A mathematical model for RTM process simulations in geometries with irregular shapes

Transcription

1 Int. Jnl. of Multihysics Volume 8 Number A mathematical model for RTM rocess simulations in geometries with irregular shaes Brauner Gonçalves Coutinho 1, Vanja Maria França Bezerra 2, Severino Rodrigues Farias Neto 3, Antonio Gilson Barbosa de Lima 4* 1 Center of Human and Exact Sciences, State University of Paraiba (UEPB), Zi Code , Monteiro-PB, Brazil 1 braunergc@yahoo.com 2 Chemical Engineering Deartment, Federal University of Rio Grande do Norte (UFRN), Zi Code , Natal-RN, Brazil 2 vanja@eq.ufrn.br 3 Chemical Engineering Deartment, Federal University of Camina Grande (UFCG), P.O. Box 10069, Zi Code , Camina Grande-PB, Brazil 3 fariasn@deq.ufcg.edu.br 4 Mechanical Engineering Deartment, Federal University of Camina Grande (UFCG), P.O. Box 10069, Zi Code , Camina Grande-PB, Brazil 4* gilson@dem.ufcg.edu.br ABSTRACT Comosites made by using RTM rocesses have attractive roerties to aerosace and automotive industries. However, it is mandatory otimize all rocess arameters. Some mathematical models develoed to redict fluid flow in orous media may be used as a suort tool for laboratory studies. These models resent comlex sets of differential artial equations that may be solved numerically, via discretization methods. This work resents a mathematical model to redict resin and air flow in RTM rocesses. The finite volume method was used to discretize the equations written in boundary fitted coordinates, without the need for any front tracking algorithm. To validate the methodology, numerical results for flow front ositions in rectilinear flows were comared to analytical data. The model was also emloyed to describe the fluid flow in geometries with arbitrary irregular boundaries. Keywords: RTM, Simulation, Finite volume, Boundary Fitted coordinates 1. INTRODUCTION Comosite materials are used by humans since ancient times. Such materials are obtained from the combination of two or more materials and share the characteristics of its constituents. The ossibility of creating a material that meets a market demand is the main advantage of comosites, esecially for automotive, aeronautical and aerosace alications. The resin transfer molding (RTM) is a roduction method characterized by filling the mold that contains the reinforcing material, by means of the injection of a fluid resin. The technique allows the roduction of large quantities of arts, with good quality finishing, at low cost. The otimization rocess requires the understanding of the arameters involved during the filling stage: mold geometry, material roerties and rocess arameters, such as temerature, injection flow rate or injection ressure.

2 286 A mathematical model for RTM rocess simulations in geometries with irregular shaes Comuter simulations are effective tools that can be helful to suort RTM otimization [1, 2]. Some of the studies conducted to simulate fluid flows during the mold filling stage in RTM rocesses use one-dimensional aroaches (linear or radial injection). In most cases, these investigations are alied to the injection of resin inside rectangular geometries. Most of the mathematical models for the resin injection in RTM rocesses consider only one hase, the resin. Such models are quite useful to analyze the macroscoic behavior of the fluid flow and understand some of its imortant roerties. Desite of the use, some limitations of these models can be noticed because air hase within the mold cavity is ignored. Models which are based on two hases (air-resin) take into account the interaction between them, roducing better results [3]. For more comlex roblems, including threedimensional models with multile hases, commercial software are generally referred, but will imly higher costs. Commercial comutational ackages also don t allow access to the source code, making difficult or imossible to imlement deeer changes to the mathematical and numerical modeling by user. A wide variety of ractical alications lies in multihase flow of immiscible fluids in orous media, roviding an overview around the models used, highlighting the alication of certain techniques [15]. Some works resent a review regarding aroaches to model and simulate RTM rocesses [1, 15]. In some works, researchers have used discretization methods that need some secific technique for location of the resin front [4 13]. When using the finite element method (FEM) or finite volume based on elements (MEFVC) some authors usually consider the transient roblem as a series of quasi static roblems, which imlies limitations to the magnitude of the time ste used [5]. The finite volume method (FVM) is used in fluid flow roblems because it ensures the conservation of the hysical quantities [14]. Several researchers have referred the use of Cartesian coordinates to study RTM rocess. Although the use of Cartesian grids is relatively easy, these grids are limited for the modeling of roblems with irregular geometries. In such cases, it is referable to use boundary fitted coordinates. In this sense, resin injection simulations in RTM rocesses using the MVF and boundary fitted coordinates are the goal of the resent work. 2. METHODOLOGY 2.1 MATHEMATICAL MODELING In simulations of RTM rocesses, the imregnation of the reform is usually modeled as a fluid resin flow through a homogeneous orous media in a macroscoic scale [1]. In many cases, the resin injection rocess may be considered isothermal [17]. Here, we use a model for a two-hase (resin air) immiscible flow with no gravitational and caillarity effects in a homogeneous orous media. The mass conservation equation for any hase (air / resin) of the system may be given by: ( φρ m Z ) = ( ρ v ) m (1) t where suerscrit indicates the hase, φ is the orosity of the media, ρ, Z and v are, resectively, density, mass fraction and velocity of hase, and ρ m is the average density of the mixture. In Equation (1), m reresents the mass flow rate er unit of volume, defined as follows: m =ρ q (2) where q reresents the volumetric flow rate of the hase.

3 Int. Jnl. of Multihysics Volume 8 Number The nature of the resin flow in RTM rocesses, allows the use of Darcy s law as the momentum equation [1]. For horizontal multihase systems, the mathematical formulation of the Darcy s law for hase is written as [18]: v = λ K P (3) where K reresents the ermeability tensor, an intrinsic roerty of the orous media, λ is defined as the hase mobility, given by: λ = k rel, (4) μ In Equation (4), μ is the hase viscosity and k rel, is an imortant arameter that will aear when multile hases are resent in the system. It reresents the relative ermeability of the hase, which deends directly on its saturation. In the resent work, it was assumed: k rel, r = S r and k rel, a = 1 k rel, r. Darcy s equation may be combined with mass conservation resulting in the following equation: ( φρ m Z ) = ( λ K P ) m (5) t where ~ λ is mobility rewritten to include hase density, as follows: rel, λ ρ k = (6) μ Primary variables in system of equations reresented by Equation (5) are the mass fractions (Z a, Z r ) and ressures (P a, P r ) of the hases air and resin. Neglecting caillary effects, we have P a = P r = P. The closing equation needed to solve this system ensures that the entire ore volume is comletely filled by hases (Z a + Z r = 1). 2.2 NUMERICAL SOLUTION Some terms in the governing equations introduce nonlinearities, for instance the hase mobility. So, to find an analytical solution for these equations would be quite imossible. In these cases, numerical solution may be a very interesting alternative. Among the different techniques cited, the finite volume method may be used, once it ensures the conservation of the hysical quantities. Boundary-fitted coordinates make mathematical model more flexible to be alied to roblems with irregular geometries Transformation of the domain The use of boundary fitted coordinates requires that the nonrectangular mesh in the hysical sace be transformed to a rectangular hyothetical comutational reresentation for it, as ictured in Figure 1. Figure 1: Schematic diagram of the transformation from hysical to comutational domain

4 288 A mathematical model for RTM rocess simulations in geometries with irregular shaes Transformation of the governing equations Governing equations also must be transformed and solved in the comutational domain. After transformation, equations can be written as follows: ( φρ ) = ξ + ξ η + Z D P D P + η D P D P 1 J t ξ η + m m J (7) where: D 1 λ = ξ x + ξ y J 2 2 ( ) (8) λ D (9) 2 = ( ξη x x + ξη y y) J λ η 2 η 2 D (10) 3 = ( x + y ) J Equations (10) have some terms defined as the metrics of the transformation which bring all the grid information to the transformed equations and relate the hysical domain to its comutational reresentation, given by: ξ x = Jy η, ξ y = Jx η, η x = Jy ξ and η y = Jx ξ [14]. For 2D grids, J reresents the relation between areas in comutational and hysical domains and can be calculated as follows: 1 J = ( xy yx) ξ η ξ η Integration of the governing equations The use of the finite volume method requires the integration of the Equation (7) in sace and time for an elementary volume shown on Figure 2. After integration, equation becomes: ΔV ( φρ ) φρ η γ ( ) = + ξ η Z Z D P D P D P + D P m m o ξ η Δ Δ Δ t + P P J e w (12) D P + D P ξ γ ξ η D P + D P ξ η ΔΔΔ m t J Δ V Δ t n where DV = DξDηDg is the volume dimensions on boundary-fitted coordinates system. s (11) Figure 2: Elementary volume for integration

5 Int. Jnl. of Multihysics Volume 8 Number Some terms with artial derivatives are resent in the integrated equation. These terms may be aroximated using finite differences, as shows the following samles for the west face: Φ (13) ξ = Φ Φ P W Δξ w Φ Φ (14) N + ΦNW ΦS ΦSW η = 4Δη w Phase mobility in Equation (12) must be evaluated in volume faces (w, e, n and s). In the resent work it was used the Uwind Differencing Scheme. Calculation of the hase mobility in face e, as an examle, is given by the hase velocity in that face calculated by the following Darcy s equation, rewritten in boundary-fitted coordinates, as follows: where: = λ Φ Φ Φ + Φ Φ Φ ( E P ) ( N NE S SE ) v G + G e e 1e 2e Δξ 4Δη Di Gi = i = w, e, n, s λ According to Uwind scheme, for v, mobility in face e will be e > 0 λe = λp. If resin flows in the oosite direction, i. e.,, then λ = λ v < 0 e E. In the other faces, a similar idea e may be used. More details about the numerical treatment can be found in [19] Initial and boundary conditions The boundary condition for the elementary volumes adjacent to the boundaries is no flow, i.e., it is assumed that all the boundaries are imermeable everywhere inside domain. Thus, the interaction with the surrounding is given by secifying the source / sink terms q used in the governing equation. This condition will occur only for volumes with resin inlets or air vents. At these oints, it is assumed that the volumetric flow rates of the hases are roortional to their mobility: r a T q (17) λ = q λ = q r a T λ where suerscrit T is the total mobility of hases, resin and air. For the resin inlets, we consider that only resin hase flows. There are two conditions ossible for resin injection: constant flow rate or constant ressure. For constant injection flow rate, we have: q r = q def (18) q a = 0 (19) For injections under constant ressure, the boundary condition at the resin inlets is: P r = P def (20) where subscrit def indicates a redefined value. At the air vents, we can define total flow rate (resin + air) as the boundary condition as follows: q T = q def (21) λ T q = q T (22) λ (15) (16)

6 290 A mathematical model for RTM rocess simulations in geometries with irregular shaes Usually, in RTM rocesses, the condition for the air vent involves the value of the ressure. P = P def (23) 3. RESULTS AND DISCUSSIONS 3.1 VALIDATION WITH ANALYTICAL SOLUTION (REGULAR GEOMETRY) The first case studied here is related to an unidirectional flow with resin injection under constant flow rate. This roblem describes the resin flow through an isotroic fibrous medium [20]. Figure 3 shows mold cavity dimensions and how resin flows inside the mold. Simulation arameters are shown in Table 1. Figure 4 shows the distribution of resin injection oints and air vents as well as the grid with 50x10 control volumes that discretizes the geometry. To simulate correctly the resin flow across the borders, oints were distributed along all the volumes, as illustrated in Figure 4. The total flow rate in each border is divided by the quantity of volumes. Figure 3: Geometry for the unidirectional flow Table 1: Simulation arameters for unidirectional flow Parameter Magnitude Unit Injection flow rate, q inj [m 3 /s] Viscosity, μ 0.1 [Pa.s] Porosity, φ 0.7 Permeability, K [m 2 ] Figure 4: Numerical grid and distribution of oints for the unidirectional flow simulation

7 Int. Jnl. of Multihysics Volume 8 Number Figure 5 shows the similarities between numerical and analytical results for the flow front advance. The analytical solution for the flow front osition as a function of time can be calculated as [8]: qt x f = (24) φhl The curve shows that the comlete cavity filling occurs aroximately at t = 155 s, what can be calculated analytically as follows [8]: Vmoldeφ t = (25) q Figure 6 illustrates the comarison between the numerical and analytical values for the injection ressure as a function of the time. In this case, analytical results were obtained by the following model [8]: P inj qinj μx = AK f (26) Figure 5: Numerical and analytical results of the resin front osition along the time Figure 6: Numerical and analytical results for the injection ressure along the time

8 292 A mathematical model for RTM rocess simulations in geometries with irregular shaes 3.2 APPLICATION (IRREGULAR BORDERS) The geometry with irregular borders used in this case is 2 mm dee, as shown in Figure 7. The values of the arameters used in the simulation are listed in Table 2. To discretize the domain a boundary fitted grid was generated, containing 44 x 35 elementary volumes, illustrated in Figure 8. This figure also shows schematically the distribution of resin inlets and air vents, located at the two oosite sides of the cavity. Here, as in the first case, total flow rate is divided by the number of volumes. Figure 7: Geometry with irregular shae for the multidirectional flow Table 2: Simulation arameters for the irregular geometry case Parameter Magnitude Unit Injection flow rate, q inj [m 3 /s] Viscosity, μ 0.1 [Pa.s] Porosity, φ 0.7 Permeability, K [m 2 ] Figure 8: Comutational grid for the irregular geometry

9 Int. Jnl. of Multihysics Volume 8 Number The evolution of the flow front osition may be seen in Figure 9, for three different times. The flow front shae changes according to the border edge. Afterwards, the flow front reaches the end of the cavity unevenly, hitting the corners faster. The ressure field inside cavity is according to the front osition, as can be seen in Figure 10 for an elased time of 35 s. From the analysis of those figures we can see that the ressure increases from atmosheric level only after resin reaches that area. Note that ressure level is higher at the location where resin is being injected, i. e., along the border, being even higher in the center of that area. For investigate how injection ressure changes with time during the whole rocess, some virtual sensors were distributed as shown in Figure 11. Pressure rofile along time for both sensors is seen in Figure 12. In the beginning of the simulation, ressure levels for both sensors are similar. As more resin is injected in the cavity, ressure in the sensor 2 (center) becomes higher and this difference ersists until the end of the rocess. Figure 11 also shows a longitudinal line used to evaluate ressure levels inside mold cavity. Pressure rofiles along this line may be seen in Figure 13, for three elased times (5, 35 and 85s). These curves show that ressure decreases linearly from the injection oints (x = 0 m), where ressure is maximum, until atmosheric value in regions where there is only air. For the others time values, curve is not comletely straight, indicating that ressure rofile may deend on the local geometry shae. Figure 9: Flow front osition at three times: 5, 35 and 85 s Figure 10: Pressure field inside mold cavity, t = 35 s

10 294 A mathematical model for RTM rocess simulations in geometries with irregular shaes Figure 11: Virtual sensors and longitudinal line used to evaluate ressure along the time Figure 12: Evolution of ressure with rocessing time virtual sensors 1 and 2 Figure 13: Pressure variation along longitudinal line for three elased times: 5, 35 and 85 s

11 Int. Jnl. of Multihysics Volume 8 Number CONCLUSIONS This work resented a mathematical model based on the Darcy s law and its numerical solution to simulate a two-hase flow (air-resin) through an isotroic orous media. The set of governing equations was discretized by the finite volume method using boundary fitted coordinates. The solution for the two unknowns of the roblem, ressure and mass fraction, was obtained imlicitly via Newton s method. Some numerical results for unidirectional flow were validated with analytical model. Methodology was also able to simulate resin flow within irregular shaed geometries, redicting estimation of the flow front shae, osition evolution in time and ressure levels inside the mold cavity during the filling stage. We conclude that the model was able to redict correctly resin flow behavior inside the orous media. The use of boundary fitted coordinates allowed alication of the model to geometries with irregular shaes. ACKNOWLEDGEMENTS The authors thank to CNPq, CAPES and FINEP (Brazilian research agencies) for the financial suort and the authors referenced in the text. REFERENCES [1] Shojaei A.; Ghaffarian S. R.; Karimian S. M. H. Modeling and simulation aroaches in the resin transfer molding rocess: a review. Polymer Comosites, 24 (4): , Aug 2003a. [2] Long A. C. Design and manufacture of textile comosites. CRC Press, 1 ed., [3] Chui W.; Glimm J.; Tangerman F.; Jardine A.; Madsen J.; Donnellan T.; Leek R. Porosity migration in RTM. Numerical methods in thermal roblems., 9: , [4] Phelan F. R.; Frederick R. Simulation of the injection rocess in resin transfer molding. Polymer Comosites, 18(4): , [5] Shojaei A.; Ghaffarian S. R.; Karimian S. M. H. Numerical simulation of threedimensional mold filling in resin transfer molding. Journal of Reinforced Plastics and Comosites, 22(16): , Nov 2003b. [6] Lee S. H.; Yang M.; Song Y. S.; Kim S. Y.; Youn J. R. Three-dimensional flow simulation of resin transfer molding utilizing multilayered fiber reform. Journal of Alied Polymer Science, 114(3): , [7] Shi F.; Dong X. 3D numerical simulation of filling and curing rocesses in nonisothermal RTM rocess cycle. Finite Elements in Analysis and Design, 47(7): , [8] Dai F.; Zhang B.; Du S. Analysis of uer- and lower- limits of fill time in resin transfer mold filling simulation. Journal of Comosite Materials, 38(13): , [9] Chang R.-Y.; Yang W.-H. Numerical simulation of mold filling in injection molding using a three-dimensional finite volume aroach. International Journal for Numerical Methods in Fluids, 37(2): , [10] Yang J.; Jia Y.; Sun S.; Ma D.; Shi T.; An L. Mesoscoic simulation of the imregnating rocess of unidirectional fibrous reform in resin transfer molding. Materials Science and Engineering: A, : , [11] Schmidt F. M.; Lafleur P.; Berthet F.; Devos P. Numerical simulation of resin transfer molding using linear Boundary Element Method. Polymer Comosites, 20 (6): , 1999.

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