Numerical simulation of resin transfer molding using linear boundary element method

Using mass conservation and Darcy's law to describe flow through isotropic porous media leads to a Laplace equation, which may be solved numerically at each time step using the boundary element method. For anisotropic porous media in which the permeability matrix is symmetric, the problem can be solved in the same way by rotating and stretching the coordinates. The numerical model has been compared with analytical solutions and experimental data in the case of radial front flows and with finite element for a frontal injection.     

Darcy's law‐based model for wicking in paper‐like swelling porous media

The wicking of liquid into a paper‐like swelling porous medium made from cellulose and superabsorbent fibers was modeled using Darcy's law. The work is built on a previous study in which the Washburn equation, modified to account for swelling, was used to predict wicking in a composite of cellulose and superabsorbent fibers. In a new wicking model proposed here, Darcy's law for flow in porous media is coupled with the mass conservation equation containing an added sink or source term to account for matrix swelling and liquid absorption. The wicking‐rate predicted by the new model compares well with the previous experimental data, as well as the modified Washburn equation predictions. The effectiveness of various permeability models used with the new wicking model is also investigated. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Interpretation of 3-D permeability measurements for RTM modeling

The resin transfer molding (RTM) technique often utilizes reinforcement with a complex fiber architecture. Several parameters, including the permeability tensor, are necessary to characterize the flow behavior in these intricate fibrous porous media. In this paper, a general procedure for extracting three-dimensional permeability tensors from data is presented. A general procedure is warranted if the permeability tensor lies out of the material plane. An approximate solution to Darcy's law was employed to relate the components of the permeability tensor to experimental measurements. The procedure requires inverting six nonlinear equations with a robust binary search algorithm. The accuracy of the approximate solution to Darcy's law was checked and found to be in close agreement with a nearly exact solution to Darcy's law obtained by finite element methods.     

SINGLE FLUID FLOW IN POROUS MEDIA

A comprehensive study on single fluid flow in porous media is carried out. The volume averaging technique is applied to derive the governing flow equations. Additional terms appear in the averaged governed equations related to porosity ε, tortuosity τ, shear factor F and hydraulic dispersivity D _h. These four parameters are uniquely contained in the volume averaged Navier-Stokes equation and not all of them are independent. The tortuosity can be related to porosity through the Brudgemann equation, for example, for unconsolidated porous media.

The shear factor models are reviewed and some new results are obtained concerning high porosity cases and for turbulent flows. It is known that there are four regions of flow in porous media: pre-Darcy's flow, Darcy's flow, Forchheimer flow and turbulent flow. The transitions between these regions arc smooth. The first region, the pre-Darcy's flow region represents the surface-interactive flows and hence is strongly dependent on the porous media and the flowing fluid. The other flow regions are governed by the flow strength of inertia. For Darcy's flow, the pressure gradient is found to be proportional to the flow rate. The Forchheimer flow, however, is identified by a strong inertia! effects and the pressure gradient is a parabolic function of flow rate. Turbulent flow is unstable and unsteady flow characterized by chaotic flow patterns. The pressure drop is slightly lower than that predicted using the laminar flow equation.

The hydraulic dispersivity is a property of the porous media. It may be considered as the connectivity of the pores in a porous medium. It characterizes the dispersion of mementum, heat and mass transfer. In this paper, only the dispersion of momentum is studied.

Single fluid flow through cylindrical beds of fibrous mats and spherical particles has been used to show how to solve the single fluid flow problems in porous media utilizing the knowledge developed in this communication. Both the pressure drop and axial flow velocity profiles are computed using the developed shear factor and hydraulic dispersion models. Both the predicted velocity profile and pressure drop compare fairly well with the published experimental data.     

Comparing velocities and pressures in redox flow batteries with interdigitated and serpentine channels

Serpentine channels adjacent to a thin, porous medium are a potentially attractive alternative to a conventional thick flow-through electrode for redox flow batteries. The hydrodynamics of serpentine flow fields were investigated with computational fluid dynamics, a two-dimensional model of the porous electrode based on Darcy's law, and a resistance network model at the scale of the active area. Predictions from the three models were used to map the available design space. The optimal electrode thickness, in terms of minimizing nonuniformity, was identified and compared to the result for an interdigitated flow field. Serpentine favors thicker electrodes and higher flows than interdigitated, in qualitative agreement with experimental findings. Furthermore, interdigitated designs deliver more uniform intraelectrode velocities and lower overall pressure drops than serpentine flow fields.     

A model for resin flow during composite processing: Part 1—general mathematical development

A generalized three-dimensional model for resin flow during composite processing has been developed. The model is based on a theory of consolidation and flow through a porous medium, which considers that the total force acting on a porous medium is countered by the sum of the opposing forces, including the force due to the spring-like effect of the fiber network and the hydrostatic force due to the pressure of the liquid within the porous medium. The flow in the laminate is described in terms of Darcy's Law for flow in a porous medium, which requires a knowledge of the fiber network permeability and the viscosity of the flowing fluid. Unlike previous resin flow models, this model properly considers the flows in different directions to be coupled and provides a unified approach in arriving at the solution. Comparison of numerical solutions with the closed form analytical solution shows good agreement. Resin pressure profiles show that the pressure gradients in the vertical and horizontal directions are not linear, unlike the assumption of linearity made in several previous resin flow models. The effects on the resin pressure of both linear and nonlinear stress-strain behavior of the porous fiber network were considered. The nonlinear behavior simulates a rapidly stiffening spring and the resin pressure decreases much more rapidly after a given initial period compared to the linear stress-strain behavior.     

Concurrent methods for permeability measurement in resin transfer molding

The numerical simulation of the resin transfer molding process (RTM) requires knowledge of the physical properties of the fibrous material. In particular, the resistance to the resin flow is measured by the permeability of the preform in the mathematical model of Darcy's law. A concurrent method for low-cost permeability estimation is proposed. The method uses a rectangular mold for the numerical determination of the principal permeabilities. The experimental data include a built-in correlation with Darcy's law and allow an estimation of both experimental and numerical errors. Since the experimental procedure can introduce a significant uncertainty on the estimated permeability, practical considerations are pointed out and some relevant parameters such as the minimum injected length and maximum injection pressure are introduced to increase the reliability of permeability measurements.     

Flow through porous media of a shear-thinning liquid with yield stress

Darcy's law for the laminar flow of Newtonian fluids through porous media has been modified to a more general form which will describe the flow through porous media of fluids whose flow behavior can be characterized by the Herschel-Bulkley model. The model covers the flow of homogeneous fluids with a yield value and a power law flow behavior. Experiments in packed beds of sand were carried out with solutions of paraffin wax in two oils and with a crude oil from the Peace River area of Canada. The model fitted the data well. A sensitivity analysis of the fitting parameters showed that the model fit was very sensitive to errors in the flow behavior index, n , of the Herschel-Bulkley model. A comparison of the “n” values calculated from viscometer measurements and from flow measurements agreed well. A more general Reynolds number for flow through porous media, which includes a fluid yield value, was developed. The data were fitted to a Kozeny-Carman type equation using this Reynolds number. The constant in the Kozeny-Carman equation was determined for the two packed beds studied using Newtonian oils. The data could all be represented, within the experimental error, by the relationship f^* = 150/Re^*. Since the mean volume to surface diameter of the packing was determined by the measurement of its permeability to a Newtonian oil, assuming C' = 150, the new definition of the Reynolds number allows the direct use of the Kozeny-Carman equation with Herschel-Bulkley type fluids.     

Permeability characterization. Part 2: Flow behavior in multiple-layer preforms

Within the resin transfer molding (RTM) process, flow is generally characterized by the progression of a distinct, nonuniform flow front into the preform as a function of time. The flow front progression introduces unsaturated characteristics into RTM flow fields. As a result, the definition of an effective in-plane permeability (K_eff) is used to determine the permeability of actual preforms as they fill with fluid. This K_eff expression expands upon the original definition of Darcy's law by generalizing its applicability to unsaturated creeping flows. Results of experimentally obtained K_eff for flow in single-layer preforms have been detailed for two common RTM materials, a random mat and a 3-D weave, in Part 1. In this paper (Part 2), we characterize the unsaturated and saturated permeabilities of multiple-layer preforms constructed from the random mat and 3-D weave materials characterized in Part 1. This work identifies the apparent permeability characteristics of a specific unsaturated multiple-layer flow that demonstrates behavior inherent to this important class of heterogeneous flows. Also, parallels are drawn between the unsaturated permeability behavior of complex 3-D weave materials and multiple-layer preforms.     

Wicking and evaporation of liquids in porous wicks: A simple analytical approach to optimization of wick design

Wicking and evaporation of volatile liquids in porous, cylindrical wicks is investigated where the goal is to model, using simple analytical expressions, the effects of variation in geometrical parameters of a wick, such as porosity, height and bead‐size, on the wicking and evaporation processes, and find optimum design conditions. An analytical sharp‐front flow model involving the single‐phase Darcy's law is combined with analytical expressions for the capillary suction pressure and wick permeability to yield a novel analytical approach for optimizing wick parameters. First, the optimum bead‐radius and porosity maximizing the wicking flow‐rate are estimated. Later, after combining the wicking model with evaporation from the wick‐top, the allowable ranges of bead‐radius, height and porosity for ensuring full saturation of the wick are calculated. The analytical results are demonstrated using some highly volatile alkanes in a polycarbonate sintered wick. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1930–1940, 2014     

POWER-LAW FLUIDS IN POROUS MEDIA

An analysis is presented to describe the parallel flow of power-law fluids within a channel bounded by porous media. It is shown that there is an excess flow above the Darcy's law prediction for the porous medium region adjacent to the channel/ porous medium boundary. This also leads to a higher flow rate in the channel. The excess flow increases with a decreasing value of the power law index, and with increasing permeability. The excess flow is found to reach a maximum at an intermediate value of the dimensionless channel width (=½H/K^½and it vanishes in the limit of h→∞and h→0. Experimental evidence is also presented to demonstrate the excess flow. The experimental data are found to be in reasonable agreement with the proposed flow model.     

Lattice‐Boltzmann simulation of sago (Metroxylon sagu) starch solutions in porous media

From rheological experiments in gelatinized sago starch solution already reported in the literature and a Lattice‐Boltzmann simulation, we provide some insight into the understanding of the non‐Newtonian fluid dynamics of sago‐starch‐type solutions in porous media. In this paper, permeability and wall shear stress in arbitrarily generated and randomly generated porous media are predicted in the range of the modified Darcy's law. Additional results on flow paths, velocity, shear‐stress tensor, and pressure fields are provided. We prove that our LBE model for sago starch solutions reproduces Blake‐Kozeny and Ergun laws. The model presented in this paper is intended to be used for simulating packed beds.     

Spontaneous imbibition of liquids in glass‐fiber wicks. Part I: Usefulness of a sharp‐front approach

Spontaneous imbibition of a liquid into glass‐fiber wicks is modeled using the single‐phase Darcy's law after assuming a sharp flow‐front marked by full saturation behind the front occurring in a transversely isotropic porous medium. An analytical expression for the height of the wicking flow‐front as a function of time is tested through comprehensive experiments involving using eight different wicks and one oil as the wicking liquid. A good fit with experimental data is obtained without using any fitting parameter. The contact‐angle is observed to be important for the success of the model—lower contact angle cases marked by higher capillary pressures were predicted the best. The proposed model provides a nice upper bound for all the wicks, thereby establishing its potential as a good tool to predict liquid absorption in glass‐fiber wicks. However, the sharp‐front model is unable to explain region of partial saturation, thereby necessitating the development of part II of this article series (Zarandi and Pillai, Spontaneous Imbibition of Liquid in Glass fiber wicks. Part II: Validation of a Diffuse‐Front Model. AIChE J, 64: 306–315, 2018) using Richard's equation. © 2017 American Institute of Chemical Engineers AIChE J, 63: 294–305, 2018     

Measurement of principal permeability with the channel flow experiment

A framework for channel flow permeability measurement in resin transfer molding (RTM) is developed in this paper. For the channel flow experiment, five possible experimental configurations are identified, of which two have not yet been mentioned in the literature. Starting from effective permeability, the entity measured in the experiment, explicit formulas for principal permeability and its orientation in two and three dimensions are developed. These formulas are applied to (published) experimental results that demonstrate their validity. The practical problems that prevent three-dimensional permeability measurement are discussed. Next, a framework for two-dimensional channel flow permeability measurement is proposed. All known channel flow permeability measurement methods are classified according to this new framework, and where appropriate, differences are discussed. An important finding emerging from this comparison is that two different definitions of Darcy's law are used today in permeability measurement.     

Viscoelastic flow in packed beds or porous media

An analysis of viscoelastic flow in packed beds or porous media is presented based on a capillary hybrid model of the flow which incorporates a viscous mode and an elongational mode. The development includes modelling of the elongational mode of the flow to obtain the elongational flow contribution to the potential drop for a viscoelastic fluid. A general expression describing viscoelastic flow in porous media is developed which utilizes the viscous response determined by the fluid model equation and an elongational flow response characterized by an elongational viscosity difference for the fluid. The expression applies to all three traditional bed models employing the tortuosity and Kozeny constant. The relationship yielded extensions of Darcy's law applicable to viscoelastic flow in porous media and an expression representing the flow of a viscoelastic fluid in a packed bed or porous core of length L. The relationship of the friction factors and respective Reynolds numbers is also presented.     

OSCILLATORY FLOW OF FOURTH-ORDER FLUID IN A POROUS HALF SPACE

This work describes the incompressible flow of fourth-order fluid in a porous half space. The flow in the porous space is caused by the porous plate oscillations in its own plane. Modified Darcy's law has been taken into account to discuss the flow characteristics in a porous space. Numerical solution of the governing nonlinear problem is obtained and the effects of various pertinent parameters are discussed.     

Flow near the permeable boundary of an aligned fiber preform: An experimental investigation using laser doppler anemometry

The flow of fluid at the interface of an aligned fiber bed and an open flow is the governing phenomenon in a number of processes of industrial importance. Traditionally, this has been modeled by applying Brinkman's modification of Darcy's law to obtain the velocity profile in planar geometries in terms of an additional parameter called “apparent viscosity.” to test this ad hoc approach, a detailed experimental investigation of the flow was conducted using Laser Doppler Anemometry (LDA) in the close vicinity of the permeable boundary of aligned fiber preforms. The performs used in the experiments consisted of cylindrical fibers aligned in one direction. Two cases were investigated. In the first case, the axis of alignment was in the direction of flow and in the second case the axis of alignment was perpendicular to the flow direction. A Hele-Shaw cell is partially filled with a fibrous preform such that an open channel flow is coupled to the Darcy flow inside the fiber bed through the permeable interface of the bed. The unfilled portion of the Hele-Shaw cell acts as an ideal porous medium of known in-plane permeability, which is much higher than the permeability of the fibrous/porous bed. Modeling this flow situation using a Hele-Shaw cell is appropriate because most composite parts are long and wide in comparison with their thickness. When a viscous fluid is injected at a constant flow rate through the above arrangement, a steady state coupled flow is created. This coupling of the open flow and the Darcy flow through the fibrous bed occurs through the boundary layer zone inside the fibrous bed. Using LDA, steady state velocity profiles are accurately measured in the boundary layer zone by traversing the fibrous bed at a suitable location. For aligned fiber beds, the depth of the boundary layer zone inside the bed was found to be of the order of the mold depth, which is much larger as compared to the Brinkman's prediction. This finding indicates the presence of a length scale that is much larger than the known length scale $ \sqrt K $, where K is the permeability of the bed made up of aligned cylindrical fibers. Based on this finding, the depth of the boundary layer thickness is incorporated in the Brinkman's solution through a boundary condition. This results in a model that compares well with the experimental data for the planar geometry and the fibrous beds considered here.     

Laminar flow past a permeable sphere

Flow past an isolated permeable sphere has been studied. The complete Navier-Stokes equation governs the fluid motion outside the sphere, while Brinkman's extension of Darcy's Law is assumed to hold within the porous sphere. The Navier-Stokes equation is solved using a finite difference scheme. The flow within the porous sphere is solved in two different ways, each being efficient over a particular range of Reynolds number. Drag Coefficients are presented for dimensionless permeability, β, of 5, 10, 15, and 30 and for Reynolds numbers up to 50. The computed drag coefficients are within 10% of the experimental values observed by Masliyah and Polikar for 15 < β > 33, the range covered in their work. Separation was observed only for β > 10. The onset of separation is delayed considerably in porous spheres.     

Darcy's law–based numerical simulation for modeling 3D liquid absorption into porous wicks

Liquid imbibition into polymer wicks, where a clear liquid front can be seen rising during the wicking process, is modeled using the concepts of flow in porous media. The flow of liquid behind the moving liquid front is modeled using the physics of single‐phase flow in a porous medium where the Darcy's law is combined with the continuity equation and a capillary suction pressure is imposed at the liquid front. A novel numerical simulation PORE‐FLOW^© based on the finite element/control volume method is proposed to model such imbibitional flows in wicks of complex shapes. A validation of the simulation is obtained by achieving an excellent comparison of its predictions with an experimental result, an analytical solution, and the Washburn equation for the case of wicking against gravity in a cylindrical wick. The simulation is also used to predict a case of two‐dimensional (2D) wicking in the altered cylindrical wicks with two different cross‐sectional areas. Once again an excellent match is obtained with the experimental results, while analytical solutions for the single and double cross‐section cases along with the Washburn equation fail to predict the 2D wicking. Later, some other types of altered wicks with sharp changes in their cross‐sectional areas were analyzed numerically for their wicking behavior. It was observed that the height of liquid front in a vertical wick as a function of time, which is proportional to the history of liquid imbibed, is strongly dependent on the extent of reduction in the wick cross‐sectional area as well as its location vis‐à‐vis the wick entrance. © 2010 American Institute of Chemical Engineers AIChE J, 2011