive ‘stiffness’ parameter.
Fig. 2. Flow near the impermeable solid and permeable porous media interfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
appears a thin gap of fluid layer, from which slow drainage is produced. The hydrodynamic force due to this slow draining opposes the particle approach to the wall significantly, which inversely scales with the distance between the particle center and the wall. This force makes the particle approach velocity normal to the wall to be very small. However, once the gap becomes extremely thin, the attractive van der Waals force increases very rapidly to overcome the repulsive hydrodynamic force facilitating the deposit of the particle on the solid surface [14]. In case of deposition on a permeable surface, one has to consider additional effects from the penetrating seepage flow into the porous surface, although basic principles of the hydrodynamic repulsion and deposition by the van der Waals force are still applicable. As described in Fig. 2, it is expected that the drainage through the thin gap would be significantly weak with the permeable wall, in comparison with the impermeable surface, though not completely ignorable. Indeed, deposition on the permeable surface is partly driven by the hydrodynamic force due to the presence of the penetrating flow into the permeable surface. Therefore, considering another effect of the van der Waals force that exerts in extremely close proximity to the porous wall, it might be a plausible approximation to regard the particle as deposited, when it approaches very close to the surface (or numerically when it actually touches the surface). We employ this deposition scheme, the forced particle deposition, in the present work, since the forced attachment by the van der Waals force may act more effectively on the porous surface due to the penetrating flows. Admittedly, unphysical acceleration of particle deposition might be expected especially for very low permeable (almost non-penetrating) media, as the strong repulsive force from hydrodynamic drainage cannot be accurately accounted for, mainly due to the relatively coarse mesh size and the finite time step with explicit time marching scheme. We remark that the present scheme is significantly different from the conventional trajectory method (for example, see [14,15]) in that in our scheme the hydrodynamic interaction is fully considered, which is important especially as the particle approaches the porous surface. We will compare our method to the trajectory method when presenting the results for single particle deposition.
3. Numerical methods 3.1. Level-set description of dual-scale porous media
2.3. Particle-porous wall interaction: forced particle deposition scheme Resolving the particle-porous wall interaction is a critical issue in this meso-scale deposition and filtration problem. First, let us consider general aspects of particle deposition on the solid impermeable surface. As is well described in a review paper (Spielman [14]), deposition of non-Brownian particles on the solid surface in a low Reynolds number flow is governed by the balance of the (repulsive) hydrodynamic force and the (adhesive) van der Waals force. When a particle in a flow is in close proximity to a solid surface, it must deviate from the undisturbed streamline and there
We employ the level-set finite-element method developed by Hwang and Advani [9] in this work to deal with flows in dual-scale porous media. We summarize their level-set scheme briefly for completeness and details can be found in [9]. The geometry of the porous media in a fluid domain can be conveniently described by the level-set function /(x) (Fig. 3). The zero-level set indicates the interface and it takes a positive value within the fluid domain and a negative value inside the porous domain. The level-set function is a signed distance function such that the magnitude of /(x) is determined by the shortest distance from a point x to the interface.
W.R. Hwang et al. / Composites: Part A 42 (2011) 1344–1352
Fig. 3. The level-set representation of the porous media in a fluid. The interface between two medium Cpf is defined by the zero level set (/ = 0). The level-set function is positive in the fluid domain and negative in the porous media.
Based on the level-set function, an approximate material index function a(x) in Eq. (2) can be expressed as follows:
8 / 6 e > < 1; að/Þ ¼ 12 ð1 /e p1 sinðpe/ÞÞ; j/j < e : > : 0; /Pe
ð10Þ
where e is the half thickness of ‘diffuse’ interface (Fig. 3) and is usually taken 1.5h with h being the mesh size. 3.2. Finite-element formulation and implementation Following the weak form of the rigid-ring description [10], we combine the equivalent momentum equation (Eq. (2)) with the continuity (Eq. (1)) and constitutive equation (Eq. (3)), the biperiodic boundary constraint (Eq. (4)) and the rigid-ring constraint for freely suspended particles (Eq. (5)) to constitute the weak form. For this purpose three Lagrangian multipliers are introduced: kp;i on the particle boundary @Pi(L2(@Pi)) for the rigid-ring constraint, kh on the domain boundary C2(L2(C2)) for the horizontal periodicity and kv on the boundary C1(L2(C1)) for the vertical periodicity. Then the weak form can be written as follows: Find u e H1(X)2, p e L2(X), kh 2 L2 ðC2 Þ, kv 2 L2 ðC1 Þ, U i 2 R2 , xi 2 R and kp;i 2 L2 ð@Pi Þ such that
Z
Z p ðr v ÞdX þ 2g Dðu Þ X X Z Z gf : Dðv ÞdX þ a u v dX þ kh ðv jC2 v jC4 ÞdC X Kp C2 Z N Z X þ kv ðv jC1 v jC3 ÞdC þ kp;i ½v fV i þ ni k C1
i¼1
C1
Z
4.1. Flow in dual-scale porous media
ð11Þ
i¼1
qðr u ÞdX ¼ 0;
ð12Þ
X
Z C2
Z C1
Z
we used a continuous linear interpolation for the Lagrangian multiplier kh and a nodal collocation method for kv . The uniform pressure difference Dp in the vertical direction is assigned as a usual natural boundary condition in Eq. (11) and, along with the horizontal periodicity constraint (Eqs. (11) and (14)), this ensures both the velocity continuity (Eq. (4)) and the pressure difference in the weak form. For the Lagrangian multiplier kp;i for the rigidbody constraint (Eqs. (11) and (15)), the point collocation method has been used and approximately one collocation point in an element appears to yield the most accurate results [10]. As the discretization method in the present study is the same as one of the author’s previous works, many details on discretization methods can be found therein [9–12]. At t = 0, initial particle position Xi is set for (i = 1, , N). Then at every time step we solve a large matrix equation from discretization of the weak form (Eqs. (11)–(15)) for a given particle position X ni using a direct method based on a sparse multi-frontal variant of Gaussian elimination (HSL/MA41) [16]. Using the particle velocity U ni , we update the particle position X nþ1 for the next time step i according to Eq. (6) using the explicit 2nd order Adams–Bashforth method. If the position X nþ1 of a particle Pi is identified as deposi ited, we enforce the rigid body velocity ðU nþ1 ; xnþ1 Þ to be zero i i in solving the discretized weak form for the next time step, along with a slight relocation of the particle attached exactly to the surface, if necessary. In the filtration problem using many particles, a certain amount of particles need to be introduced periodically into the computational domain; some of them will be filtered by the porous media inside and the remaining ones that pass through the porous media without deposition should be removed from the computation. This process requires book-keeping the status of each particle, whether it is deposited, newly introduced or removed. Moreover, newly introduced particles though periodic should be randomly distributed at each addition event. 4. Numerical Results
@Pi
ðx X i Þgds Z N X ¼ Dpðn v ÞdC þ F Pi V;
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lh ðu jC2 u jC4 ÞdC ¼ 0;
ð13Þ
lv ðu jC1 u jC3 ÞdC ¼ 0;
ð14Þ
lp;i ½u fU i þ xi k ðx X i Þgds ¼ 0; ði ¼ 1; ; NÞ;
ð15Þ
First we consider flow in dual-scale porous media without particles to show typical flow behavior. A model system is presented in Fig. 4, which consists of two elliptic fiber tows, one at the center and the other at the corner (split into four). The computational domain is 5 [mm] 1 [mm] in size and the major and minor axes of the tow are 2.2 [mm] and 0.25 [mm], respectively, and it is discretized by the uniform regular rectangular finite element mesh. Due to the bi-periodicity, it represents a squeezed hexagonal arrangement of tow in a bounded domain. Assuming a typical radius of fiber filament Rf = 0.023 [mm] with the porosity u = 0.53, the transversal permeability of the tow Kp can be estimated by the prediction of Gebart [17] as Kp = 1.1718 105 [mm2], as was done in [10]. The choice of the pressure difference and the viscosity is important in particle-filled flow, since the hydrodynamic drag force scales with the viscosity, the (relative) velocity and the particle size: i.e., FH = 6pgfRV / R(rp) as the velocity scales with the
@P i
for all admissible functions v e H1(X)2, q e L2(X), lh e L2(C2), lv e L2(C1), V i 2 R2, vi 2 R and lp,i e L2(@Pi). We discretized the computational domain uniformly by regular square elements and employed the bi-quadratic interpolation for the velocity and the linear discontinuous interpolation for the pressure. For the periodic constraints in Eqs. (11), (13), and (14),
Fig. 4. A sketch of a model dual scale porous media with two elliptic fiber tows in a bi-periodic domain along with the uniform regular rectangular finite-element discretization.
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Fig. 6. Particle trajectories in the single particle deposition problem using the initial particle position (X, Y) = (1.59, 0.565) for three different permeability value Kp of the porous media from 1.1718 104 [mm2] to 1.1718 106 [mm2]. Decoupled particle paths (blue curves), from the conventional trajectory approach using the unperturbed (without particle) velocity field, are presented for each case as well. The initial particle position is denoted by a black circle and the upper and lower curves around the channel represent the porous boundary. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Streamlines and the shear rate distributions (in [1/s]) in the model problem for three-different permeability values: (from top to bottom) Kp = 1.1718 104 [mm2], 1.1718 105 [mm2] and 1.1718 106 [mm2]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
pressure gradient. With this in mind, we choose the viscosity of the uncured epoxy resin gf to be 0.1 [Pa s], keeping gp = gf, and the pressure difference of 103 [Pa], which yields (1/gf@p/@x) 104 in average and, as a result, the velocity becomes O(1) [mm/s] in magnitude in the fluid domain and O(0.1) [mm/s] in the porous media. Fig. 5 shows typical streamlines and shear rate distributions for three different permeability values Kp = 1.1718 104 [mm2], 1.1718 105 [mm2] and 1.1718 106 [mm2]. We use a 500 100 mesh, correspondingly the mesh size h = 0.01 [mm], for this computation (and all others computations, except the final filtration example) and the shear rate is definedpas the second invariant ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi of the rate-of-deformation tensor II2D ¼ 2D : D. One can clearly observe a noticeable change in streamlines and in the shear rate distributions, mainly due to the different velocity of the penetrating flow inside the porous tow depending on the tow permeability. The existence of the penetrating intra-tow flow, though small, changes the flow dynamics and affects particle deposition significantly. According to the particle deposition mechanism discussed earlier, one may expect increased particle deposition in the highly permeable media.
Fig. 7. Particle trajectories in the single particle deposition problem using the initial particle position (X, Y) = (2.51, 0.63) for four different permeability value Kp of the porous media from 1.1718 104 [mm2] to 1.1718 107 [mm2]. Decoupled particle paths (blue curves), from the conventional trajectory approach using the unperturbed (without particle) velocity field, are presented for each case as well. The initial particle position is denoted by a black circle and the lower curve represent the upper porous boundary of the central fiber tow in Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4.2. Single particle deposition In this section, we investigate the single particle deposition on the porous media, focusing on coupled flow effects of this phenomenon near the interface. We use the same tow geometry and materials properties as in the previous case (Fig. 5) and additionally we have a single particle of size R = 20 [lm], initially located at (X, Y) = (1.59, 0.565) inside the channel between two porous media. The time step in the simulation is 0.01 [s]. Plotted in Fig. 6 are paths of particles from the initial position until deposition occurs, if any, for three different permeability values of the porous media Kp = 1.1718 104 [mm2],1.1718 105 [mm2] and 1.1718 106 [mm2]. In Fig. 6, we present the prediction using the conventional trajectory method as well, which is denoted as ‘decoupled’ as this particle trajectory is obtained by integrating the unperturbed velocity without particles. (Note that we solved the full computational domain with the porous tow geometry, as shown in Fig. 5, but present only a small region of interest near the particle in Fig. 6.) Observation from Fig. 6 can be summarized as follows:
Fig. 8. A sketch of the velocity decomposition of flow around the particle in simple shear flow. The total velocity can be decomposed into the given simple shear flow ussf and the hyperbolic perturbation flow contribution u0 due to the presence of the particle.
(i) From the coupled direct simulation result, particle deposition is delayed as the permeability decreases and, in case of the lowest permeability (Kp = 1.1718 106 [mm2]), no particle deposition is observed at all. (ii) As the permeability becomes lower, direct coupled flow simulations predict earlier particle deposition than the decoupled trajectory approach. The first observation is important and can be easily expected, since there appears a seepage flow into the porous surface whose magnitude varies inversely with the permeability and this toward-surface flow enhances the particle deposition by overcoming the hydrodynamic resistance due to the drainage flow. However, the second observation is somewhat confusing, because it has been
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Fig. 9. Decomposition of velocity fields around the particle near the porous interface for four different permeability values of the porous surface: (A) Kp = 1.1718 104 [mm2]; (B) 1.1718 105 [mm2]; (C) 1.1718 106 [mm2]; (D) 1.1718 107 [mm2]. The total velocity field u is presented on the first column. The unperturbed velocity field u0 and the perturbed velocity contribution u0 are presented in the second and the last columns. The black circle indicates the particle and the upper and lower (orange) lines indicate the porous surface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
well known that particle deposition to the impermeable solid surface is significantly delayed due to the repulsive hydrodynamic force from the drainage, in comparison with the conventional trajectory approach [14]. Obviously the solid surface is a limiting
case of the low permeable porous surface and this implies that there must be a break-even point such that further decrease in the permeability, may lead to delay in the particle deposition. To check this we performed another set of simulations with the
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Fig. 10. Effects of the particle size on particle deposition. Three different particles (10, 20 and 30 [lm] in radius) are initially located at the same position, denoted by black circles. The permeability of the porous media in this case is Kp = 1.1718 105 [mm2]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
different initial particle position at (X, Y) = (2.51, 0.63) near the middle of the central fiber tow. Particle paths are presented in Fig. 7 for four different permeability values from 1.1718 104 [mm2] to 1.1718 107 [mm2]. As expected, particle deposition is found to be significantly delayed on the lowest permeable surface (Kp = 1.1718 107 [mm2]) in the coupled simulation and in fact no deposition is observed at all, in comparison with the conventional trajectory result. In the two cases with the high permeability Kp = 1.1718 104 [mm2] and 5 2 1.1718 10 [mm ], earlier particle deposition is observed with the coupled method than with the decoupled approach, though it
can be hardly identified by naked eyes, probably because of the impinging flow field in this region. Further, we observe that, in case of Kp = 1.1718 106 [mm2], the particle initially deviates from the unperturbed (decoupled) trajectories away from the porous surface, but then it gradually turns its direction toward the surface, resulting in the final deposition, which looks exactly the same position predicted by the decoupled trajectory analysis. Although the coincidence in final deposition position happens by accident, this case with Kp = 1.1718 106 [mm2] seems to be very close to the break-even point mentioned above. That is, for the permeability lower than this, the coupled direct simulation predicts earlier deposition than the decoupled trajectory approach, while the higher permeability yields slower or delayed deposition in comparison to the trajectory method. The results in Fig. 7 confirm this hypothesis. 4.3. Flow field around the particle near the porous surface Turning of the particle trajectory toward the wall, when it approaches in close proximity of the porous surface with the intermediate permeability, is interesting. In fact it is found to be that in general all particle trajectories from the coupled direct simulation in Figs. 6 and 7, show such a turning trajectory toward the surface, when the particle deposition occurs. In this section, we investigate
Fig. 11. Six consecutive particle distributions in a model dual-scale porous media from a numerical filtration and deposition experiment with randomly inserted 500 particles of the radius 20 [lm] for 2 s. Blue curves indicate the boundary of porous tows and red circles are particles. The black box in the initial configuration is the region, where five initial particles are randomly distributed every 0.1 [s], and particles below the lower violet line are removed from the computational domain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
W.R. Hwang et al. / Composites: Part A 42 (2011) 1344–1352
flow fields around particles near the porous surface, in order to understand the hydrodynamic mechanisms behind faster and/or slower depositions of the particle, depending on the permeability. To do so, let us consider the sole effect of the presence of a freely suspended particle in the flow around the particle, which was investigated already by Hwang et al. [12]. When a particle is subjected to simple shear flow with the shear rate c_ , the angular velocity of the particle appears 0:5c_ [12]. Then the velocity field inside the particle can be simply decomposed into up = ussf + u0 , where up ¼ ð0:5c_ y; 0:5c_ xÞ is the rigid-body motion of the particle, ussf ¼ ðc_ y; 0Þ is the given shear flow and u0 ¼ ð0:5c_ y; 0:5c_ xÞ is a perturbation due to the presence of the freely suspended particle. The perturbation is of hyperbolic nature: stretching in the direction 135° and 45° from the flow direction and contraction in the remaining orthogonal direction (see Fig. 8). This hyperbolic flow, solely generated by the presence of the particle, is not restricted to simple shear flow but is present in many general flow fields [12]. In Fig. 9, we present local velocity fields around the particle near the porous surface: the unperturbed velocity field u0 and the perturbed velocity contribution u0 as well as the total velocity field u, which satisfies u = u0 + u0 . Four different values of permeability of the porous media are tested: (A) Kp = 1.1718 104 [mm2], (B) 1.1718 105 [mm2], (C) 1.1718 106 [mm2] and (D) 1.1718 107 [mm2]. The velocity contribution u0 is the sole effect of the particle presence and u0 is the velocity field without particles. Let us investigate the two limiting cases. In the highly permeable surface case (Fig. 9A), there appears a hyperbolic velocity field – stretching in two directions and contraction in other orthogonal directions – around the particle in the perturbed velocity contribution because of the highly permeable surface, although there is a wall. On the other hand, formation of such a hyperbolic flow field due to the particle cannot be observed in case of the lowest permeable surface (Fig. 9D) and the velocity near the porous wall almost vanishes, as the wall behaves like a non-penetrating solid surface. The additional hyperbolic flow generates downward flow in the direction of the particle movement near the gap (at the lower left region of the particle in Fig. 9A–C) and the strength of this flow increases with the permeability of the wall. As this downward flow is not present in the absence of the particle and yields additional downward penetrating flow through the porous media, one may conclude that the downward flow in the gap between the particle and porous wall in the direction of particle movement is responsible for the early deposition of the particle, in comparison with the conventional trajectory prediction. In summary, particle deposition is enhanced in the highly permeable media and there are two mechanisms for this enhancement. One is the existence of the penetrating flow into the porous tow, which obviously facilitates approach of the particle towards the porous surface due to the presence of the penetrating seepage flow. This conclusion can be easily drawn by investigating the unperturbed velocity field (without particle) with the conventional trajectory method. The second mechanism is the formation of the downward flow due to the presence of the particle near the porous media, which yields even faster particle deposition than the prediction from the conventional trajectory method. As the permeability decreases, the relative contribution from the drainage flow – the hydrodynamic repulsive force – between thin gap between particle and porous surface increases significantly and, in the limiting case with very low permeability, particle deposition is seriously delayed, much slower than prediction from conventional trajectory analyses.
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by taking the particle radius of 10, 20 and 30 [lm]. The particle is initially located at (X, Y) = (1.59, 0.565) and the permeability of the porous tow is set Kp = 1.1718 105 [mm2]. From Fig. 10, one can observe that the coupled direct simulation always predicts earlier deposition than the conventional trajectory method, which can be now well understood from the previous investigation, and that a larger particle is deposited faster than a smaller one. The obvious reason for this fast deposition of a large particle is the fact that the large particle touches the porous boundary faster and this attachment is considered as the primary mechanism of deposition in the present study. Though not presented here, we remark that there is no effect of the pressure on the particle path until deposition. Of course, the time for deposition is inversely proportional to the assigned pressure drop for a given initial position: i.e. tdeposit / 1/Dp. This is due to the fact that the velocity in both fluid and porous domains scales linearly with the assigned pressure drop. 4.5. Particle filtration As for the final example, we performed a numerical particle filtration experiment with a large number of particles, in order to check the feasibility of this direct simulation scheme in studying the particle filtration and deposition in LCM. Plotted in Fig. 11 are consecutive particle distributions in a model porous media. The model porous media consists of three elliptic porous tows located in the middle of the computational domain of the size 3 [mm] 2 [mm]. The major and minor axes of elliptic tows are 1.32 [mm] and 0.25 [mm], respectively. As in the previous case, the permeability of the tow is specified as Kp = 1.1718 105 [mm2] and the viscosity of a fluid is 0.1 [Pa s]. The flow is driven by the pressure drop of 103 [Pa] between the upper and lower boundaries. The computational domain is discretized by a 300 200 finite element mesh and the time step is 0.005 [s]. Every 0.2 s, we add five randomly distributed particles in a small box in the upper region, indicated by the box in black in Fig. 11 (initial), and particles are removed when they move into the region below the violet line, and we compute for 2 s and the total number of particles involved in the computation is 500.1 The coefficients Cp and q in the particle–particle interaction modeling (Eq. (9)) are 1 and 0.015 (1.5h), respectively. In this particular case, only 17 particles out of total 500 are observed to be deposited on the porous surface. Most of other particles pass through the channel between porous tows, although they sometimes go by the porous tow in close proximity. Interestingly, most particle deposition occurs in the middle region of the central fiber tow, where fluid velocity is very small, which is consistent with the experimental finding [7,8]. Once a few particles are deposited in a certain region, near the middle of the central tow, the flow though the tow in neighboring regions is significantly accelerated, due to shielding (or blocking) effect of already deposited particles (lowering the local permeability), and this results in further deposition of particles nearby. As a result, deposited particles form a single layer on the top surface of the central porous tow. 5. Conclusions
4.4. Effects of the particle size and the pressure drop
In this work, we present a new numerical tool in simulating the particle deposition in dual-scale porous media, especially tailored for the LCM process. We consider the particle–fluid hydrodynamic interaction, the particle deposition onto the permeable surface, at the individual particle level, and addressed it with the coupled
In Fig. 10, we investigate the effect of particle size on the particle deposition rate, presenting particle trajectories until deposition
1 The corresponding movie file of this numerical filtration experiment is accompanied with the manuscript as a supplementary data.
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dual-scale flow. The particle hydrodynamics are incorporated with a fictitious domain method, the so-called ‘rigid-ring description,’ and the coupled flow in dual scale porous media is treated by the Stokes–Brinkman coupling with the level-set method. Reasonable modeling for particle–particle interactions and for particleporous wall interaction was introduced. The single particle deposition phenomenon was extensively investigated especially for the effects of the permeability. We report that particle deposition is enhanced in the highly permeable media and there are two mechanisms for this enhancement. One is the existence of the penetrating flow into the porous tow and the other is the formation of the downward flow due to the presence of the particle near the porous media. Particle filtration simulation was also performed using a large number of particles to show the feasibility of this scheme in investigating the particle filtration and deposition in LCM. Further parametric study needs to be conducted for the filtration and deposition due to the flow of a particle suspension through a porous media. For example, effects of the size and aspect ratio of fiber tows, their geometrical arrangement and permeability, the particle size, particle volume fraction (or concentration), etc., which is of great significance in manufacturing advanced multifunctional composite materials. Acknowledgements This work was supported by the National Research Foundation (NRF) of Korea funded by the Ministry of Education, Science and Technology (2009-0094015 and 2010-0021614) and by the Army Research Laboratory (Grant Number W911NF-06-2-011). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.compositesa.2011.05.017.
References [1] Chohra M, Advani SG, Yarlagadda S. Filtration of particles through a single layer of dual scale porous media. Adv Compos Lett 2007;16:205–21. [2] Chodra M, Advani SG, Gokce A, Yarlagadda S. Modeling of filtration through multiple layers of dual scale porous media. Polym Compos 2006;27:570–81. [3] Steggall-Murphy C, Simacek P, Advani SG, Yariagadda S, Walsh S. A model for thermoplastic melt impregnation of fiber bundles during consolidation of powder-impregnated continuous fiber composites. Composites Part A 2010;41:93–100. [4] Erdal M, Güçeri SI, Danforth SC. Impregnation molding of particle-filled preceramic polymers: process modeling. J Am Ceram Soc 1999;82:2017–28. [5] Lefevre D, Comas-Cardona S, Binétruy C, Krawczak P. Modelling the flow of particle-filled resin through a fibrous preform in liquid composite molding technologies. Composites Part A 2007;38:2154–63. [6] Lefevre D, Comas-Cardona S, Binétruy C, Krawczak P. Coupling filtration and flow during liquid composite molding: experimental investigation and simulation. Compos Sci Technol 2009;69:2127–34. [7] Nordlund M, Fernberg SP, Lundström TS. Particle deposition mechanisms during processing of advanced composite materials. Composites Part A 2007;38:2182–93. [8] Fernberg SP, Sandlund EJ, Lundström TS. Mechanisms controlling particle distribution in infusion molded composites. J Reinf Plast Compos 2006;25:59–66. [9] Hwang WR, Advani SG. Numerical simulations of Stokes–Brinkman equation for permeability prediction of dual-scale porous media. Phys Fluids 2010;22:113101. [10] Hwang WR, Hulsen MA, Meijer HEH. Direct simulations of particle suspensions in a sliding bi-periodic frame. J Comput Phys 2004;194:742–72. [11] Hwang WR, Hulsen MA, Meijer HEH. Direct simulations of particle suspensions of a viscoelastic fluid in a sliding bi-periodic frame. J Non-Newt Fluid Mech 2004;121:15–33. [12] Hwang WR, Anderson PD, Hulsen MA. Chaotic advection in a cavity flow with rigid particles. Phys Fluids 2005;17:043602. [13] Glowinski R, Pan TW, Hesla TI, Joseph DD. A distributed Lagrangian multipliers/fictitious domain method for particulate flows. Int J Multip Flow 1999;25:755–94. [14] Spielman LA. Particle capture from low-speed laminar flow. Ann Rev Fluid Mech 1977;9:297–319. [15] Eimelech M, Gregory J, Jia X, Williams RA. Particle deposition and aggregation: measurement, modeling and simulation. Butterworth-Heinenmann; 1995. [16] The HSL mathematical software library, UK.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Direct simulations of particle deposition and filtration in dual-scale porous media Wook Ryol Hwang a, Suresh G. Advani b,⇑, Shawn Walsh c a
School of Mechanical Engineering, Research Center for Aircraft Parts Technology (ReCAPT), Gyeongsang National University, Jinju 660-701, Republic of Korea Department of Mechanical Engineering, Center for Composite Materials, University of Delaware, Newark, DE 19716, USA c Army Research Laboratory, Aberdeen Proving Grounds, Aberdeen, MD 21005, USA b
a r t i c l e
i n f o
Article history: Received 3 March 2011 Received in revised form 17 May 2011 Accepted 21 May 2011 Available online 27 May 2011 Keywords: A. Particle-reinforcement B. Rheological properties C. Computational modeling E. Resin transfer molding (RTM)
a b s t r a c t A two dimensional direct numerical simulation technique is developed to describe particulate flows in dual-scale porous media to predict particle deposition on the permeable porous surface in liquid composite molding processes. This individual particle level simulation accounts for hydrodynamic interaction between particles and the fluid, especially near a porous wall (fiber tow), and can predict the deposition of the particles on solid or porous surfaces. A Stokes–Brinkman coupling is employed to describe the flow in dual-scale porous media and a fictitious domain approach is used to deal with freely suspended particles in the fluid stream. A single particle deposition process is investigated extensively along with effects of the permeability of porous media, the particle size and the pressure drop. Mechanisms leading to accelerated or delayed deposition of particles are analyzed by investigating the velocity fields around the particle in close proximity of the porous surface. Finally, particle filtration simulation are performed with a large number of particles to demonstrate the feasibility of this scheme to address particle deposition and filtration during manufacturing of composites using liquid composite molding processes in which the particles are mixed with the resin and the suspension is injected into a stationary dual scale preform. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Particle deposition and filtration within suspensions through porous media is of great significance in several natural and industrial processes: for example, formation of river beds, ground water contamination, deep bed filtration and manufacturing advanced composite materials. Among them, we are particularly interested in liquid composite manufacturing (LCM) processes, such as the resin transfer molding (RTM) or vacuum assisted resin transfer molding (VARTM), where particles (or fillers) are often added to achieve a specific property like fire resistance, abrasion resistance, electromagnetic shielding, or even for reduction in material costs. An interesting example is in manufacturing functionally graded composites (FGC) with the VARTM process in which particles are added to introduce multifunctional properties [1,2]. Depending on the specific application, the goal of infiltration in this case may be to entrap all particles in one layer, to achieve uniform distribution of particles, or to create a particle concentration gradient, usually in the thickness direction. The preforms are usually dual scale fabrics in which yarns or tows (bundles of thousands of micron size fibers) are woven, stitched or knitted together. Understanding particle deposition and filtration phenomena in
⇑ Corresponding author. Tel.: +1 302 831 8975; fax: +1 302 831 3619. E-mail address: [email protected] (S.G. Advani). 1359-835X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2011.05.017
such a heterogeneous dual-scale fibrous porous media will provide guidance to obtain a desired gradient of particle deposition in this application. Another example is a recently proposed manufacturing process for thermoplastic matrix fiber-reinforced composites by introducing thermoplastic powder as an aqueous suspension [3]. This novel process combines the VARTM process to impregnate the preform with the thermoplastic powder suspension together with the compression molding process to consolidate the thermoplastic matrix with the fabrics under pressure and heat. Again physical understanding of particle deposition and filtration within dual-scale fabrics will play a key role in designing proper material and process parameters such as geometrical architectures and the permeability of the fabrics, the particle size and concentration, the pressure or flow rate during impregnation. Few researchers have addressed particle filtration in liquid molding processes, in most cases intended to propose macroscopic constitutive models for the particle concentration and particle retention during flow. For example, Erdal et al. [4] formulated an evolution equation for the particle concentration during particlefilled resin infusion and filtration kinetics, considering resuspension of particles. Using these models for single scale porous media, they performed impregnation flow simulations for complex geometries and reported that irregular geometry of the mold greatly influences the local rate of filtration. Lefevre et al. [5,6] argued that a proper retention function needs to be introduced to account for their experimental observation in typical particle
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concentration profiles. For dual-scale porous media, Chohra et al. [1,2] modeled the filtration phenomena to predict the particle concentration distribution through single- and multi-layer fibrous tow geometries, employing a flow resistance network model with the viscosity variation due to the presence of particles. However, as was indicated in Nordlund et al. [7], such process models based on simple empirical relationship do not describe the phenomena causing particle depositions itself. Indeed micro-scale optical measurements and PIV experiments by the same group showed that particle filtration patterns are highly non-homogeneous and irregular and are closely related to the local flow fields [7,8]. In this regard, in-depth knowledge about a specific mechanism is necessary that leads to particle deposition and collectively to filtration within dual-scale porous media. Hence, it is important to examine individual particle behavior considering hydrodynamic interactions between particles and resin and specific flow characteristics in dual-scale porous media. In this work, we develop a direct solid/liquid meso-scale flow simulation to understand particle deposition and filtration in dual-scale porous media. The term ‘direct’ has been used, since hydrodynamic, inter-particle and particle-porous wall interactions are considered at the individual particle level. In addition, it is a meso-scale simulation because the flow in dual scale porous media is described by the Stokes–Brinkman coupling with the bi-periodic boundary condition that has been verified to successfully capture flow phenomena in such a system, especially near the interface between fluid and porous medium which invokes the particle deposition criteria. We first model the system by a set of governing equations, identifying important forces that govern the flow and particle deposition in this problem. Then we present our numerical method and implementation techniques. The numerical scheme is examined for the single-particle deposition on the porous surface case to understand the interplay between the particle deposition and flow in dual-scale porous media. This scheme is then extended to simulate the filtration and deposition process by introducing many particles in the flow that are randomly generated.
2. Modeling There are two distinct pore scales in dual scale porous media of liquid composite molding: one is the inter-tow space (macro-pore) between fiber bundles and the other is the intra-tow space (micropore) between micron-sized fibers within a tow. For a typical woven fabric, the inter-tow spacing is of the order of several hundreds of microns or more and the intra-tow spacing is of the order of one micron. The particle size of interest in the present study is in between, i.e. of the order of tens of microns, considering applications to manufacturing of functionally graded composites or thermoplastic matrix composites [1–3], as mentioned earlier. In this case, no surface filtration is expected in the macro-pores, but it takes places around the fiber tow boundary, as particles cannot penetrate into the micro-pores. We note that such particles in a fluid are nonBrownian: i.e., random displacement of particles due to the thermal fluctuation of fluid particles can be safely neglected. Fig. 1 describes the system of interest. The computational domain X is composed of the fixed region occupied by the porous material (Xp), the region occupied by the surrounding fluid (Xf) and the region occupied by particles. The interface between the porous and fluid regions is denoted by Cpf and the boundary of the computational domain is denoted by Ci(i = 1, 2, 3, 4). The horizontal (and vertical) periodicity in x (and y) is applied between C1 and C3 (and C2 and C4) to introduce the unit cell flow problem in X. The pressure difference Dp is assigned in the vertical direction. We consider only circular particles in this study and particles
Fig. 1. A schematic description of solid/liquid flow in dual-scale porous media in a bi-periodic unit cell. The subscripts ‘p’ and ‘f’ denote the porous media and the surrounding fluid, respectively. Particles are indicated by filled black circle and the i particle is denoted by Pi(i = 1, , N). The pressure drop in the vertical direction is taken into account.
S are denoted by Pi(t)(i = 1, , N) with P(t) for Ni¼1 Pi ðtÞ and N is the total number of particles. For a particle Pi, Xi = (Xi, Yi), Ri, Ui = (Ui, Vi) and xi = xik are used for the coordinates of the particle center, the particle radius, the translational velocity and the angular velocity respectively; and k is the unit normal vector out of the plane. 2.1. Governing equations for flows in dual-scale porous media Assuming that the inertia of the fluid is often neglected at this scale, the fluid domain can be modeled by the Stokes flow. For the porous media, we use the Brinkman equation as a momentum equation, considering the porous media as a continuum body that can be characterized by the permeability. This meso-scale description should be sufficient for our specific purpose, as the particle penetration through the micro-pore is excluded. Recently, we developed a single momentum equation called equivalent momentum equation, for the Stokes–Brinkman coupling with both the continuous stress and stress jump conditions that is valid everywhere in the domain, including the interface between the two media (for details, see Hwang and Advani [9]). In the present work, we employ this approach to model the coupled flow in dual-scale porous media. The single set of equations for both the regions (except the region occupied by particles, i.e. X n P) with the interface condition can be written as follows:
r u ¼ 0; r r a
gf Kp
in X n P;
ð1Þ
u ¼ 0;
ð2Þ
r ¼ p I þ 2g D;
in X n P;
ð3Þ
where u, r, p, g, Kp, I and D are the velocity, the stress, the pressure, the viscosity, the permeability of the porous medium, the identity tensor and the rate-of-deformation tensor, respectively. The superscript ‘’ defines the domain to which all physical quantities belongs to: For example in Xp, u = up, r = rp, p = pp and g = gp; in Xf, u = uf, r = rf, p = pf and g = gf. The function a is a material index which takes a value of ‘1’ in Xp (porous media) and ‘0’ in Xf (fluid domain). Eq. (2) is a special case of the Stokes– Brinkman coupling as described in Hwang and Advani [9] with the continuous stress condition on the interface Cpf. For the boundaries of the computational domain, we assign the bi-periodic boundary condition such that flow defined in a single unit cell could represent much larger periodic arrays of the cell structures. The periodicity in the horizontal and vertical directions can be expressed by the velocity continuity over the periodic boundary respectively:
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ujC2 ¼ ujC4
and ujC1 ¼ ujC3 :
ð4Þ
In addition to the periodicity, a uniform pressure difference Dp is applied in the vertical direction which is the driving force of the flow. 2.2. Particle domain by the rigid-ring description and hydrodynamic interaction Following the work by Hwang and co-authors [10–12], we consider the circular particle as a rigid ring, filled with the same fluid as in the fluid domain, and the rigid body condition is imposed on the particle boundary only. From the rigid-ring description, we use exactly the same set of equations for the particle domain as those of the fluid domain and it allows the implicit treatment of hydrodynamic interaction between particles and the fluid. Indeed, one can use a single set of governing equations (Eqs. (1)–(3)) for the entire domain Xð¼ Xf [ Xp [ PÞ and the discretization of the particle needs to be done only along its boundary. The rigid-body condition on the particle boundary states that the velocity of a fluid particle attached on the particle boundary is the same as that of the particle:
u ¼ U i þ xi k ðx X i Þ on @Pi ðtÞ:
ð5Þ
In addition, the movement of particle is given by kinematics
dX i ¼ Ui; dt
X i jt¼0 ¼ X i;0 :
ð6Þ
To determine the unknown particle rigid-body motion (Ui, xi), one needs balance conditions for the drag force and the torque on the particle boundary. In the absence of inertia and external forces or torques, the particle is subjected to the following force and torque balances:
F Hi þ F Pi ¼ 0;
Ti ¼
Z
F Hi ¼
Z
r n ds;
ð7Þ
@Pi ðtÞ
ðx X i Þ r n ds ¼ 0;
ð8Þ
@P i ðtÞ
where Ti = Tik and n is a unit normal vector on the particle boundary pointing out of the particles. Eq. (7) describes the balance between the hydrodynamic drag force F Hi and the repulsive particle–particle collision force F Pi , which is an artificial force to avoid (numerical) particle overlap. (The particle-porous wall interaction will be discussed in the next section separately.) We choose the following expression of Glowinski et al. [13] for this short-range particle– particle repulsive force: F Pi
¼
N X j¼1;i–j
( F Pi;j ;
F Pi;j
¼
0;
di;j > Ri þ Rj þ q;
C p ðX i X j ÞðRi þ Rj þ q di;j Þ2 ; di;j 6 Ri þ Rj þ q; ð9Þ
where di,j = |Xi Xj| is the distance between the centers of the ith and jth particle and q is the force range and Cp is a positive ‘stiffness’ parameter.
Fig. 2. Flow near the impermeable solid and permeable porous media interfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
appears a thin gap of fluid layer, from which slow drainage is produced. The hydrodynamic force due to this slow draining opposes the particle approach to the wall significantly, which inversely scales with the distance between the particle center and the wall. This force makes the particle approach velocity normal to the wall to be very small. However, once the gap becomes extremely thin, the attractive van der Waals force increases very rapidly to overcome the repulsive hydrodynamic force facilitating the deposit of the particle on the solid surface [14]. In case of deposition on a permeable surface, one has to consider additional effects from the penetrating seepage flow into the porous surface, although basic principles of the hydrodynamic repulsion and deposition by the van der Waals force are still applicable. As described in Fig. 2, it is expected that the drainage through the thin gap would be significantly weak with the permeable wall, in comparison with the impermeable surface, though not completely ignorable. Indeed, deposition on the permeable surface is partly driven by the hydrodynamic force due to the presence of the penetrating flow into the permeable surface. Therefore, considering another effect of the van der Waals force that exerts in extremely close proximity to the porous wall, it might be a plausible approximation to regard the particle as deposited, when it approaches very close to the surface (or numerically when it actually touches the surface). We employ this deposition scheme, the forced particle deposition, in the present work, since the forced attachment by the van der Waals force may act more effectively on the porous surface due to the penetrating flows. Admittedly, unphysical acceleration of particle deposition might be expected especially for very low permeable (almost non-penetrating) media, as the strong repulsive force from hydrodynamic drainage cannot be accurately accounted for, mainly due to the relatively coarse mesh size and the finite time step with explicit time marching scheme. We remark that the present scheme is significantly different from the conventional trajectory method (for example, see [14,15]) in that in our scheme the hydrodynamic interaction is fully considered, which is important especially as the particle approaches the porous surface. We will compare our method to the trajectory method when presenting the results for single particle deposition.
3. Numerical methods 3.1. Level-set description of dual-scale porous media
2.3. Particle-porous wall interaction: forced particle deposition scheme Resolving the particle-porous wall interaction is a critical issue in this meso-scale deposition and filtration problem. First, let us consider general aspects of particle deposition on the solid impermeable surface. As is well described in a review paper (Spielman [14]), deposition of non-Brownian particles on the solid surface in a low Reynolds number flow is governed by the balance of the (repulsive) hydrodynamic force and the (adhesive) van der Waals force. When a particle in a flow is in close proximity to a solid surface, it must deviate from the undisturbed streamline and there
We employ the level-set finite-element method developed by Hwang and Advani [9] in this work to deal with flows in dual-scale porous media. We summarize their level-set scheme briefly for completeness and details can be found in [9]. The geometry of the porous media in a fluid domain can be conveniently described by the level-set function /(x) (Fig. 3). The zero-level set indicates the interface and it takes a positive value within the fluid domain and a negative value inside the porous domain. The level-set function is a signed distance function such that the magnitude of /(x) is determined by the shortest distance from a point x to the interface.
W.R. Hwang et al. / Composites: Part A 42 (2011) 1344–1352
Fig. 3. The level-set representation of the porous media in a fluid. The interface between two medium Cpf is defined by the zero level set (/ = 0). The level-set function is positive in the fluid domain and negative in the porous media.
Based on the level-set function, an approximate material index function a(x) in Eq. (2) can be expressed as follows:
8 / 6 e > < 1; að/Þ ¼ 12 ð1 /e p1 sinðpe/ÞÞ; j/j < e : > : 0; /Pe
ð10Þ
where e is the half thickness of ‘diffuse’ interface (Fig. 3) and is usually taken 1.5h with h being the mesh size. 3.2. Finite-element formulation and implementation Following the weak form of the rigid-ring description [10], we combine the equivalent momentum equation (Eq. (2)) with the continuity (Eq. (1)) and constitutive equation (Eq. (3)), the biperiodic boundary constraint (Eq. (4)) and the rigid-ring constraint for freely suspended particles (Eq. (5)) to constitute the weak form. For this purpose three Lagrangian multipliers are introduced: kp;i on the particle boundary @Pi(L2(@Pi)) for the rigid-ring constraint, kh on the domain boundary C2(L2(C2)) for the horizontal periodicity and kv on the boundary C1(L2(C1)) for the vertical periodicity. Then the weak form can be written as follows: Find u e H1(X)2, p e L2(X), kh 2 L2 ðC2 Þ, kv 2 L2 ðC1 Þ, U i 2 R2 , xi 2 R and kp;i 2 L2 ð@Pi Þ such that
Z
Z p ðr v ÞdX þ 2g Dðu Þ X X Z Z gf : Dðv ÞdX þ a u v dX þ kh ðv jC2 v jC4 ÞdC X Kp C2 Z N Z X þ kv ðv jC1 v jC3 ÞdC þ kp;i ½v fV i þ ni k C1
i¼1
C1
Z
4.1. Flow in dual-scale porous media
ð11Þ
i¼1
qðr u ÞdX ¼ 0;
ð12Þ
X
Z C2
Z C1
Z
we used a continuous linear interpolation for the Lagrangian multiplier kh and a nodal collocation method for kv . The uniform pressure difference Dp in the vertical direction is assigned as a usual natural boundary condition in Eq. (11) and, along with the horizontal periodicity constraint (Eqs. (11) and (14)), this ensures both the velocity continuity (Eq. (4)) and the pressure difference in the weak form. For the Lagrangian multiplier kp;i for the rigidbody constraint (Eqs. (11) and (15)), the point collocation method has been used and approximately one collocation point in an element appears to yield the most accurate results [10]. As the discretization method in the present study is the same as one of the author’s previous works, many details on discretization methods can be found therein [9–12]. At t = 0, initial particle position Xi is set for (i = 1, , N). Then at every time step we solve a large matrix equation from discretization of the weak form (Eqs. (11)–(15)) for a given particle position X ni using a direct method based on a sparse multi-frontal variant of Gaussian elimination (HSL/MA41) [16]. Using the particle velocity U ni , we update the particle position X nþ1 for the next time step i according to Eq. (6) using the explicit 2nd order Adams–Bashforth method. If the position X nþ1 of a particle Pi is identified as deposi ited, we enforce the rigid body velocity ðU nþ1 ; xnþ1 Þ to be zero i i in solving the discretized weak form for the next time step, along with a slight relocation of the particle attached exactly to the surface, if necessary. In the filtration problem using many particles, a certain amount of particles need to be introduced periodically into the computational domain; some of them will be filtered by the porous media inside and the remaining ones that pass through the porous media without deposition should be removed from the computation. This process requires book-keeping the status of each particle, whether it is deposited, newly introduced or removed. Moreover, newly introduced particles though periodic should be randomly distributed at each addition event. 4. Numerical Results
@Pi
ðx X i Þgds Z N X ¼ Dpðn v ÞdC þ F Pi V;
1347
lh ðu jC2 u jC4 ÞdC ¼ 0;
ð13Þ
lv ðu jC1 u jC3 ÞdC ¼ 0;
ð14Þ
lp;i ½u fU i þ xi k ðx X i Þgds ¼ 0; ði ¼ 1; ; NÞ;
ð15Þ
First we consider flow in dual-scale porous media without particles to show typical flow behavior. A model system is presented in Fig. 4, which consists of two elliptic fiber tows, one at the center and the other at the corner (split into four). The computational domain is 5 [mm] 1 [mm] in size and the major and minor axes of the tow are 2.2 [mm] and 0.25 [mm], respectively, and it is discretized by the uniform regular rectangular finite element mesh. Due to the bi-periodicity, it represents a squeezed hexagonal arrangement of tow in a bounded domain. Assuming a typical radius of fiber filament Rf = 0.023 [mm] with the porosity u = 0.53, the transversal permeability of the tow Kp can be estimated by the prediction of Gebart [17] as Kp = 1.1718 105 [mm2], as was done in [10]. The choice of the pressure difference and the viscosity is important in particle-filled flow, since the hydrodynamic drag force scales with the viscosity, the (relative) velocity and the particle size: i.e., FH = 6pgfRV / R(rp) as the velocity scales with the
@P i
for all admissible functions v e H1(X)2, q e L2(X), lh e L2(C2), lv e L2(C1), V i 2 R2, vi 2 R and lp,i e L2(@Pi). We discretized the computational domain uniformly by regular square elements and employed the bi-quadratic interpolation for the velocity and the linear discontinuous interpolation for the pressure. For the periodic constraints in Eqs. (11), (13), and (14),
Fig. 4. A sketch of a model dual scale porous media with two elliptic fiber tows in a bi-periodic domain along with the uniform regular rectangular finite-element discretization.
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Fig. 6. Particle trajectories in the single particle deposition problem using the initial particle position (X, Y) = (1.59, 0.565) for three different permeability value Kp of the porous media from 1.1718 104 [mm2] to 1.1718 106 [mm2]. Decoupled particle paths (blue curves), from the conventional trajectory approach using the unperturbed (without particle) velocity field, are presented for each case as well. The initial particle position is denoted by a black circle and the upper and lower curves around the channel represent the porous boundary. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Streamlines and the shear rate distributions (in [1/s]) in the model problem for three-different permeability values: (from top to bottom) Kp = 1.1718 104 [mm2], 1.1718 105 [mm2] and 1.1718 106 [mm2]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
pressure gradient. With this in mind, we choose the viscosity of the uncured epoxy resin gf to be 0.1 [Pa s], keeping gp = gf, and the pressure difference of 103 [Pa], which yields (1/gf@p/@x) 104 in average and, as a result, the velocity becomes O(1) [mm/s] in magnitude in the fluid domain and O(0.1) [mm/s] in the porous media. Fig. 5 shows typical streamlines and shear rate distributions for three different permeability values Kp = 1.1718 104 [mm2], 1.1718 105 [mm2] and 1.1718 106 [mm2]. We use a 500 100 mesh, correspondingly the mesh size h = 0.01 [mm], for this computation (and all others computations, except the final filtration example) and the shear rate is definedpas the second invariant ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi of the rate-of-deformation tensor II2D ¼ 2D : D. One can clearly observe a noticeable change in streamlines and in the shear rate distributions, mainly due to the different velocity of the penetrating flow inside the porous tow depending on the tow permeability. The existence of the penetrating intra-tow flow, though small, changes the flow dynamics and affects particle deposition significantly. According to the particle deposition mechanism discussed earlier, one may expect increased particle deposition in the highly permeable media.
Fig. 7. Particle trajectories in the single particle deposition problem using the initial particle position (X, Y) = (2.51, 0.63) for four different permeability value Kp of the porous media from 1.1718 104 [mm2] to 1.1718 107 [mm2]. Decoupled particle paths (blue curves), from the conventional trajectory approach using the unperturbed (without particle) velocity field, are presented for each case as well. The initial particle position is denoted by a black circle and the lower curve represent the upper porous boundary of the central fiber tow in Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4.2. Single particle deposition In this section, we investigate the single particle deposition on the porous media, focusing on coupled flow effects of this phenomenon near the interface. We use the same tow geometry and materials properties as in the previous case (Fig. 5) and additionally we have a single particle of size R = 20 [lm], initially located at (X, Y) = (1.59, 0.565) inside the channel between two porous media. The time step in the simulation is 0.01 [s]. Plotted in Fig. 6 are paths of particles from the initial position until deposition occurs, if any, for three different permeability values of the porous media Kp = 1.1718 104 [mm2],1.1718 105 [mm2] and 1.1718 106 [mm2]. In Fig. 6, we present the prediction using the conventional trajectory method as well, which is denoted as ‘decoupled’ as this particle trajectory is obtained by integrating the unperturbed velocity without particles. (Note that we solved the full computational domain with the porous tow geometry, as shown in Fig. 5, but present only a small region of interest near the particle in Fig. 6.) Observation from Fig. 6 can be summarized as follows:
Fig. 8. A sketch of the velocity decomposition of flow around the particle in simple shear flow. The total velocity can be decomposed into the given simple shear flow ussf and the hyperbolic perturbation flow contribution u0 due to the presence of the particle.
(i) From the coupled direct simulation result, particle deposition is delayed as the permeability decreases and, in case of the lowest permeability (Kp = 1.1718 106 [mm2]), no particle deposition is observed at all. (ii) As the permeability becomes lower, direct coupled flow simulations predict earlier particle deposition than the decoupled trajectory approach. The first observation is important and can be easily expected, since there appears a seepage flow into the porous surface whose magnitude varies inversely with the permeability and this toward-surface flow enhances the particle deposition by overcoming the hydrodynamic resistance due to the drainage flow. However, the second observation is somewhat confusing, because it has been
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Fig. 9. Decomposition of velocity fields around the particle near the porous interface for four different permeability values of the porous surface: (A) Kp = 1.1718 104 [mm2]; (B) 1.1718 105 [mm2]; (C) 1.1718 106 [mm2]; (D) 1.1718 107 [mm2]. The total velocity field u is presented on the first column. The unperturbed velocity field u0 and the perturbed velocity contribution u0 are presented in the second and the last columns. The black circle indicates the particle and the upper and lower (orange) lines indicate the porous surface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
well known that particle deposition to the impermeable solid surface is significantly delayed due to the repulsive hydrodynamic force from the drainage, in comparison with the conventional trajectory approach [14]. Obviously the solid surface is a limiting
case of the low permeable porous surface and this implies that there must be a break-even point such that further decrease in the permeability, may lead to delay in the particle deposition. To check this we performed another set of simulations with the
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Fig. 10. Effects of the particle size on particle deposition. Three different particles (10, 20 and 30 [lm] in radius) are initially located at the same position, denoted by black circles. The permeability of the porous media in this case is Kp = 1.1718 105 [mm2]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
different initial particle position at (X, Y) = (2.51, 0.63) near the middle of the central fiber tow. Particle paths are presented in Fig. 7 for four different permeability values from 1.1718 104 [mm2] to 1.1718 107 [mm2]. As expected, particle deposition is found to be significantly delayed on the lowest permeable surface (Kp = 1.1718 107 [mm2]) in the coupled simulation and in fact no deposition is observed at all, in comparison with the conventional trajectory result. In the two cases with the high permeability Kp = 1.1718 104 [mm2] and 5 2 1.1718 10 [mm ], earlier particle deposition is observed with the coupled method than with the decoupled approach, though it
can be hardly identified by naked eyes, probably because of the impinging flow field in this region. Further, we observe that, in case of Kp = 1.1718 106 [mm2], the particle initially deviates from the unperturbed (decoupled) trajectories away from the porous surface, but then it gradually turns its direction toward the surface, resulting in the final deposition, which looks exactly the same position predicted by the decoupled trajectory analysis. Although the coincidence in final deposition position happens by accident, this case with Kp = 1.1718 106 [mm2] seems to be very close to the break-even point mentioned above. That is, for the permeability lower than this, the coupled direct simulation predicts earlier deposition than the decoupled trajectory approach, while the higher permeability yields slower or delayed deposition in comparison to the trajectory method. The results in Fig. 7 confirm this hypothesis. 4.3. Flow field around the particle near the porous surface Turning of the particle trajectory toward the wall, when it approaches in close proximity of the porous surface with the intermediate permeability, is interesting. In fact it is found to be that in general all particle trajectories from the coupled direct simulation in Figs. 6 and 7, show such a turning trajectory toward the surface, when the particle deposition occurs. In this section, we investigate
Fig. 11. Six consecutive particle distributions in a model dual-scale porous media from a numerical filtration and deposition experiment with randomly inserted 500 particles of the radius 20 [lm] for 2 s. Blue curves indicate the boundary of porous tows and red circles are particles. The black box in the initial configuration is the region, where five initial particles are randomly distributed every 0.1 [s], and particles below the lower violet line are removed from the computational domain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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flow fields around particles near the porous surface, in order to understand the hydrodynamic mechanisms behind faster and/or slower depositions of the particle, depending on the permeability. To do so, let us consider the sole effect of the presence of a freely suspended particle in the flow around the particle, which was investigated already by Hwang et al. [12]. When a particle is subjected to simple shear flow with the shear rate c_ , the angular velocity of the particle appears 0:5c_ [12]. Then the velocity field inside the particle can be simply decomposed into up = ussf + u0 , where up ¼ ð0:5c_ y; 0:5c_ xÞ is the rigid-body motion of the particle, ussf ¼ ðc_ y; 0Þ is the given shear flow and u0 ¼ ð0:5c_ y; 0:5c_ xÞ is a perturbation due to the presence of the freely suspended particle. The perturbation is of hyperbolic nature: stretching in the direction 135° and 45° from the flow direction and contraction in the remaining orthogonal direction (see Fig. 8). This hyperbolic flow, solely generated by the presence of the particle, is not restricted to simple shear flow but is present in many general flow fields [12]. In Fig. 9, we present local velocity fields around the particle near the porous surface: the unperturbed velocity field u0 and the perturbed velocity contribution u0 as well as the total velocity field u, which satisfies u = u0 + u0 . Four different values of permeability of the porous media are tested: (A) Kp = 1.1718 104 [mm2], (B) 1.1718 105 [mm2], (C) 1.1718 106 [mm2] and (D) 1.1718 107 [mm2]. The velocity contribution u0 is the sole effect of the particle presence and u0 is the velocity field without particles. Let us investigate the two limiting cases. In the highly permeable surface case (Fig. 9A), there appears a hyperbolic velocity field – stretching in two directions and contraction in other orthogonal directions – around the particle in the perturbed velocity contribution because of the highly permeable surface, although there is a wall. On the other hand, formation of such a hyperbolic flow field due to the particle cannot be observed in case of the lowest permeable surface (Fig. 9D) and the velocity near the porous wall almost vanishes, as the wall behaves like a non-penetrating solid surface. The additional hyperbolic flow generates downward flow in the direction of the particle movement near the gap (at the lower left region of the particle in Fig. 9A–C) and the strength of this flow increases with the permeability of the wall. As this downward flow is not present in the absence of the particle and yields additional downward penetrating flow through the porous media, one may conclude that the downward flow in the gap between the particle and porous wall in the direction of particle movement is responsible for the early deposition of the particle, in comparison with the conventional trajectory prediction. In summary, particle deposition is enhanced in the highly permeable media and there are two mechanisms for this enhancement. One is the existence of the penetrating flow into the porous tow, which obviously facilitates approach of the particle towards the porous surface due to the presence of the penetrating seepage flow. This conclusion can be easily drawn by investigating the unperturbed velocity field (without particle) with the conventional trajectory method. The second mechanism is the formation of the downward flow due to the presence of the particle near the porous media, which yields even faster particle deposition than the prediction from the conventional trajectory method. As the permeability decreases, the relative contribution from the drainage flow – the hydrodynamic repulsive force – between thin gap between particle and porous surface increases significantly and, in the limiting case with very low permeability, particle deposition is seriously delayed, much slower than prediction from conventional trajectory analyses.
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by taking the particle radius of 10, 20 and 30 [lm]. The particle is initially located at (X, Y) = (1.59, 0.565) and the permeability of the porous tow is set Kp = 1.1718 105 [mm2]. From Fig. 10, one can observe that the coupled direct simulation always predicts earlier deposition than the conventional trajectory method, which can be now well understood from the previous investigation, and that a larger particle is deposited faster than a smaller one. The obvious reason for this fast deposition of a large particle is the fact that the large particle touches the porous boundary faster and this attachment is considered as the primary mechanism of deposition in the present study. Though not presented here, we remark that there is no effect of the pressure on the particle path until deposition. Of course, the time for deposition is inversely proportional to the assigned pressure drop for a given initial position: i.e. tdeposit / 1/Dp. This is due to the fact that the velocity in both fluid and porous domains scales linearly with the assigned pressure drop. 4.5. Particle filtration As for the final example, we performed a numerical particle filtration experiment with a large number of particles, in order to check the feasibility of this direct simulation scheme in studying the particle filtration and deposition in LCM. Plotted in Fig. 11 are consecutive particle distributions in a model porous media. The model porous media consists of three elliptic porous tows located in the middle of the computational domain of the size 3 [mm] 2 [mm]. The major and minor axes of elliptic tows are 1.32 [mm] and 0.25 [mm], respectively. As in the previous case, the permeability of the tow is specified as Kp = 1.1718 105 [mm2] and the viscosity of a fluid is 0.1 [Pa s]. The flow is driven by the pressure drop of 103 [Pa] between the upper and lower boundaries. The computational domain is discretized by a 300 200 finite element mesh and the time step is 0.005 [s]. Every 0.2 s, we add five randomly distributed particles in a small box in the upper region, indicated by the box in black in Fig. 11 (initial), and particles are removed when they move into the region below the violet line, and we compute for 2 s and the total number of particles involved in the computation is 500.1 The coefficients Cp and q in the particle–particle interaction modeling (Eq. (9)) are 1 and 0.015 (1.5h), respectively. In this particular case, only 17 particles out of total 500 are observed to be deposited on the porous surface. Most of other particles pass through the channel between porous tows, although they sometimes go by the porous tow in close proximity. Interestingly, most particle deposition occurs in the middle region of the central fiber tow, where fluid velocity is very small, which is consistent with the experimental finding [7,8]. Once a few particles are deposited in a certain region, near the middle of the central tow, the flow though the tow in neighboring regions is significantly accelerated, due to shielding (or blocking) effect of already deposited particles (lowering the local permeability), and this results in further deposition of particles nearby. As a result, deposited particles form a single layer on the top surface of the central porous tow. 5. Conclusions
4.4. Effects of the particle size and the pressure drop
In this work, we present a new numerical tool in simulating the particle deposition in dual-scale porous media, especially tailored for the LCM process. We consider the particle–fluid hydrodynamic interaction, the particle deposition onto the permeable surface, at the individual particle level, and addressed it with the coupled
In Fig. 10, we investigate the effect of particle size on the particle deposition rate, presenting particle trajectories until deposition
1 The corresponding movie file of this numerical filtration experiment is accompanied with the manuscript as a supplementary data.
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dual-scale flow. The particle hydrodynamics are incorporated with a fictitious domain method, the so-called ‘rigid-ring description,’ and the coupled flow in dual scale porous media is treated by the Stokes–Brinkman coupling with the level-set method. Reasonable modeling for particle–particle interactions and for particleporous wall interaction was introduced. The single particle deposition phenomenon was extensively investigated especially for the effects of the permeability. We report that particle deposition is enhanced in the highly permeable media and there are two mechanisms for this enhancement. One is the existence of the penetrating flow into the porous tow and the other is the formation of the downward flow due to the presence of the particle near the porous media. Particle filtration simulation was also performed using a large number of particles to show the feasibility of this scheme in investigating the particle filtration and deposition in LCM. Further parametric study needs to be conducted for the filtration and deposition due to the flow of a particle suspension through a porous media. For example, effects of the size and aspect ratio of fiber tows, their geometrical arrangement and permeability, the particle size, particle volume fraction (or concentration), etc., which is of great significance in manufacturing advanced multifunctional composite materials. Acknowledgements This work was supported by the National Research Foundation (NRF) of Korea funded by the Ministry of Education, Science and Technology (2009-0094015 and 2010-0021614) and by the Army Research Laboratory (Grant Number W911NF-06-2-011). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.compositesa.2011.05.017.
References [1] Chohra M, Advani SG, Yarlagadda S. Filtration of particles through a single layer of dual scale porous media. Adv Compos Lett 2007;16:205–21. [2] Chodra M, Advani SG, Gokce A, Yarlagadda S. Modeling of filtration through multiple layers of dual scale porous media. Polym Compos 2006;27:570–81. [3] Steggall-Murphy C, Simacek P, Advani SG, Yariagadda S, Walsh S. A model for thermoplastic melt impregnation of fiber bundles during consolidation of powder-impregnated continuous fiber composites. Composites Part A 2010;41:93–100. [4] Erdal M, Güçeri SI, Danforth SC. Impregnation molding of particle-filled preceramic polymers: process modeling. J Am Ceram Soc 1999;82:2017–28. [5] Lefevre D, Comas-Cardona S, Binétruy C, Krawczak P. Modelling the flow of particle-filled resin through a fibrous preform in liquid composite molding technologies. Composites Part A 2007;38:2154–63. [6] Lefevre D, Comas-Cardona S, Binétruy C, Krawczak P. Coupling filtration and flow during liquid composite molding: experimental investigation and simulation. Compos Sci Technol 2009;69:2127–34. [7] Nordlund M, Fernberg SP, Lundström TS. Particle deposition mechanisms during processing of advanced composite materials. Composites Part A 2007;38:2182–93. [8] Fernberg SP, Sandlund EJ, Lundström TS. Mechanisms controlling particle distribution in infusion molded composites. J Reinf Plast Compos 2006;25:59–66. [9] Hwang WR, Advani SG. Numerical simulations of Stokes–Brinkman equation for permeability prediction of dual-scale porous media. Phys Fluids 2010;22:113101. [10] Hwang WR, Hulsen MA, Meijer HEH. Direct simulations of particle suspensions in a sliding bi-periodic frame. J Comput Phys 2004;194:742–72. [11] Hwang WR, Hulsen MA, Meijer HEH. Direct simulations of particle suspensions of a viscoelastic fluid in a sliding bi-periodic frame. J Non-Newt Fluid Mech 2004;121:15–33. [12] Hwang WR, Anderson PD, Hulsen MA. Chaotic advection in a cavity flow with rigid particles. Phys Fluids 2005;17:043602. [13] Glowinski R, Pan TW, Hesla TI, Joseph DD. A distributed Lagrangian multipliers/fictitious domain method for particulate flows. Int J Multip Flow 1999;25:755–94. [14] Spielman LA. Particle capture from low-speed laminar flow. Ann Rev Fluid Mech 1977;9:297–319. [15] Eimelech M, Gregory J, Jia X, Williams RA. Particle deposition and aggregation: measurement, modeling and simulation. Butterworth-Heinenmann; 1995. [16] The HSL mathematical software library, UK.