Multiscale modeling of unsaturated flow in dual-scale fiber preforms of liquid composite molding I: Isothermal flows

Abstract

Mold-filling simulation of unsaturated flows in LCM is important for optimizing mold design quickly and cost-effectively in the virtual space. For the first time, a true multiscale approach is developed for simulating the unsaturated flow under isothermal conditions in the dual-scale fiber-mats of RTM. To solve the coupled macro–micro equation-set, a coarse global mesh is used to solve the global flow equations over the entire domain while fine local meshes in form of the periodic unit-cells of fabrics are employed to solve the local tow-impregnation process. A multiscale algorithm based on hierarchical computational grids has been proposed to simulate the unsaturated flow in the dual-scale fiber mats under isothermal conditions. The predictions are compared with measurements for a 1-D flow experiment which indicates that the proposed approach can be used to simulate the unsaturated flow accurately through dual-scale fiber mats in LCM without the use of any fitting parameters.

Introduction

Liquid composite molding (LCM) has been recognized as a cost-effective and promising process for manufacturing net-shaped parts from fiber reinforced polymer composites [1]. LCM is a generic set of technologies for manufacturing composite parts that include processes such as Resin Transfer Molding (RTM), structural reaction injection molding (SRIM), Vacuum Assisted Resin Transfer Molding (VARTM), and Seemann Composites Resin Infusion Molding Process (SCRIMP). Although one particular LCM process may be different from the other in terms of their specific details, the LCM processes share many commons aspects which allow us to categorize these processes into a single category of the composite-manufacturing technologies.² In a typical LCM processes, long reinforcing fibers in the form of preforms made from fiber mats are placed in a closed mold. The matrix material (polymeric resin) in liquid form is then either injected under pressure or sucked into the mold due to vacuum to infiltrate the fiber mat. When the mold is filled with resin, the thermosetting polymer matrix is allowed to undergo a cross-linking reaction before the final part is removed from the mold.

The quality of LCM products and the efficiency of the process depend strongly on the wetting of fiber preform during the mold-filling stage of LCM. The mold-filling in the hard-mold LCM processes such as RTM and SRIM is affected by several parameters including the location of resin inlet-gates and air vents, the resin infusion pressure, the applied clamp force, and the temperature of the resin mixture. The traditional trial-and-error methods to optimize the mold and process design can be too time-consuming and economically prohibitive. As a consequence, the numerical mold-filling simulations emerge as one of the most effective ways to optimize the LCM technologies. Successful computer simulations are able to improve the mold design in virtual space without the need for the expensive and time-consuming trial-and-error approach to mold design.

Traditionally, the fibrous preforms are viewed as porous media with uniform pore-size distributions, i.e. all pore diameters fall within a narrow range. Such porous media have also been called the single-scale porous media. Assuming that the pores in a fiber preform behind the moving flow front are fully saturated with resin, the liquid resin flow impregnating the dry fiber preform during the mold-filling stage of LCM can hence be modeled using the Darcy’s law as

$$\mathbf{v}=-\frac{\mathbf{K}}{\mu }·\nabla p$$

where v is the volume-averaged velocity of resin in the fibrous porous medium, p is the pore-averaged resin pressure, K is the permeability tensor for the fiber preform, and μ is the resin viscosity. Since the resin is incompressible, the continuity equation can be expressed as

$$\nabla ·\mathbf{v}=0$$

Inserting Eq. (1) into Eq. (2) leads to an elliptic partial different equation (i.e. the Laplace equation) with the resin pressure as the unknown variable; this equation governs the pressure field in the region wetted by the resin. After introducing appropriate boundary conditions, the pressure is obtained from the Laplace equation which can then be used to estimate resin velocity from the Darcy’s law.

Although the above given approach of using the Darcy’s law for the single-scale porous flow has been successfully applied to model the isothermal/non-isothermal mold-filling of LCM over past two decades [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], researchers found that there are some obvious discrepancies between the numerical predictions and experiments for some types of the fiber mats including the woven and stitched fabrics [13], [14], [15], [16], [17], [18], [19]. A careful examination of the micro-structure of fiber preforms gives clues to the cause of such a discrepancy. The fiber tows (or bundles) are made of thousands of individual micron-size fibers are either woven or stitched to form the fabrics used in LCM. The inter-fiber distance within the intra-tow region is of the order of micrometers, whereas the distance between the fiber bundles, also called gaps and constituting the inter-tow region, is of the order of millimeters. This order of magnitude difference in the pore length-scales in the same medium leads to its classification as a ‘dual-scale’ or ‘dual-porosity’ porous medium. Because of this dual-scale nature of fiber mats, when resin is injected into a mold, it quickly passes through the inter-tow channels (gaps) without impregnating the tows due to high flow resistance from the tows. After the front has passed, resin from the surrounding gap region continues to impregnate the tows gradually. The delayed impregnation of tows leads to the presence of partially-saturated region behind the macroscopic flow front during the mold-filling of LCM process as shown schematically in Fig. 1a. Since the pore size within a tow (the inter-fiber space) is only on the order of micrometers, resin flow in the intra-tow region occurs at the micro-scale, whereas the flow in the inter-tow region occurs at the macro-scale. The coexistence of the macro- and micro-flows due to the dual-scale nature of the preforms has some well-documented influence on mold-filling in LCM such as the formation of voids [13], [20], [21], [22], the creation of a partially-saturated region behind the front [13], [15], [16], [17], [19], and the development of a ‘droop’ in the inlet-pressure history under the constant injection-rate 1-D flow experiments [15], [17], [18], [19], [23], [24], [25]. The presence of the partially-saturated region behind the flow front in the dual-scale porous medium contradicts the key assumption of full saturation behind the flow front which is employed in the conventional model for mold-filling in LCM.

Since the delayed impregnation of tows at the micro-level can be viewed as sinks of liquid in the macroscopic flow field, a general form of the continuity equation has been proposed through various means to include a non-zero sink term [13], [20], [24] as

$$\nabla ·\mathbf{v}=-S$$

where S is the sink term representing the disappearance of resin from the gap into the tows due to their delayed impregnation. This new continuity equation combined with the Darcy’s law, Eq. (1), allows one to model the macroscopic flow in dual-scale porous media.

The sink term plays a key role in modeling the transient mold-filling in LCM with dual-scale preforms. Parnas and Phelan [13] assumed the impregnation of liquid into the tows to be transversely radial and derived a mathematical expression for the sink term in terms of the penetration of micro-fronts into cylindrical tows. Using this model, they succeeded in predicting the experimentally observed phenomenon of jump of the inlet pressure for directional fiber mats at the end of the 1-D mold-filling process. Chan and Morgan [20] have extended the same model to the case of bidirectional mats for integrating the tow-impregnation process into the 1-D macroscopic flow. Pillai and Advani [24], [26] developed sink functions for a variety of simplified gap-tow geometries, and applied the model to a full-scale 2-D mold-filling simulation and successfully replicated the drooping inlet-pressure profile typically witnessed in 1-D flow experiment under constant injection-rates in the dual-scale fiber mats. Wang and Grove [27] determined the function of tow-saturation rate by conducting the transient tow-impregnation simulation in a local 2-D unit cell that represented the architecture of the plain woven fabrics. The entire flow domain was viewed as a combination of macro- and micro-pore networks (a concept similar to the one followed in the fractured porous-media studies). The tow-saturation rate was added into the continuity equations for networks of macro-pores and micro-pores as the sink and source terms, respectively. The two sets of governing equations, coupled through the tow-saturation rate term, were solved alternately using one FE mesh for both porous networks. Instead of explicitly including the sink term in the continuity equation as required by the regular sink-term based formulation, Simacek and Advani [28] associated 1-D bar elements with the nodes of FEM mesh to represent the tow-impregnation (and hence the loss of resin), and thus avoided changing the governing equations.

Although some progress in modeling unsaturated flow in LCM for dual-scale fiber mats has been made through the efforts of researchers in this field, there are several obvious weaknesses in published studies. First, the governing equations (Eqs. (1), (3)) used by many investigators for describing the dual-scale porous media flow in LCM have been proposed without rigorous theoretic derivation, which leads to ambiguous definitions of basic flow quantities due to the presence of dual length-scale pores in the same media.³ The establishment of flow equations using the sound knowledge of the porous media flow physics, as was done by Pillai [29], can clarify these blurred definitions and give insight on how the macro-and micro-flows are coupled, especially for non-isothermal reactive flow in LCM [30]. Second, the sink term of Eq. (3) in previous studies have often been estimated by significantly simplifying the tow-impregnation process (typically modeled as a 1-D micro-flow) in the often complex unit-cell geometry of a woven or stitched fiber mat that merits a full 3-D flow treatment; this definitely has an impact on the accuracy of the unsaturated flow simulation. Although the global flow was modeled without the sink term by using 1-D elements to represent tow-impregnation in [28], flow inside the tows was still much simplified in order to estimate the equivalent parameters of the 1-D elements. Third, there is a lack of proper experimental validations of models proposed so far. The unsaturated flow predictions using the proposed method were indeed compared with the experiment by Kuentzer et al. [18]; however, it was more of a permeability characterization study than a true experimental validation. This is because the permeability of tows was used as a fitting parameter while matching the numerical predictions with the experimental results.⁴

Apart from the aforementioned sink-term based approach, another widely adopted method to model the unsaturated flow in dual-scale fiber mats is to assign a permeability called unsaturated permeability K_unsat to the fiber mats that is different from its saturated permeability K_sat (measured after the mats have been fully wetted and steady-state conditions have been established) [31]. K_unsat is often estimated by fitting the analytical pressure profiles obtained from Eqs. (1), (2) onto the experimental results. The ratio of K_unsat to K_sat falling in a wide range from 4 to ¼ has been reported in the published studies [31]. Except for directly estimating through experimental means, there exists no other reliable method for predicting K_unsat of the dual-scale fiber mats in LCM. What is more important is that the K_unsat approach completely ignores the effect of liquid absorption by fiber tows on the distribution of temperature and cure. (As was pointed out by Jadhav and Pillai [32], [33], trapping of earlier-injected resin leads to rise in temperature and cure inside the tows.) Therefore, the use of K_unsat to capture the effects of unsaturated flow near the resin front is more of an ad hoc, albeit convenient, approach rather than an exhaustive scientific one.

Recently, Tan and Pillai [34] developed a fast mold-filling simulation in the dual-scale porous-media (formed from the woven and stitched fabrics) of LCM under isothermal conditions. The microscopic tow-impregnation process in a 3-D periodic unit cell of dual-scale fabric was first used to tune a sink function with three parameters. The sink function was then incorporated into the global continuity equation. Only a single computational domain representing the porous network formed by the inter-tow gaps of the dual-scale fiber mats was employed in the mold-filling simulation. A good match between numerical simulation and experiment without the use of any fitting parameters demonstrated the accuracy of their approach for simulating isothermal flows in the dual-scale fabrics of LCM. Although the proposed ‘fast’ method was shown to effectively model the isothermal flow in dual-scale fiber mats, it is not possible to extend the method to non-isothermal conditions. The reason is that it is not possible to formulate the unit-cell based sink functions for the mass, energy and cure balance equations during the reactive, non-isothermal flow due to the nonlinear dependence of sink functions on multiple flow quantities including pressure, saturation, temperature, and cure.

In this paper, instead of using one computational domain, we develop a multiscale framework for modeling unsaturated flow, in which a global grid and a local grid are employed to solve the global mold-filling flow and the local tow-impregnating flow, respectively. In this way, one does not need to formulate a lumped equation to characterize the tow impregnation; hence the method can be easily extended to non-isothermal case—in fact, solving reactive, non-isothermal flows remains the main motivation for proposing the multiscale method. According to the two distinct length-scales involved in unsaturated flow through dual-scale fabrics, a two-stage volume-averaging method to derive flow governing equations at two levels was proposed by Pillai [29]. In order to solve the coupled equations acting at two different scales, a global and local FE models were created separately and targeted at the global flow domain and the local tow-impregnation regions, respectively. A coarse global mesh is used to solve the global flow over the entire LCM mold, whereas a fine local mesh in the form of a periodic unit-cell of woven or stitched fabrics is employed to solve the local tow-impregnation so as to compute sink terms that are required for solving the global flow. The unit-cell mesh is associated with each node of global mesh behind the flow front, so there is one global FE model associated with a number of unit-cell FE models in the solution. The finite element/control volume (FE/CV) method is used to solve the transient filling flows on the global as well as the local unit-cell FE meshes. Since the local tow-impregnation conditions vary at different nodes in the global mesh, each time step involves solving for one global-domain (global mesh) and numerous local-domains (unit-cell meshes) affiliated with the wetted FE nodes of the global mesh. A multiscale algorithm based on the hierarchical framework of computational grids has then been proposed to solve the unsaturated flow in dual-scale fiber mats under non-isothermal conditions. Key aspects of the multiscale algorithm including association of unit-cell FE models to global FE model, nonlinearity and coupling of global and local flows, synchronization of advancement of flow fronts in the global- and local-flows have been presented in detail in the present first part of the three-part paper series. A prediction from the proposed multiscale model for a simple 1-D flow has been validated through experiments.