Multi-scale computational fluid dynamics of impregnation die for thermoplastic carbon fiber prepreg production

Accepted Manuscript Title: Multi-scale computational fluid dynamics of impregnation die for thermoplastic carbon fiber prepreg production Author: Son Ich Ngo Young-Il Lim Moon-Heui Hahn Jaeho Jung Yun-Hyuk Bang PII: DOI: Reference:

S0098-1354(17)30114-X http://dx.doi.org/doi:10.1016/j.compchemeng.2017.03.007 CACE 5757

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

18-7-2016 26-2-2017 8-3-2017

Please cite this article as: Ngo, S. I., Lim, Y.-I., Hahn, M.-H., Jung, J., and Bang, Y.H.,Multi-scale computational fluid dynamics of impregnation die for thermoplastic carbon fiber prepreg production, Computers and Chemical Engineering (2017), http://dx.doi.org/10.1016/j.compchemeng.2017.03.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Multi-scale computational fluid dynamics of impregnation die for thermoplastic carbon

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fiber prepreg production

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Son Ich Ngoa, Young-Il Lima,*, Moon-Heui Hahnb, Jaeho Jungb, and Yun-Hyuk Bangb

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Center of Sustainable Process Engineering (CoSPE), Department of Chemical Engineering,

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Hankyong National University, Gyonggi-do Anseong-si Jungangno 327, 17579 Korea

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Hyosung R&D business Lab., Hyosung Research Institute, Anyang, 14080 Korea

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*

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[email protected]

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Corresponding Author: Tel.: +82 31 670 5207, Fax: +82 31 670 5445, E-mail address:

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Highlights

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- A multi-scale CFD model was developed for a carbon fiber (CF) impregnation process.

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- Resin permeability to CF filaments was obtained from a micro-scale CFD model.

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- Uniformity index (UI) of the resin velocity was obtained from a macro-scale CFD model.

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- Tow speed showing maximum UI was expressed as a linear function of the slip velocity.

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Abstract

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A multi-scale computational fluid dynamics (CFD) model of a pultrusion process was proposed

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for unidirectional carbon fiber (UD-CF) prepreg production. Polyamide 6 (PA6) and

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polyacrylonitrile-based CF were used as the thermoplastic polymer matrix and reinforcement,

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respectively. The non-Newtonian viscosity of PA6 was expressed by Carreau's model. A micro-

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scale CFD model was constructed to obtain a proper resin permeability to CF filaments, while

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the tow domain was treated as sliding porous media in the macro-scale CFD. The resin velocity

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profile showed a similar shape to the relative resin amount experimentally measured in the UD-

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CF prepreg. The uniformity index of the resin velocity (UIv) on the outlet surface was calculated

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for 45 case studies with several tow speeds and resin flow rates. The tow speed showing a

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maximum UIv was remarkably well expressed as a linear function of the slip velocity, which is

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the difference between the tow speed and resin velocity.

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Keywords: Unidirectional carbon fiber prepreg (UD-CF prepreg); Polyamide 6 (PA6);

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Thermoplastic pultrusion; Computational fluid dynamics (CFD); Multi-scale simulation.

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1. Introduction

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High strength fiber-reinforced composites have received much attention for various applications,

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such as aerospace, automotive, sporting goods, industrial equipment, wind turbines, compressed

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gas storage, buildings, and infrastructure (Chang & Lees, 1988; DOE, 2014; Han et al., 2015;

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Mitschang et al., 2003; Soutis, 2005). Particular interest has been focused on the carbon fiber

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(CF) as a reinforcement of engineering polymers due to its high strength and modulus, high

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thermal conductivity, and low thermal expansion (Soutis, 2005; Yokozeki et al., 2007). Carbon

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fiber-reinforced composites (CFRCs) show excellent mechanical properties compared with

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metals or ceramics (Kim et al., 2011). Moreover, CFRCs with a thermoplastic matrix have high

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environmental resistance, high impact strength, good damage tolerance, and recyclability (Chang

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& Lees, 1988; Sakaguchi et al., 2000). Among available thermoplastic polymers, polyamide

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(PA) is considered a good candidate for the thermoplastic composite matrix owing to its low cost

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and easy handling (Botelho et al., 2003).

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The impregnation processes for thermoplastic CFRCs can be divided into melt, powder, and

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solvent impregnations (Gibson & Månson, 1992). The prepreg technique is a method used to

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fabricate CFRC intermediates (Mitschang et al., 2003). Commonly used techniques to make the

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thermoplastic prepreg are film lamination, pultrusion impregnation, and powder impregnation

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(Gibson & Månson, 1992; Miller et al., 1998). In pultrusion, which is a continuous and linear

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manufacturing process, the fiber is consolidated by a matrix resin and pulled through a heated die

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to shape and cure the composite into a final product (Carlone et al., 2006; Miller et al., 1998;

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Ströher et al., 2013). The fiber pulling speed, die temperature, pressure, and resin volume

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fraction are important parameters for obtaining a high quality final product (Ströher et al., 2013).

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Since high viscosity of thermoplastics makes the resin impregnation difficult (Gibson & Månson,

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1992), a uniform spreading of resin is critical for high quality of the thermoplastic CFRC.

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The effects of thermal conductivity and heat capacity were numerically studied for a

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thermosetting composite pultrusion process (Ströher et al., 2013). The influences of pulling

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speed and die temperature were analyzed in the two dimensional (2D) domain. The study

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focused on the heat transfer and curing of thermosetting composites (Ströher et al., 2013). A

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dual-scale porous media model for microscopic and macroscopic resin flows was proposed to

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investigate the degree of impregnation of the thermoplastic matrix composite during the

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pultrusion process (Kim et al., 2001).

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The rise in computational capacity has allowed improved modeling and simulation capabilities

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for CFRC impregnation processes with complex geometry. Computational fluid dynamics (CFD)

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is a good method for modeling the resin and tow flows during impregnation. CFD was applied

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for both Newtonian and non-Newtonian fluids in porous media (Tosco et al., 2013), in which the

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Darcy-Forchheimer law was validated with a non-Newtonian rheological model. In addition, the

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influence of fiber distribution characteristics on the transverse permeability was examined for an

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in-depth understanding of the resin impregnation in a 2D CFD simulation (Bechtold & Ye,

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2003). A 2D dual scale CFD simulation was presented to investigate the effect of inter- and

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intra- tow porosity on only the overall permeability (Tahir et al., 2014). However, those studies

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focused on microscopic transport phenomena in porous systems. A multi-scale CFD model

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including both the full 3D impregnation die and the microscopic creeping flow in a

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representative element volume is necessary to design the impregnation process and optimize its

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operating conditions.

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Multi-scale modeling is an approach that combines models of different scales of a system in

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order to obtain an overall model of desired quality or computation efficiency which is difficult to

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achieve by a single scale model (Zhao et al., 2012). The finding of relation between microscopic

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and macroscopic properties is an essential problem confronted by the computation mechanics

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community (Nguyen et al., 2011) as well as engineering community (Vlachos et al., 2006; Yang

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& Marquardt, 2009).

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Increasing the production speed decreases the impregnation quality. Thus, there is a trade-off

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between processing speed and impregnation quality, referred to as the 'speed-quality dilemma'

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(Marissen et al., 2000). Ruiz et al. (Ruiz et al., 2006) demonstrated the existence of an optimal

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resin velocity that minimizes voids in the reinforcement during resin transfer molding. However,

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few researchers have examined the relation between the tow pulling speed and resin flow rate in

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terms of the impregnation quality.

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This article aims to develop a multi-scale 3D CFD model of a pultrusion impregnation die for the

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production of unidirectional carbon fiber (UD-CF) prepreg with a thermoplastic polymer matrix,

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polyamide 6 (PA6 or Nylon 6). The non-Newtonian viscosity based on the Carreau model with

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parameters obtained from experiment data is used. The macro-scale CFD model includes a

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moving tow domain of porous media, and a continuum resin phase. The homogeneous

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permeability of a porous tow is calculated from the micro-scale CFD model considering steady-

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state fluid flow inside a microscopic tow structure. The permeability computed in the micro-scale

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CFD is passed to the macro-scale CFD where a resin penetration velocity through the porous tow

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domain is obtained with the permeability. The resin penetration velocity to the CF bundle is

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chosen as a convergence criterion for the iteration between the macro-scale and micro-scale CFD

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models. Hydrodynamics of the resin flow in the microscopic and macroscopic scales, such as

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velocity, pressure, and temperature, are obtained from the CFD simulations. The uniformity of

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the resin velocity is investigated by the parametric study of various tow speeds and resin mass

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flow rates to identify their optimum combination.

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2. UD-CF prepreg pultrusion process

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UD-CF prepreg was produced via a pultrusion process, as shown in Fig. 1a. Fifteen CF tows

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were pulled through a heated stationary die where the resin was impregnated into the fiber. One

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thin and long CF tow is composed of 12,000 CF filaments, each of about 7 µm in diameter. The

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tow has a width of 6 mm and a thickness of 0.13 mm, as shown in Fig. 1b. The resin, PA6, was

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injected through a hanger-shaped tube from the bottom of the impregnation die. PA6 (1011BRT,

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Hyosung, Korea) and polyacrylonitrile (PAN)-based CF (H2550 12K, Hyosung, Korea) were

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purchased as the base polymer matrix and reinforcement, respectively.

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Fig. 1. Impregnation die structure for carbon fiber (CF) tow and polyamide 6 (PA6) resin.

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2.1. Material properties

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The material properties of the resin (PA6) and the tow (CF) are listed in Table 1. The density,

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heat capacity, and thermal conductivity of the resin are given at 260 oC. The non-Newtonian

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resin viscosity (η) is modeled by Carreau's formula (Tanner, 2000) as a function of shear rate (γ)

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and temperature (T).

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1

(1)

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where η0, η∞, λ, and n are the zero shear viscosity, infinite shear viscosity, time constant, and

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power-law index, respectively. The temperature dependence term, H(T), is expressed as:

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exp

!

(2)

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where α, T0, and Tα are the activation, zero, and reference temperatures, respectively. The

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viscosity of PA6 was measured at four temperatures (T = 220, 240, 260, and 280 °C) with shear

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rate values ranging from 10 to 1000 s-1. Fig. 2 compares experimental data to the curves fitted by

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Carreau's model. The permeability (K) of the tow is calculated from a micro-scale CFD model

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for a representative volume of tow.

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Table 1. Material properties of resin and tow. Resin (PA6) 3 932a Density (ρ, kg/m ) Heat capacity (cp, J/kg/K) 3,728a Thermal conductivity (k, W/m/K) 0.264a Carreau's model Viscosity (η, kg/m/s) Porosity (ε) 2 Permeability (K, m ) a values at 260 °C.

Tow (CF) 1,800 1,075 0.357 0.408 Micro-scale CFD model

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Experimental data

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Model at 220 oC Model at 240 oC

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Viscosity (η, Pa.s)

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Model at 260 oC

200 150 100 50 0 0

Model at 280 oC

200

400

600

800

136 137 138 139 140 141

Shear rate (γ , 1/s)

480 500

520

540

1000

560

Temperature (T, K)

Fig. 2. Comparison of experimental data and curves fitted by Carreau’s model for resin viscosity.

Table 2 shows the model parameters of the non-Newtonian viscosity. The correlation coefficient between the experimental data and the model values is 0.9961.

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Table 2. Parameter values of Carreau's viscosity model depending on shear rate (γ) and

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temperature (T) for PA6.

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Value 145 0.0058 -3.556 330.9 1.89 493 0 5148

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Parameter Time constant ( , s) Power-law index ()) Zero shear viscosity ( , kg/m/s) Infinite shear viscosity ( , kg/m/s) Reference temperature ( * , K) Zero temperature ( , K) Activation temperature (α, K)

2.2. Mesh structure and boundary conditions

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Two CFD domains are considered in the multi-scale simulation. The micro-scale domain

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represents a microscopic tow structure with randomly-distributed CF filaments having 7 µm in

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diameter, where the permeability is predicted for a given resin velocity. The macro-scale 3D

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domain includes the impregnation die structure (see Fig. 1) with the moving tow.

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2.2.1. Micro-scale structure

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A representative structure of tow is illustrated in Fig. 3. The UD-CF filaments are randomly

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distributed in a volume with Width×Depth×Height of 120 × 130 × 60 µ( (see Fig. 3b), where

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the depth of tow is the same value as the tow thickness (see Fig. 1b). The mesh structure consists

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of about 1.85 million polyhedral cells. The inlet and outlet temperatures of the resin are fixed at

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260 oC. The wall temperature is given at 235 oC. The inlet boundary condition of the micro-scale

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CFD model is set as a mass flow rate which was calculated from the macro-scale CFD model for

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the resin penetration velocity in the y-direction (or gravity direction). The pressure at the outlet is

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fixed at 0.0 Pa.

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Fig. 3. Mesh structure of micro-scale CFD domain.

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2.2.2. Macro-scale structure

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The macro-scale mesh structure of the impregnation die is shown in Fig. 4. A non-conformal

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mesh is constructed at the interface between the resin and tow domains in which mesh node

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locations are not identical, as illustrated in Fig. 4a. Since the tow domain with a rectangular

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shape is relatively thin compared to the fluid domain, the hexahedral mesh structure is used for

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the tow domain while the polyhedral mesh for the resin domain. The non-conformal mesh

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structure reduces the number of cells to the half of that of a standard mesh structure, keeping the

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mesh quality.

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However, the non-conformal mesh structure can lead some accuracy loss because of

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interpolation at the interface. The non-conformal interface calculation is handled using a virtual

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polygon approach, which stores the area and centroid of polygon faces (ANSYS Fluent User's

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guide, ANSYS Inc., 2016). It is expected that the accuracy loss is negligible, because the node

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mismatch is minimized, as seen in Fig. 4a.

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The mesh density analysis was carried out in the macro-scale CFD simulation, as shown in Fig. 5,

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to find the effect of mesh numbers on the accuracy of numerical results. The number of meshes

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used for the mesh density analysis was 1.0, 1.4, 4.0, 5.8, 8.0, and 11.2 million cells. The volume-

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averaged resin velocities with respect to the mesh number in the whole domain, resin domain and

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tow domain are displayed in Fig. 5a, 5b, and 5c, respectively. The area-averaged resin velocity

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on the outlet surface is shown in Fig. 5d. It is observed that the stable CFD solution can be

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achieved with over than 4 million cells. Therefore, a mesh structure of 4.2 million cells is chosen

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to save the computational time, keeping the accuracy of CFD simulation. The resin and tow

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domains have 2.8 million polyhedral cells and 1.4 million hexahedral cells, respectively.

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The quality of the mesh plays a significant role in the accuracy and stability of the numerical

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computation (ANSYS Fluent User's guide, ANSYS Inc., 2016). The minimum orthogonal

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quality of the mesh structure was 0.404, which should be more than 0.01. The maximum

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skewness was 0.596, which should be kept below 0.95 for most mesh structures.

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In this CFD domain, there are two inlets and one outlet (see Fig. 1a). Table 3 presents the

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boundary conditions of the base case, which is named E2-27 because the tow speed (uT) is 2

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m/min, the resin mass flow rate (qR) is about 27 g/min, and the inlet conditions are the same as

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the experimental operating conditions. The tow is inserted at a lower temperature (220 °C) than

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the resin temperature (260 °C).

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Fig. 4. Mesh structure of macro-scale CFD domain.

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-3

-3

2.96

3.205 3.204

3.202

(a) Whole domain

0

2

4 6 8 Number of meshes

10

2.94 2.92 2.9

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(b) Resin domain

0

2

6

4 6 8 Number of meshes

x 10

-3

0.03 0.028

0

2

4 6 8 Number of meshes

10

7.5

(d) Outlet surface

0

2

x 10

4 6 8 Number of meshes

10

12 6

x 10

d te

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Table 3. Boundary conditions of macro-scale CFD simulation (E2-27). Boundary type Resin inlet Tow inlet Resin/tow outlet Wall

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6

x 10

Fig. 5. Mesh density analysis for macro-scale CFD simulation.

200 201

8

7

12 6

198 199

8.5

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0.032

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Resin velocity (m/s)

(c) Tow domain

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Resin velocity (m/s)

x 10

0.034

10

cr

3.203

x 10

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Resin velocity (m/s)

Resin velocity (m/s)

3.206

x 10

uT (m/min) 2 -

qR (g/min) 26.65 -

T (°C) 260 220 260 260

P (atm) 1 -

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To determine the optimal combination between the tow speed and resin mass flow rate, 45 case

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studies were performed at nine tow speeds from 2 to 10 m/min with increments of 1, and five

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resin mass flow rates from 16 to 32 g/min with increments of 4. The cases were named according

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to the tow speed and resin mass flow rate (e.g., C2-16). For all 45 cases, the temperature and

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pressure were set to the same values as those of E2-27.

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2.3 Relative resin amount of UD-CF prepreg

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The thermoplastic UD-CF prepreg passes through a grooved area for further resin impregnation

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and spreading. Then, it is compressed by the pressing roller. After cooling to room temperature,

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the UD-tape is wound into a roll.

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To measure the uniformity of the resin impregnation, ten segments of the UD-tape were cut at a

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width of 10 mm and length of 45 mm. Each segment was sintered at 400 °C for 14 hours in a

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muffle furnace (AJ-MB8, Ajeon Heating Industrial Co., Ltd., Korea). The resin weight, which is

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the mass difference before and after sintering, was measured. The relative resin amount was

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obtained from the ratio of the segment resin weight to the average resin weight. Fig. 6 shows the

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10 segments and the relative resin amount along the segment number. A rather uniform resin

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distribution is observed in the center with a lower resin amount at the two edges.

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Fig. 6. Experimental measurement of relative resin amount for UD-CF prepreg. 13 Page 13 of 35

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3. Multi-scale CFD modeling

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Both microscopic and macroscopic fluids are assumed as an incompressible continuum phase at

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steady-state. The Carreau model (see Table 2) is applied for the non-Newtonian viscosity of resin.

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3.1. Macro-scale CFD model

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There are two domains in the continuum CFD model: the tow and resin domains. The former is

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assumed as a porous medium moving at a constant speed, uT, which is called the tow speed. It is

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supposed that the porous tow domain is filled 40.8% with resin (ε = 0.408, see Table 1). The

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microscopic creeping-flow inside the porous medium (Whitaker, 1986) is neglected in the

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macro-scale CFD. However, the resistance force between the tow and resin domains is

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considered using Darcy's law (Phelan et al., 1994). The ratio of the void area inside the tow to

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the cross-sectional area of the exit is only 0.047, which means that 4.7% of the inlet resin can

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impregnate (or penetrate) into the CF tow. The continuity, momentum, and energy equations are

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applied to the continuous resin phase of both the tow and resin domains.

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3.1.1. Continuity equation

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Considering a moving coordinate system of the tow domain with a linear velocity, + ,,,,-, the fluid

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phase inside the tow can be transformed from a stationary frame to a moving frame:

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Tow domain: . ∙ 0 + ,-

+ ,,,,-

0

(3)

where ρ is the resin density. The continuity equation of the resin domain is expressed as: Resin domain: . ∙ 0+ ,-

0

(4)

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The resin flow is assumed to be incompressible (i.e., constant ρ) and non-reactive at steady-state.

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Eqs. (3) and (4) represent the overall mass conservation of the resin in the tow and resin domains,

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respectively.

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3.1.2. Momentum conservation equation

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The momentum equation of the fluid at steady-state in the two domains leads to the following

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equations. The resin flow inside the sliding tow domain is expressed by a porous media model

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provided by ANSYS Fluent (ANSYS Inc., 2016). + ,,,,- + ,-

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Resin domain: ∇ ∙ 0+ ,-+ ,-

∇4

+ ,,,,- 3

∇4

∇ ∙ 5̿

an

Tow domain: ∇ ∙ 20 + ,-

∇ ∙ 5̿

07-

8-

07-

(5) (6)

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where 4 is the static pressure, 07- is the gravitational force, and 5̿ is the stress tensor. 8- in the tow

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domain includes the forces of resistance and inertia (Wang et al., 2015): 9: ;

+ ,-

+ ,,,,-

<

0|+ ,-

+ ,,,,-| + ,-

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8-

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d

255

+ ,,,,-

(7)

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where > is the permeability in each direction, < is the inertial resistance factor, and |+ ,-

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the magnitude of the relative velocity. The first term on the right-hand side in Eq. (7) refers to

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the viscous loss defined by Darcy's law (Phelan et al., 1994). The second term is the inertial loss.

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The viscous resistance coefficient attributed to the porous tow is known as the inverse of the

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permeability (1/K). In this study, the inertial resistance factor was neglected for the thin tow

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domain (C2 = 0). This assumption is acceptable for liquid flow through the permeable region,

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because the flow rate is very slow (Dake, 1983; Higdon & Kojima, 1981).

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The left-hand side of Eq. (5) is the convection term of momentum. The stress tensor (5̿) is given

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by

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+ ,,,,-| is

5̿

∇+ ,-

∇+ ,-

?

∇∙+ ,-@ !̿

(8) 15 Page 15 of 35

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where @ ̿ is the identity tensor and ∇+ ,-

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non-Newtonian viscosity ( ) expressed by Carreau’s model (Tanner, 2000) is previously

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described in Eqs. (1) and (2).

∇+ ,-

is the strain rate tensor for the resin flow. The

ip t

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3.1.3. Energy conservation equation

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The energy transport equation is introduced into the resin phase of the two domains, which

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includes the heats of convection and conduction, mass diffusion, and viscous dissipation. Under

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the assumption of thermal equilibrium inside the porous tow, the steady-state energy

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conservation equations are given as (ANSYS Fluent User's Guide, ANSYS Inc., 2016):

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Tow domain: 0AB + ,-

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Resin domain: 0AB + ,- ∙ ∇

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where TT and T are the resin temperature in the tow and resin domains, respectively, cp is the heat

280

capacity of the resin, and kf is the thermal conductivity of the resin (see Table 1). The effective

281

thermal conductivity (kT,eff) in the tow domain is computed as follows:

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us

+ ,,,,- ∙ ∇C

an

+ ,-

∇∙ D

,FGG ∇

M

+ ,,,,- ∙ ∇

∇ ∙ DG ∇

5̿ ∙ + ,-

25̿ ∙ + ,-

+ ,,,,- 3

(9) (10)

te

d

+ ,- ∙ .C

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cr

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D

,FGG

HDG

1

H D

(11)

where kT is the thermal conductivity of the tow.

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3.2. Micro-scale CFD model

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The governing equations of the micro-scale CFD model are the same as those of the resin

287

domain in the macro-scale CFD model, that is, Eqs. (4), (6), and (10). It is noted that the micro-

288

scale CFD model was solved by the density-based solver, while the macro-scale CFD model by

289

the pressure-based solver. As the density-base solver calculates continuity, momentum, energy,

290

and species equations simultaneously, and then turbulence and other scalar equations are solved 16 Page 16 of 35

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sequentially (ANSYS Fluent Theory guide, ANSYS Inc., 2016), this solver is adequate for

292

strongly-coupled nonlinear conservation equations. The micro-scale CFD model was easily

293

converged with the density-based solver in our observation.

ip t

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3.3. Coupling strategy of multi-scale CFD model

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According to Darcy’s law, the permeability of a given porous medium is calculated by the

297

following relation (Choi et al., 1998): I

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cr

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;JΔK

ηL

(12)

where I (? /N is the total discharge of flow through a given volume, > (

300

permeability of porous media, O (

301

and Q ( is the length over the pressure drop. Let Q/A the average penetration velocity of resin

302

to the porous media (up, m/s). The permeability (K) is simplified:

is the

>

L

+B η ΔK

d

M

is the cross sectional area, ΔC CP is the pressure drop,

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(13)

As seen in Eq. (7), the permeability is required to solve the macro-scale CFD model. When up is

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given and ∆P is calculated in the micro-scale model, K is obtained from Eq. (13). In the macro-

306

scale CFD model, the penetration velocity (up) can be defined as a volume-averaged velocity in

307

the y-direction through the porous media.

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Fig. 7 illustrates the coupling strategy of the present multi-scale approach. The calculation

309

algorithm is carried out by the following procedure:

Ac ce p

304

310

(a) Initially, K is chosen as a common value from the literature.

311

(b) The macro-scale CFD simulation with a given K(i) is performed.

312

(c) up(i) is calculated as a volume-averaged velocity magnitude in the y-direction through the

313

tow domain. 17 Page 17 of 35

315 316 317

(d) The relative error (%) between the old and new values of RB is calculated. If it is greater than 5%, continue the loop. If not, stop and use K(i) for the final K. (e) The micro-scale CFD simulation with an inlet velocity given by up(i) is performed and ∆P is obtained.

ip t

314

(f) A new permeability, K(i+1), is calculated from Eq. (13).

319

(g) Return to the macro-scale CFD model and a new penetration velocity, up(i+1), is calculated with K(i+1).

us

320

cr

318

It is noted that the micro-scale CFD model is solved just once at each iteration, assuming that

322

the micro-scale tow structure is homogeneous for the entire tow domain.

an

321

M

323

Macro-scale CFD

i=i+1

Ac ce p

η(T,γ), K(i)

te

d

Initialization i=1 (i) K = Kinit

u (pi )

= uy

(i)

tow

Error(%) = 100

K ( i +1 ) = u (pi )η

u (pi +1 ) − u (pi ) u (pi )

< 5%

No

L ∆P

Micro-scale CFD uinlet = up(i) η(T,γ)

Yes

Break

324 325

Fig. 7. Iterative and sequential procedure between macro-scale and micro-scale CFD simulations.

326

18 Page 18 of 35

4. Results and discussion

328

The continuum macro- and micro-scale CFD models were solved using a finite volume-based

329

solver, ANSYS Fluent R15.0 (ANSYS Inc., USA). The least squares cell-based gradient

330

evaluation was used for both the micro- and macro-scales CFD. In the macro-scale CFD model,

331

the first-order spatial derivative of pressure (∇P) was discretized by the second-order upwind

332

scheme, while those of velocity (∇ ∙ + ,-), and temperature (∇T) by the first-order upwind scheme.

333

In the micro-scale CFD model, the second-order upwind scheme was used for the discretization

334

of all first-order derivatives. The calculation was performed on a workstation with 24 cores of

335

2.7 GHz CPU and 128 GB RAM.

an

us

cr

ip t

327

M

336

4.1 Permeability from multi-scale CFD simulation

338

To find a proper permeability, the macro-scale CFD simulation was carried out with an initial

339

permeability of 1.08×10-13 m2 that was taken elsewhere (Lam & Kardos, 1991). In the exit of the

340

impregnation die, the velocity magnitude of resin in the y-direction (or gravity direction) is

341

shown in Fig. 8 along the impregnation die width. The volume-averaged velocity is drawn by a

342

horizontal solid line, which is used for the inlet velocity in the micro-scale CFD model.

te

Ac ce p

343

d

337

19 Page 19 of 35

10

-8

ip t

10

-7

-9

cr

10

p

u (m/s)

10

-6

-10

0.04

0.05

0.06

0.07

0.1

0.11

0.12

0.13

0.14

an

344

0.08 0.09 Die width (m)

us

10

Fig. 8. Penetration velocities (up) with respect to die width obtained from macro-scale CFD

346

model.

M

345

347

The relative error gradually decreases with the iteration number, as shown in Fig. 9. It reaches

349

less than 5% after five iterations and the reciprocal of K (1/K) converges to 1.78×1012 1/m2.

350

When the resin penetration velocity changes from 1×10-8 to 5.5×10-8 m/s in the micro-scale CFD

351

simulation, the volume-averaged resin viscosity (η=151.1 Pa⋅s) inside tow is almost constant, as

352

shown in Fig. 10. The reciprocal permeability (1/K) converges with the increase of up, because

353

the pressure drop (∆P) also increases with up (see Eq. (13)). In the multi-scale CFD simulation, a

354

converged permeability (K) of 5.61×10-13 m2 is found, which is in good agreement with that

355

predicted by a coupled-flow model (Choi et al., 1998).

Ac ce p

te

d

348

20 Page 20 of 35

80

6.0E+12

60

Relative error (%)

8.0E+12

ip t

100

4.0E+12

40

cr

1/K (1/m2 )

1.0E+13

20

0.0E+00 0

1

2

3

Iterations (i)

4

0

5

an

356

us

2.0E+12

Fig. 9. Convergence of reciprocal permeability (1/K) with respect to iterations (i) of multi-scale

358

CFD simulation.

M

357

d

151.138

te

2.0E+12

Ac ce p

1/K (1/m2 )

2.5E+12

151.14

151.136

1.5E+12

151.134

1.0E+12

151.132

5.0E+11 1.0E-08

2.0E-08

3.0E-08

4.0E-08

5.0E-08

η (Pa.s)

3.0E+12

151.13 6.0E-08

Resin penetration velocity (up , m/s)

359 360

Fig. 10. Reciprocal permeability (1/K) and viscosity (η) with respect to resin penetration velocity

361

(up).

362

21 Page 21 of 35

Fig. 11 depicts the penetration velocity (up) and pressure (P) contours inside the microscopic tow

364

domain. The inlet penetration velocity and pressure drop are 5.37×10-8 m/s and 1.56×103 Pa,

365

respectively.

366 367 368

Ac ce p

te

d

M

an

us

cr

ip t

363

Fig. 11. Velocity (up) and pressure (P) contours in micro-scale CFD.

369

4.2 Macro-scale CFD results

370

The macro-scale CFD simulation was performed with K=5.61×10-13 m2 to find an optimum

371

combination between the resin flow rate and the tow speed. Fig. 12 shows the velocity, pressure,

372

temperature, and viscosity contours of the center slice obtained at uT = 2 m/min and qR = 26.65

373

g/min (E2-27). It is observed that the hydrodynamics of this system are divided into before-

374

contacting and after-contacting regions of the tow and resin. The fifteen tows move at 0.033 m/s

22 Page 22 of 35

(or 2 m/min). The resin injected from the bottom of the impregnation die contacts the tow in the

376

top side of the structure at a mean resin velocity of 1.6 × 10-3 m/s (see Fig. 12a). It is noted that

377

the high velocity bands of the top side are not the tow velocity, but the resin velocity moving at

378

the tow speed. The exit pressure is set to 1.013 × 105 Pa (1 atm), and the pressure drop of the

379

impregnation die is about 2.61 × 105 Pa (2.6 atm), as shown in Fig. 12b.

380

The tow and resin domains have the same pressure in the top side. The resin enters at 533 K

381

(260 °C) and contacts the tow of 493 K (220 °C) in the top side. The resin temperature of the tow

382

domain decreases by a few degrees of Celsius because of heat transfer (see Fig. 12c). The

383

viscosity of the resin is 155 Pa⋅s in the entrance at 260 °C. The viscosity between the tows is

384

very low because of the high shear rate (see also Fig. 2). The resin viscosity decreases by about

385

120 Pa⋅s due to the increase in shear rate, as depicted in Fig. 12d.

M

an

us

cr

ip t

375

d

386

4.3 Resin velocity

388

The amount of resin impregnation (mimp) to carbon fiber (CF) may be expressed as follows: mimp = ρAT u p t

(14)

Ac ce p

389

te

387

390

where ρ is the resin density, AT is the tow area, up is the resin penetration velocity, and t is the

391

impregnation time. mimp is proportional to up and the others (ρ, AT, and t) are constants. Thus, it is

392

expected that the resin velocity (u) on the outlet surface is linearly related to mimp.

393

Fig. 13 shows the velocities of the resin obtained at the centerline of the outlet surface. These

394

velocities are divided into three groups: the resin velocity at the tow speed (0.033 m/s), resin

395

velocity near the tow (about 0.014 m/s), and resin velocity (about 0.0045 m/s). The area-

396

averaged resin velocity must be 0.0054 m/s according to mass conservation. The resin velocity

23 Page 23 of 35

near the tow is raised by the drag effect between the tow and resin. The velocity contour is

398

amplified near the tow, as shown in Fig. 13b.

Fig. 12. CFD results at uT = 2 m/min and qR = 26.65 g/min.

Ac ce p

400

te

399

d

M

an

us

cr

ip t

397

401

24 Page 24 of 35

402

Fig. 13. Resin velocities in centerline of outlet surface obtained from macro-scale CFD

403

simulation.

404

The reduced resin velocity (ur,i) excluding the tow domain along the centerline on the outlet

406

surface is calculated for 10 segments of the die width. u x ,i ux

cr

ur ,i =

, i = 1, 2, 3, ..., 10

(15)

us

407

ip t

405

where ux,i is the average resin velocity of x-direction for a segment i, and u x is the mean resin

409

velocity of ux over the entire centerline. Since one segment has many computational cells with

410

different velocities, as seen in Fig. 13a, two reduced resin velocities (ur) near the tow and

411

throughout the entire range are considered in Fig. 14. The relative resin amount experimentally

412

measured for the final UD-CF prepreg (see Section 2.3) is also shown in Fig. 14.

413

The reduced resin velocity near the tow is smooth like the experimental data. However, the resin

414

velocity of the two edges is higher than that of the center, which differs from the experimental

415

profile. The reduced resin velocity obtained over the entire range varies considerably because of

416

the low near-wall velocity. Its shape is similar to that of the relative resin amount profile. Since

417

the prepreg is subsequently compressed by the pressing roller, as mentioned earlier, this variation

418

on the exit surface will be smoothed. Fig. 14 indicates that the resin velocity profile along the die

419

width can be useful to investigate the uniformity of the impregnated resin amount of the prepreg.

Ac ce p

te

d

M

an

408

420 421

4.4 Uniformity of resin velocity

422

As previously mentioned, the key issue of UD-CF prepreg production is to maximize the

423

production rate while maintaining a high quality product. The productivity at a given resin mass

25 Page 25 of 35

424

flow rate increases with the increase of the tow speed. A uniform distribution of impregnated

425

resin is required for quality control.

426

ip t cr

1.2

1

us

0.8

Relative resin amount (exp.) Reduced resin velocity near tow Reduced resin velocity

0.6

an

Relative resin amount or reduced resin velocity (ur)

1.4

0.4 0

1

2

3

4

5

6

7

8

9

10

Segment number of UD-CF prepreg

M

427

Fig. 14. Comparison of relative resin amount (experimental) and reduced resin velocities on

429

outlet surface (CFD simulation) at uT = 2 m/min and qR = 26.65 g/min.

d

428

te

430

In order to calculate the uniformity of the impregnated resin quantitatively, a uniformity index

432

(UIv) of the resin velocity on the outlet surface except the tow domain is proposed as follows:

Ac ce p

431

n

(

∑ um ,i − um Ai

433

UI v = 1 − i =1

)

(16)

n

2um ∑ Ai i =1

434

where um,i and Ai are the resin velocity magnitude and the facet area, respectively, of each cell

435

element i. um is the average velocity magnitude through the cross-sectional area having n cell

436

elements:

26 Page 26 of 35

∑ (um ,i Ai ) n

um = i =1

437

(17)

n

∑ Ai

i =1

The area-averaged uniformity index ranges from 0 to 1, and a value of 1 indicates the highest

439

uniformity.

440

Fig. 15 illustrates the UIv for the 45 case studies with respect to the tow speed (uT) and resin mass

441

flow rate (qR). The upper and lower limits of UIv are 0.91 (C4-20) and 0.65 (C2-16), respectively.

an

us

cr

ip t

438

1 0.95 0.9

0.8

M

UIv

0.85

0.75 0.7 0.65

d

0.6 10 8

25

4

20

15

Resin mass flow rate ( qR , g/min)

Ac ce p

2

442

te

Tow speed ( uT , m/min)

35

30

6

443

Fig. 15. Uniformity index of resin velocity (UIv) on outlet surface with respect to tow speed (uT)

444

and resin mass flow rate (qR).

445 446

When the resin mass flow rate is given, a maximum UIv is determined according to the tow

447

inserting speed. It seems that uT increases linearly with the increase of qR. Fig. 16a depicts the

448

tow speed with respect to the resin mass flow rate, which is taken at the maximum UIv. The

449

linear regression line between uT (m/min) and qR (g/min) is obtained with a correlation

450

coefficient of 0.9809 as follows:

27 Page 27 of 35

451

uT = 0.1175qR + 2.42

(18)

r r The momentum equation in the tow domain is expressed as the relative velocity ( u − uT ) as seen

453

r in Eq. (5). Furthermore, the porous resistance force ( F ) in Eq. (7) is proportional to the relative

454

velocity. The drag force between the tow and resin may have a main influence on the uniformity

455

of the resin velocity (UIv) along the die width. The relative velocity is translated to the slip

456

velocity (uT-ux) in the axial direction. The theoretical resin velocity (ux,th) in the longitudinal

457

direction, x, is calculated on the exit surface as follows:

qR ρ( A − (1 − ε) AT )

cr

us

u x ,th =

(19)

an

458

ip t

452

where A and AT are the entire cross-sectional area and the tow area on the exit surface,

460

respectively. Fig. 16b illustrates the tow speed with respect to the slip velocity, which

461

demonstrates a remarkable linear relationship with a correlation coefficient of 0.9998:

d

uT = 1.1049 (uT − u x , th ) − 0.2475

(20)

te

462

M

459

For the pultrusion impregnation die, the uniformity of the resin velocity may be interpreted as the

464

relation between the tow speed and slip velocity. It is noteworthy that in practice, the tow speed

465

and resin flow rate are determined to meet the mass ratio of CF to prepreg. Therefore, the

466

impregnation die should be designed so that the mass ratio coincides with a suitable combination

467

of tow speed and slip velocity.

468

Ac ce p

463

28 Page 28 of 35

6.5 y = 0.1175x + 2.42 R² = 0.9809

6 5.5 5

CFD results Linear regression line

4.5 4 15

20

25

30

y = 1.1049x - 0.2475 R² = 0.9998

6 5.5 5

CFD results Linear regression line

4.5 4 4

35

4.5

5

5.5

6

Slip velocity (uT - u x,th , m/min)

Resin mass flow rate (qR, g/min)

(a) uT versus qR

(b) uT versus (uT - ux,th )

cr

469

ip t

Tow speed (u T , m/min)

Tow speed (u T , m/min)

6.5

Fig. 16. Relation between tow speed (uT) and resin flow rate (qR) or slip velocity (uT - ux) at

471

maximum uniformity index (UIv).

us

470

an

472

5. Conclusion

474

This study presented a multi-scale CFD simulation of a pultrusion die for unidirectional carbon

475

fiber (UD-CF) prepreg production. Polyamide 6 (PA6) and polyacrylonitrile (PAN)-based CF

476

were used as the base polymer matrix and reinforcement, respectively. The resin injected from

477

the bottom was impregnated into fifteen CF tows moving at a constant speed. The non-

478

Newtonian viscosity of PA6 was expressed by Carreau's model with respect to the shear rate and

479

temperature. In the macro-scale CFD, the moving tow was considered as porous media having a

480

constant permeability that was computed in the micro-scale CFD. Both the resin and tow

481

domains were assumed to be continuous and incompressible. A non-conformal mesh strategy

482

was adopted to join different mesh structures of the thin tow and the large resin domains in the

483

macro-scale CFD.

484

A reduced resin velocity, which is the ratio of the resin velocity to its mean value, was obtained

485

from the macro-scale CFD simulation at a tow speed of 2 m/min and a resin mass flow rate of

486

26.67 g/min. The reduced velocity profile showed a similar shape to that of the relative resin

Ac ce p

te

d

M

473

29 Page 29 of 35

amount experimentally measured for ten segments of UD-CF prepreg. The uniformity index of

488

the resin velocity (UIv) on the outlet surface was calculated for 45 case studies of several tow

489

speeds and resin flow rates. It was found that the tow speed showing a maximum UIv had a linear

490

relation with the resin flow rate. The velocity uniformity may be influenced by the drag force

491

having a linear relation with the slip velocity that is the difference in velocities between the tow

492

and the resin. Therefore, the tow speed showing a maximum UIv was remarkably well expressed

493

as a linear function of the slip velocity.

us

cr

ip t

487

494

Acknowledgements

496

This research was supported by Basic Science Research Program through the National Research

497

Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (Grant

498

number: NRF-2016R1A2B4010423).

d

M

an

495

te

499

Nomenclature

501

O

Cross sectional area (m2)

502

<

Inertial resistance factor (-)

503

AB

Heat capacity (J/kg/K)

504

8

Source term forces (N)

505

7-

Gravity vector (=9.81 m/s2)

Ac ce p

500

Temperature dependence term of viscosity (-)

506 507

@̿

Identity tensor (-)

508

D

Thermal conductivity (W/m/K)

509

>

Permeability (m2)

510

Q

Length (m) 30 Page 30 of 35

mimp

resin mass impregnated to carbon fiber (g)

512

)

Number of cells (-)

513

C

Pressure (Pa, atm)

514

S

Mass flow rate (g/min)

515

Q

Volume flow rate (m3/s)

516

t

time (s)

cr

ip t

511

us

Temperature (K or °C)

517

+

Velocity (m/min or m/s)

519

U@

Uniformity index (-)

520

+B

Penetration velocity of resin through tow (m/s)

an

518

522

Greek letters

523

5̿

M

521

Activation temperature (K)

525

Shear rate (1/s)

H

Ac ce p

526

Porosity (-)

527

Viscosity (kg/m/s)

528

Time constant (s)

529 530

0

te

524

d

Stress tensor (Pa)

Density (kg/m3)

531

References

532

Bechtold G, Ye L. Influence of fibre distribution on the transverse flow permeability in fibre

533

bundles. Compos Sci Technol 2003; 63:2069-79.

31 Page 31 of 35

534 535

Botelho EC, Figiel L, Rezende MC, Lauke B. Mechanical behavior of carbon fiber reinforced polyamide composites. Compos Sci Technol 2003; 63:1843-55. Carlone P, Palazzo GS, Pasquino R. Pultrusion manufacturing process development by

537

computational modelling and methods. Math Comput Model 2006; 44:701-9.

ip t

536

Chang IY, Lees JK. Recent development in thermoplastic composites: A review of matrix

539

systems and processing methods. J Thermoplast Compos Mater 1988; 1:277-96.

544 545 546 547 548 549 550 551 552 553 554 555

us

an

1983.

DOE. Fiber reinforced polymer composite manufacturing workshop: Summary report (DOE/EE-

M

543

Dake LP. Fundamentals of reservoir engineering. 1st ed. Amsterdam: Elsevier Science B.V.;

1041). Arlington; U.S. Department of Energy, Advanced manufacturing office; 2014.

d

542

processing. J Non-Newton Fluid Mech 1998; 79:585-98.

Gibson AG, Månson JA. Impregnation technology for thermoplastic matrix composites. Compos Manuf 1992; 3:223-33.

te

541

Choi MA, Lee MH, Chang J, Lee SJ. Permeability modeling of fibrous media in composite

Han SH, Cho EJ, Lee HC, Jeong K, Kim SS. Study on high-speed RTM to reduce the

Ac ce p

540

cr

538

impregnation time of carbon/epoxy composites. Compos Struct 2015; 119:50-8. Higdon JJL, Kojima M. On the calculation of Stokes' flow past porous particles. Int J Multiphase Flow 1981; 7:719-27.

Kim D-H, Lee WI, Friedrich K. A model for a thermoplastic pultrusion process using commingled yarns. Compos Sci Technol 2001; 61:1065-77. Kim SY, Baek SJ, Youn JR. New hybrid method for simultaneous improvement of tensile and impact properties of carbon fiber reinforced composites. Carbon 2011; 49:5329-38.

32 Page 32 of 35

559 560 561 562 563

Marissen R, Th. van der Drift L, Sterk J. Technology for rapid impregnation of fibre bundles with a molten thermoplastic polymer. Compos Sci Technol 2000; 60:2029-34.

ip t

558

fiber beds during processing of composites. Polym Eng Sci 1991; 31:1064-70.

Miller AH, Dodds N, Hale JM, Gibson AG. High speed pultrusion of thermoplastic matrix composites. Compos Part A: Appl Sci Manuf 1998; 29:773-82.

cr

557

Lam RC, Kardos JL. The permeability and compressibility of aligned and cross-plied carbon

Mitschang P, Blinzler M, Wöginger A. Processing technologies for continuous fibre reinforced

us

556

thermoplastics with novel polymer blends. Compos Sci Technol 2003; 63:2099-110. Nguyen VP, Stroeven M, Sluys LJ. Multiscale continuous and discountinuous modeling of

565

heterogeneous materials: A review on recent developments. J Multiscale Model 2011;

566

03:229-70.

569

M

d

568

Phelan FR, Leung Y, Parnas RS. Modeling of microscale flow in unidirectional fibrous porous media. J Thermoplast Compos Mater 1994; 7:208-18.

te

567

an

564

Ruiz E, Achim V, Soukane S, Trochu F, Bréard J. Optimization of injection flow rate to minimize micro/macro-voids formation in resin transfer molded composites. Compos Sci

571

Technol 2006; 66:475-86.

572

Ac ce p

570

Sakaguchi M, Nakai A, Hamada H, Takeda N. The mechanical properties of unidirectional

573

thermoplastic composites manufactured by a micro-braiding technique. Compos Sci Technol

574

2000; 60:717-22.

575 576 577 578

Soutis C. Carbon fiber reinforced plastics in aircraft construction. Mater Sci Eng, A 2005; 412:171-6. Ströher GR, Zaparoli EL, de Andrade CR. Parabolic modeling of the pultrusion process with thermal property variation. Int Commun Heat Mass Transfer 2013; 42:32-7.

33 Page 33 of 35

579 580

Tahir MW, Hallström S, Åkermo M. Effect of dual scale porosity on the overall permeability of fibrous structures. Compos Sci Technol 2014; 103:56-62. Tanner RI. Engineering Rheology. 2nd ed. New York: Oxford University Press; 2000.

582

Tosco T, Marchisio D, Lince F, Sethi R. Extension of the Darcy–Forchheimer law for shear-

ip t

581

thinning fluids and validation via pore-scale flow simulations. Transport Porous Med 2013;

584

96:1-20.

Vlachos DG, Mhadeshwar AB, Kaisare NS. Hierarchical multiscale model-based design of

us

585

cr

583

experiments, catalysts, and reactors for fuel processing. Comput Chem Eng 2006; 30:1712-

587

24.

591 592 593 594

M

Whitaker S. Flow in porous media I: A theoretical derivation of Darcy's law. Transport Porous Med 1986; 1:3-25.

d

590

moving porous media. Int J Heat Mass Transfer 2015; 82:357-68.

te

589

Wang L, Wang L-P, Guo Z, Mi J. Volume-averaged macroscopic equation for fluid flow in

Yang A, Marquardt W. An ontological conceptualization of multiscale models. Comput Chem Eng 2009; 33:822-37.

Ac ce p

588

an

586

Yokozeki T, Iwahori Y, Ishiwata S, Enomoto K. Mechanical properties of CFRP laminates

595

manufactured from unidirectional prepregs using CSCNT-dispersed epoxy. Compos Part A:

596

Appl Sci Manuf 2007; 38:2121-30.

597

Zhao Y, Jiang C, Yang A. Towards computer-aided multiscale modelling: A generic supporting

598

environment for model realization and execution. Comput Chem Eng 2012; 40:45-57.

599

34 Page 34 of 35

600

List of Tables

601

Table 1. Material properties of resin and tow.

602

Table 2. Parameter values of Carreau's viscosity model depending on shear rate (γ) and

604

temperature (T) for PA6. Table 3. Boundary conditions of macro-scale CFD simulation (E2-27).

605

ip t

603

List of figures

607

Fig. 1. Impregnation die structure for carbon fiber (CF) tow and polyamide 6 (PA6) resin.

608

Fig. 2. Comparison of experimental data and curves fitted by Carreau’s model for resin viscosity.

609

Fig. 3. Mesh structure of micro-scale CFD domain.

610

Fig. 4. Mesh structure of macro-scale CFD domain.

611

Fig. 5. Mesh size distribution in resin domain with polyhedral mesh structure.

612

Fig. 6. Experimental measurement of relative resin amount for UD-CF prepreg.

613

Fig. 7. Iteration procedure between macro-scale and micro-scale CFD simulations.

614

Fig. 8. Penetration velocities (up) with respect to die width obtained from macro-scale CFD

617 618 619

us

an

M

d

Fig. 9. Convergence of reciprocal permeability (1/K) with respect to iterations (i) of multi-scale CFD simulation.

te

616

model.

Fig. 10. Reciprocal permeability (1/K) and viscosity (η) with respect to resin penetration velocity

Ac ce p

615

cr

606

(up).

620

Fig. 11. Velocity (up) and pressure (P) contours in micro-scale CFD.

621

Fig. 12. CFD results at uT = 2 m/min and qR = 26.65 g/min.

622

Fig. 13. Resin velocities in centerline of outlet surface obtained from macro-scale CFD

623 624 625 626 627 628 629

simulation.

Fig. 14. Comparison of relative resin amount (experimental) and reduced resin velocities on outlet surface (CFD simulation) at uT = 2 m/min and qR = 26.65 g/min. Fig. 15. Uniformity index of resin velocity (UIv) on outlet surface with respect to tow speed (uT) and resin mass flow rate (qR). Fig. 16. Relation between tow speed (uT) and resin flow rate (qR) or slip velocity (uT - ux) at maximum uniformity index (UIv).

35 Page 35 of 35