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Prediction of degree of impregnation in thermoplastic unidirectional carbon fiber prepreg by multi-scale computational fluid dynamics
Chemical Engineering Science 185 (2018) 64–75
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Prediction of degree of impregnation in thermoplastic unidirectional carbon fiber prepreg by multi-scale computational fluid dynamics Son Ich Ngo a, Young-Il Lim a,⇑, Moon-Heui Hahn b, Jaeho Jung b a Center of Sustainable Process Engineering (CoSPE), Department of Chemical Engineering, Hankyong National University, 327 Jungang-ro, Anseong-si, Gyeonggi-do 17579, Republic of Korea b Hyosung R&D Business Lab., Hyosung Corporation, 74 Simin-daero, Dongan-gu, Anyang-si, Gyeonggi-do 14080, Republic of Korea
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Degree of impregnation (DoI) for
unidirectional carbon fiber prepreg (UD-CFP). Pressure profile of impregnation die obtained using macro-scale 3D Eulerian CFD model. DoI with time and pressure obtained from micro-scale 2D volume of fluid CFD model. Prediction of cumulative DoI for UDCFP impregnation die by multi-scale CFD model.
a r t i c l e
i n f o
Article history: Received 2 February 2017 Received in revised form 6 March 2018 Accepted 5 April 2018 Available online 7 April 2018 Keywords: Unidirectional carbon fiber prepreg (UD-CFP) Thermoplastic resin Resin impregnation die Degree of impregnation (DoI) Computational fluid dynamics (CFD) Multi-scale simulation
a b s t r a c t A multi-scale simulation approach was proposed to predict the degree of impregnation (DoI) in thermoplastic unidirectional carbon fiber prepreg (UD-CFP). The multi-scale approach included a twodimensional (2D) micro-scale computational fluid dynamics (CFD) in a representative elementary volume (REV) of carbon fiber (CF) tow, a 3D macro-scale CFD of an entire impregnation die with 15 sliding CF tows, and a process-scale simulation assembling data from the micro- and macro-scale CFDs. In the macro-scale steady-state CFD, thermoplastic resin injection and CF tow insertion were considered for an impregnation die 10 cm in width. In the micro-scale transient CFD, impregnation mechanisms of resin into CF filaments 7 lm in diameter were identified in terms of surface coverage, capillary permeation, and penetration through CF filaments. The DoI as a function of pressure and time was obtained from the micro-scale CFD within a range of pressures found in the macro-scale CFD. In the process-scale simulation, the cumulative DoI of the 15 tows was predicted along the impregnation die length with the aid
Abbreviations: 2D, two-dimensional; 3D, three-dimensional; CF, carbon fiber; CFD, computational fluid dynamics; CFRC, carbon fiber reinforced composite; CFP, carbon fiber prepreg; CSF, continuum surface force; DMSO, dimethyl sulfoxide; DoI, degree of impregnation; PA6, polyamide 6; REV, representative elementary volume; RMSD, root mean squared deviation; RoI, rate of impregnation; UD, uni-directional; VOF, volume of fluid. ⇑ Corresponding author. E-mail address: [email protected] (Y.-I. Lim). https://doi.org/10.1016/j.ces.2018.04.010 0009-2509/Ó 2018 Elsevier Ltd. All rights reserved.
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Nomenclature a, b, c A C2 Ca cp dH DoICFD DoICum DoIfunc E F ! g h H ! i I ! j k K L LP M n ! n N P R R2 Re t T T0
constants of DoI function cross sectional area (m2) inertial resistance coefficient (–) capillary number (–) heat capacity (J/kg/K) hydraulic diameter (m) DoI obtained from micro-scale CFD cumulative DoI DoI function obtained by regression of DoICFD energy (J) source term force (N) gravity vector (=9.81 m/s2) penetration depth (m) temperature dependence term of viscosity (–) unit direction vector of x-axis identity tensor (–) unit direction vector of y-axis thermal conductivity (W/m/K) permeability (m2) number index of times length of wetted perimeter (m) number index of pressures power-law index normal vector (–) number index of computational cells in 2D CFD domain gauge pressure (Pa) number index of computational cells in tow domain correlation coefficient (–) Reynolds number (–) time (s) temperature (K or °C) zero temperature (K)
Tref u V x xend
reference temperature (K) velocity (m/s) molar volume (cm3/mol) axial length (m) end of impregnation die length (m)
Greek letters a volume fraction (–) b slope limiter value (–) c shear rate (1/s) e voidage (–) g viscosity (kg/m/s) g0 zero shear viscosity (kg/m/s) g1 infinite shear viscosity (kg/m/s) h contact angle (rad or °) j surface curvature (1/m) s time constant (s) q density (kg/m3) r surface tension (N/m) s stress tensor (Pa) U molar volume constant (–) Subscripts a activation D DMSO eff effective r resin surf surface tension T tow w wall
of the micro- and macro-scale CFD results. Combining the multi-scale models gives a potential to predict the uniformity of the transverse resin amount in the final UD-CFP product. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction High strength fiber-reinforced composite materials have been widely applied in many structural and mechanical systems such as aerospace, automotive, sporting goods, biomedical, building, and infrastructure (Chang and Lees, 1988; Han et al., 2015; Ngo et al., 2017; Rodríguez-Tembleque and Aliabadi, 2016). Carbon fiber-reinforced composites (CFRCs) with thermoplastic matrices show excellent properties including light weight, high strength and modulus, good electrical and thermal conductivity, environmental resistance, and recyclability (Chang and Lees, 1988; Luo et al., 2014; Sakaguchi et al., 2000). However, the use of CFRCs with thermoplastic matrices suffers from drawbacks regarding their manufacture in industrial applications. Since the molten viscosity of thermoplastic resins is higher than that of thermoset resins, there are difficulties in impregnating thermoplastic resins into carbon fiber (CF) tows (Sakaguchi et al., 2000; Ye et al., 1995), where the tow is a bundle of strands (or filaments) having a diameter of a few micrometers. Understanding the impregnation mechanisms at the micro-scale is important to achieve a good quality of CFRCs (Hou et al., 1998; Ye et al., 1995). The prepreg used to fabricate CFRC intermediates (Ngo et al., 2017) is ready to lay into the mold without the addition of any resin,
because of pre-impregnation with a resin. Many researchers have presented impregnation and compaction behaviors of intermediate thermoplastic composites for commingled yarn (Hamada et al., 1993; Klinkmüller et al., 1995; Van West et al., 1991), powderimpregnated unidirectional fiber bundles (Kim et al., 1989), and prepreg tapes (Lee and Springer, 1987). The axial and transverse permeabilities were dependent on the microscopic characteristics and orientation of the fiber network (Gutowski et al., 1987). A comprehensive model describing the impregnation and consolidation mechanisms was developed for thermoplastic CFRCs under the assumption of an isothermal Newtonian fluid, porous media tow, and absence of capillary effect between the fluid and CF tow (Hou et al., 1998; Ye et al., 1995). Bijsterbosch and Gaymans (1993) studied the degree of impregnation (DoI) of glass fiber with polyamide 6 (PA6) in an impregnation bath, where the DoI experimentally obtained for various process and material parameters was expressed as a function of the square roots of time and pressure (Bijsterbosch and Gaymans, 1993). A resin permeability to CF filaments obtained at the micro-scale was used to calculate the resin velocity inside CFRC tows at the macro-scale, assuming that the CF filaments were entirely impregnated (Ngo et al., 2017). However, few studies have addressed the influence of the microscopic impregnation mechanism on the quality of CFRCs at the macro-scale.
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The fiber impregnation process by melting the polymer matrix is intrinsically heterogeneous at all length scales. When modeling the fluid flow at macro-scale, it is generally not possible to account for all pertinent scales in a single simulation. The heterogeneous effects of smaller scales are introduced into the homogeneous macro-scale by constitutive relations (Weinan, 2011). By simultaneously considering models at different scales, multi-scale modeling approach has the potential to share the efficiency of macroscale models as well as the accuracy of micro-scale models (Weinan, 2011; Li et al., 2013). A central issue that invariably arises in the literature is how the understanding of mechanisms at the micro-scale can enable predictions of functional behavior at the macro-scale (Yip and Short, 2013). With the increase of computational power nowadays, computational fluid dynamics (CFD) has become a viable tool for simulating multi-physics and complex geometry problems at a wide range of fluid scales (Tu et al., 2012). In small scale flow, CFD is seen to yield very detailed flow field solutions of all relevant variables such as velocities, pressure, temperature, concentrations, and volume fraction. Such detailed solutions are expected to achieve a better understanding of phenomena occurring in a micro-scale impregnation process. For larger scales in industrial applications, process and design optimization is sped up at low cost through the introduction of CFD tools (Joshi and Ranade, 2003). A multi-scale steady-state CFD simulation sequentially approaching from small to large scales of a structured-packing column was presented to examine the liquid holdup and pressure drop (Raynal and Royon-Lebeaud, 2007). A structure-based drag correlation obtained at meso-scale was incorporated into the gas-liquid Eulerian CFD model of bubble columns (Yang et al., 2011). The volume of fluid CFD at the micro- and meso-scales, and porous media Eulerian CFD at the column-scale were combined to investigate hydrodynamics of three different types of structured-packings (Li et al., 2016). A dual-scale porous media model for microscopic and macroscopic resin flows was proposed to investigate the DoI of a thermoplastic matrix composite during the pultrusion process (Kim et al., 2001). A multi-scale CFD model including both the full 3D impregnation die and the microscopic creeping flow in a representative element volume (REV) was used to optimize operating conditions such as tow speed and resin flow rate (Ngo et al., 2017). Unfortunately, these multi-scale CFD approaches focused only on evaluating a physical property (i.e., pressure drop, film thickness, or permeability) at micro-scale CFD for use in macro-scale CFD. There remains a need for an efficient method that can predict functional behaviors at the macro-scale from microscopic mechanisms. The product quality in the process-scale has to be obtained by an extensive coupling between the micro- and macro-scale CFD simulations. This article aims to develop a multi-scale CFD model to predict the DoI of an impregnation die for the production of a thermoplastic unidirectional carbon fiber prepreg (UD-CFP). In the micro-scale transient CFD within a REV of CF tow, impregnation mechanisms are found and the microscopic DoI is calculated at various resin inlet pressures. The macro-scale 3D CFD covers the entire impregnation die and identifies the pressure distribution along the die length. Combining the micro-scale CFD and macro-scale CFD simulation results, a cumulative DoI at the end of the impregnation die is obtained and compared to experimental data for the relative resin amount of the final UD-CFP tape. This article reveals how to connect the micro- and macro-scale CFDs to obtain process-scale performance required for industrial applications. 2. Material properties and geometry Fifteen CF tows were impregnated with resin in a prepreg die. One tow consisted of 12,000 CF filaments, and polyamide 6 (PA6)
was used as the thermoplastic resin. The micro- and macro-scale CFDs took into account 128 filaments of CF and the entire impregnation die, respectively. Before the tow entered the impregnation die, the void area of the tow was filled with dimethyl sulfoxide (DMSO), which is an organic solvent used in the upstream process of UD-CFP. DMSO in the liquid phase was replaced by the resin (PA6) during impregnation and volatiles at a die operating temperature over 220 °C. 2.1. Material properties The material properties of PA6, DMSO, and the CF tow are listed in Table 1. The density, heat capacity, and thermal conductivity of the resin and DMSO are given at 260 °C, and 189 °C. The nonNewtonian resin viscosity (g) was modeled by Carreau’s formula (Tanner, 2000) as a function of shear rate (c) and temperature (T):
n1
g ¼ HðTÞ g1 þ ðg0 g1 Þ½1 þ c2 s2 2
ð1Þ
where g0, g1, s, and n are the zero shear viscosity, infinite shear viscosity, time constant, and power-law index, respectively. The temperature dependence term, H(T), is expressed as:
1 1 HðTÞ ¼ exp T a T T 0 T ref T 0
ð2Þ
where Ta, T0, and Tref are the activation, zero, and reference temperatures, respectively. The model parameters (Table 2) were obtained from the regression of Carreau’s model with the viscosity experimentally measured at four temperatures (T = 220, 240, 260, and 280 °C) and shear rate values ranging from 10 to 1000 s1 (Ngo et al., 2017). 2.2. Geometry and meshing Two CFD domains are considered in the multi-scale simulation. The macro-scale 3D domain is the impregnation die (see Fig. 1) including a porous media moving tow domain and a resin fluid domain. The geometry is asymmetric because the resin is injected through a pipe introduced from the left. The micro-scale domain is
Table 1 Material properties of PA6, DMSO, and CF tow.
a b
Resin (PA6)
DMSO
CF tow
Density (q, kg/m3) Heat capacity (cp, J/kg/K) Thermal conductivity (k, W/m/K) Viscosity (g, kg/m/s)
932a 3728a 0.264a Carreau’s model
1800 1075 0.357 –
Voidage (e)
–
974b 2177b 0.0454b 0.55 103b –
0.408
Values at 260 °C. Values at 189 °C.
Table 2 Parameter values of Carreau’s viscosity model as function of shear rate (c) and temperature (T) for PA6 (Ngo et al., 2017). Parameter
Value
Time constant (s, s) Power-law index (n) Zero shear viscosity (g0 , kg/m/s) Infinite shear viscosity (g1 , kg/m/s) Reference temperature (T ref , K) Zero temperature (T 0 , K) Activation temperature (Ta, K)
0.0058 3.556 330.9 1.89 493 0 5148
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Fig. 1. Geometry and meshing of impregnation die in macro-scale 3D CFD.
a 2D microscopic tow structure with randomly-distributed CF filaments 7 lm in diameter. The micro-scale 2D domain illustrates a representative elementary volume (REV) of the tow domain (see Fig. 2). The voidage of the REV (e = 0.408) is the same as that of the CF tow. It is assumed that the micro-scale REV is periodically repeated in the macroscopic tow domain. One tow has a width of 6 mm and a thickness of 0.13 mm. The die width is 100 mm. In the macro-scale CFD, the mesh independent test was performed to find the optimal mesh number in the range from 1 to 11 million cells. A mesh structure of 4.2 million cells was chosen to save computational time while retaining the accuracy of the
CFD simulation (Ngo et al., 2017). The resin and tow domains have 2.8 million polyhedral cells and 1.4 million hexahedral cells, respectively. In the micro-scale 2D CFD domain (see Fig. 2), the triangular and rectangular mixed-mesh structure was constructed. The REV has about 27,800 meshes. It is assumed that the resin passes through in one direction from left to right.
3. CFD modeling The macro-scale CFD model is solved in one liquid phase at steady state under the assumption that the impregnation die is fully filled with resin. An unsteady-state liquid-liquid volume of fluid (VOF) model with surface tension force is used in the micro-scale CFD simulation, where the two liquid phases are DMSO and PA6.
3.1. Macro-scale CFD model
Fig. 2. Geometry and meshing of representative elementary volume (REV) in micro-scale 2D CFD.
There are two domains in the continuum macro-scale CFD model: tow and resin domains. The former is assumed to be a porous medium moving at constant speed, uT, called the tow speed. The porous tow domain is filled 40.8% with resin (e = 0.408, see Table 1). The solid phase of the tow is not modeled, and the microscopic creeping-flow inside the porous medium (Whitaker, 1986) is neglected in this study. However, the resistance force between the liquid (resin) and the solid (porous media) was considered using Darcy’s law (Ngo et al., 2017). The continuity, momentum, and energy equations are applied to the continuous resin phase in both the tow and resin domains. Considering a moving coordinate system of the tow domain with ! a linear velocity ( uT ), the fluid phase inside the tow is transformed from a stationary frame to a moving frame. The governing equations of the macro-scale steady state CFD model are listed in Table 3. The continuity equations of the resin and tow domains are shown in Eqs. (T1) and (T2). The fluid is assumed to be incompressible (i.e., constant density, q).
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S.I. Ngo et al. / Chemical Engineering Science 185 (2018) 64–75 Table 3 Macro-scale CFD model in resin and tow domains. ! ! Continuity equation (resin domain): r ðq u Þ ¼ 0 ! ! ! (tow domain): r ðqð u uT ÞÞ ¼ 0 ! ! ! !! ! Momentum equation (resin domain): r ðq u u Þ ¼ r P þ r ðsÞ þ q g ! ! ! ! ! ! ! ! ! (tow domain): r ðqð u uT Þð u uT ÞÞ ¼ r P þ r ðsÞ þ q g þ F ! where F in the tow domain includes the forces of porous resistance and inertia: ! ! ! ! ! ! ! F ¼ Kg ð u uT Þ C 2 12 qj u uT jð u uT Þ
(T1) (T2) (T3) (T4) (T5)
s is defined as ! !! ! ! u þ r u T Þ 23 r u I s ¼ g½ð r !
(T6)
! ! ! ! ! ! ! Energy equation (resin domain): qcp u r T þ u r P ¼ r ðkr r T þ s u Þ ! ! ! ! ! ! ! ! ! ! (tow domain): qcp ð u uT Þ r T T þ ð u uT Þ r P ¼ r ðkT;eff r T T þ ðs ð u uT ÞÞÞ where kT;eff ¼ ekr þ ð1 eÞkT
The momentum conservation equations for the resin and tow domains are expressed in Eqs. (T3) and (T4), respectively. P is the ! static pressure, q g is the gravitational force, and s is the stress ! tensor. F is the source term including resistance forces in Eq. (T5), where K is the permeability, C 2 is the inertial resistance coef! ! ficient, and j u uT j is the magnitude of the relative velocity. The first term on the right-hand side of Eq. (T5) refers to the viscous loss defined by Darcy’s law. The second term is the inertial loss. The viscous resistance factor is the inverse of the permeability (1/K). In this study, the inertial resistance coefficient is neglected in the thin tow domain (C2 = 0) because the resin flow rate inside the tow is very low (Ngo et al., 2017). The stress tensor (s) is expressed by Eq. (T6), where I is the identity tensor and ! !T ðr u þ ðr u Þ Þ is the strain rate tensor for the resin flow. The energy transport equation is introduced to the resin phase of the two domains. Under the assumption of thermal equilibrium inside the porous tow, the steady-state energy conservation equations of the resin and tow domains are given in Eqs. (T7) and (T8), respectively. Here, TT and T are the resin temperatures in the tow and resin domains, respectively, cp is the heat capacity of the resin, and kr is the thermal conductivity of the resin (see Table 1). The effective thermal conductivity (kT,eff) in the tow domain is calculated by Eq. (T9), where kT is the thermal conductivity of the tow. The second terms on the right-hand sides of Eqs. (T7) and (T8) represent the viscous energy dissipation. 3.2. Micro-scale 2D VOF model The volume of fluid (VOF) model relies on the assumption that two phases do not interpenetrate and is applied to the micro-scale CFD. A liquid-liquid system including DMSO and PA6 is considered in the VOF model at unsteady-state. Since the boiling point of DMSO is 189 °C, DMSO evaporates at the operating temperature (TT 260 °C). However, it is assumed that the microscopic tow is initially filled with liquid DMSO at a slightly elevated pressure (about 1.5 atm). The resin (PA6) is impregnated by extruding DMSO. The 128 CF filaments randomly distributed are supposed to be a rigid body without deformation. Thus, the applied pressure, length and width of REV during impregnation do not change in this micro-scale CFD. The surface tension effect becomes important when Reynolds number (Re) and Capillary number (Ca) are less than unity.
qu dH g u g Ca ¼ r
Re ¼
(T7) (T8) (T9)
dH = 4A/Lp, where A is the area and Lp is the wetted perimeter. Re and Ca of the resin inside the tow are about 0.044 and 0.32, respectively. In this study, the continuum surface force model (Brackbill et al., 1992) is used as a source term in the momentum equation. The governing equations are summarized in Table 4. Tracking of the interface between phases is accomplished by solving the continuity equation for the incompressible liquid. Assuming there is no mass transfer between the two phases and no source term exists in each phase, the continuity equation for DMSO (denoted as subscript D) is shown in Eq. (T10), where a is the volume frac! tion, and u is the velocity. The volume fraction will not be solved for the primary resin phase (denoted as subscript r), but ar is computed based on the relation of Eq. (T11). In this study, a special interpolation treatment of the volume fraction to the cells that contain the interface between the two phases is used, which is based on the slope limiter second-order reconstruction compressive scheme (ANSYS Fluent Theory Guide, 2016). Here, the slope limiter value (b) is set to b ¼ 2. In the VOF model, a single momentum equation shown in Eq. (T12) is solved throughout the domain, and the resulting velocity field is shared in both two phases. The left hand side of Eq. (T12) includes accumulation and convection of momentum per unit volume. In the right-hand side of Eq. (T12), P is the pressure shared in !! !! ! both phases, q g is the gravity force, gð r u þ r u T Þ is the stress tensor, and F surf is the surface tension force on the interface between two phases as an additional source term. The continuum surface force (CSF) model (Brackbill et al., 1992) is expressed by Eq. (T13), where rrD is the interfacial surface tension between the resin and DMSO, and jr is the surface curvature of the resin as defined in Eq. (T14). jr is a divergence of the unit ! ! normal vector for the gradient of resin volume fraction ( n r ¼ r ar ). By combining Young’s equation of surface tension and contact angle (Young, 1805), Fokews’ theory of adsorptive adhesion
Table 4 Micro-scale two-phase volume of fluid (VOF) model. ! ! aD Continuity equation (DMSO): @@t þ r ðaD u D Þ ¼ 0 (resin): aD þ ar ¼ 1 Single momentum equation: ! ! ! !! !!T ! !! ! @ @t ðq u Þ þ r ðq u u Þ ¼ r P þ r ½gð r u þ r u Þ þ q g þ F surf ! ar where surface tension force (F surf ): F surf ¼ rrD 1qðjqr r þqr Þ 2 D ! ! r ar Þ surface curvature (jr): jr ¼ r ð !
(T10) (T11) (T12) (T13) (T14)
j r ar j
ð3Þ
; and r are the mean velocity, mean viscosity, and sur; g where u face tension, respectively. dH is the hydraulic diameter defined as
! ! ! ! @ ðqEÞ þ r ð u ðqE þ PÞÞ ¼ r ðk r TÞ Single energy equation: @t Single properties (density): q ¼ ar qr þ aD qD (viscosity): g ¼ ar gr þ aD gD (energy): E ¼ ar qr Er þqaD qD ED (mass-averaged energy)
(conductivity): k ¼ ar kr þ aD kD
(T15) (T16) (T17) (T18) (T19)
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(Fowkes, 1964), and the Girifalco-Good correlation (Girifalco and Good, 1957), it is possible to calculate the interfacial surface tension (rrD) using the following equation:
rrD ¼ rr
ð1 þ coshÞ2 4U2
!
cosh
ð4Þ
where U is a constant defined as follows (Girifalco and Good, 1957):
U¼
4ðV r V D Þ1=3
ð5Þ
1=3 2
½V r1=3 þ V D
V r and V D are the molar volumes of the resin and DMSO, respectively. The contact angle (h) between the resin and DMSO is calculated from (Girifalco and Good, 1957):
12 rr cosh 2U 1
ð6Þ
rD
Table 5 reports the physical properties of the resin and DMSO, and the contact angle and interfacial surface tension between the two liquid phases estimated from Eqs. (4)–(6). The surface tensions of PA6 and DMSO are calculated by the group contribution method (Baum, 1997). Since the difference between the surface tensions of PA6 and DMSO is small, rrD is small. The contact angle between the two liquid phases is estimated to h = 33.5°. The energy equation is shared for the two phases in a given control volume, as shown in Eq. (T15). Single properties such as density (q), viscosity (g), energy (E), and thermal conductivity (k) are employed to solve the VOF model, as shown in Eqs. (T16–T19). Energy (E) is treated as a mass-averaged value in Eq. (T18), where Er and ED are calculated with the specific heat capacity (cp) and the shared temperature (T). In order to include the resin adhesion effect on the wall of the CF filament, a dynamic boundary condition is applied to adjust the surface curvature (jw) on the wall (Brackbill et al., 1992).
!
n wÞ jw ¼ r ð!
ð7Þ
die. The temperature of the wall is the same as the inlet resin temperature. In the micro-scale 2D CFD, the resin enters at 260 °C and at eight different pressures from 2 to 9 104 Pa, which are identified from the macro-scale CFD in the entire impregnation domain (see Section 4.2). It is assumed that the temperature of the CF surface (250 °C) is a little lower than that of the resin, because the tow comes in at 220 °C in the macro-scale domain (see Table 6). 3.4. Multi-scale coupling strategy The present multi-scale model is divided into four main parts: (i) macro-scale CFD, (ii) micro-scale CFD, (iii) regression with the micro-scale CFD data, and (iv) process-scale simulation, as illustrated in Fig. 4. The multi-scale model aims to obtain the degree of impregnation (DoI) and rate of impregnation (RoI) as a function of pressure (P) and time (t) in the production of UD-CFP. First, the macro-scale 3D CFD is performed for the entire impregnation die considering the tow as a porous media with a transverse permeability of 5.61 1013 m2 (Ngo et al., 2017), that is about twice higher than that of a laminate graphite composite (Gutowski et al., 1987). The range of resin pressures induced in the moving tow is found in the macro-scale CFD simulation. Then, the microscale 2D CFD simulation at unsteady-state is repeated at various pressure values (P 1 P M ). Each simulation provides DoI with respect to t, and DoI data at M pressures are obtained in the micro-scale CFD. The DoI at a given time is calculated by the ratio of the area occupied by resin (Ar) to the total area (A) (Kim et al., 1989):
DoICFD ¼
Ar ¼ A
PN
a
i¼1 Ai r;i PN i¼1 Ai
ð9Þ
where DoICFD is the DoI obtained from CFD, N is the number of computational cells, Ai is the area of cell i, and ar,i is the resin volume fraction within cell i. ar,i = 1.0 for a cell fully filled with resin and ar,i has a value between 0 and 1 for a cell lying at the interface of the resin and DMSO.
! where n w is the unit normal vector to the tangent to the surface that makes with the CF solid surface. When the contact angle between the resin and CF surface (hw) is known, the unit normal vector is expressed as follows:
! ! ! n w ¼ sin hw i w þ cos hw j w
ð8Þ
! ! where i w and j w are the unit vectors tangential and normal to the CF surface, respectively. Fig. 3 illustrates the unit normal vector of resin on the CF wall. The contact angle (hw ) of PA6 on the CF surface is around 42° (Scott, 1997). 3.3. Boundary conditions In the macro-scale 3D CFD, the resin enters at 260 °C and 26.65 g/min. The tow (220 °C) at the inlet moves with a speed of 2 m/min. The gauge pressure (P) is set to 0 Pa at the exit of the impregnation
Fig. 3. Contact angle and normal vector of resin on CF wall.
Table 5 Physical properties of PA6 and DMSO. Parameters
Symbol (unit)
Value
Reference
Surface tension of PA6 with air Surface tension of DMSO with air Molar volume of PA6 Molar volume of DMSO Contact angle between PA6 and DMSO Molar volume constant Interfacial surface tension
rr ðN=mÞ rD ðN=mÞ
55.7 10-3 65.7 10-3 104.39 71.03 0.83 0.996 0.769 10-3
Baum (1997)
V r ðcm3 =molÞ V D ðcm3 =molÞ cosh
U
rrD ðN=mÞ
Fowkes (1964), Girifalco and Good (1957), Young (1805)
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Table 6 Boundary conditions of macro- and micro-scale CFD models. Macro-scale CFD model Boundary area
T (°C)
P (Pa)
! j uT j (m/min)
Mass (g/min)
Resin inlet Tow inlet Resin/tow outlet Wall
260 220 260 260
– – 0 –
– 2 – –
26.65 – – –
Micro-scale CFD model Boundary area
T (°C)
P (Pa)
Resin inlet
260
Resin outlet CF surface
260 250
ð2 9Þ 104 0 –
–
Resin pressure range (P1 ~ PM) DoIcum for 15 tows
Micro-scale 2D CFD (Unsteady-state)
4
1
Micro-scale CFD at P1
Macro-scale 3D CFD (Steady-state)
Micro-scale CFD at P2
Process-scale Simulation
Positions (xi)
Micro-scale CFD at P3 …
Pressures (Pi)
Micro-scale CFD at PM 2
DoICFD
3
Data fitting of micro-scale CFD
DoIfunc and RoI
Fig. 4. Coupling strategy of multi-scale CFD models.
x uT
Once a set of DoICFD data is acquired from the micro-scale CFD, a DoI function (DoIfunc) with respect to pressure P and t is obtained by means of regression with this data set. Since the DoI of fiberreinforced composites was expressed as a function of the square pffiffiffiffiffi root of both time and pressure ( Pt) (Bijsterbosch and Gaymans, 1993; Carleton and Nelson, 1994; Kim et al., 1989), the global funcpffiffiffi pffiffi tion is proposed in terms of t and P:
Substituting Eq. (13) in Eq. (10), the final DoI function is obtained as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P DoIfunc ðP; tÞ ¼ a tþb þc 1 106 1 106
The cumulative DoI (DoIcum) for each tow from 1 to 15 is calculated by the summation of DoI in each cell:
ð10Þ
where a, b, and c are the model constants, which are found by minimizing the root mean squared deviation (RMSD) between DoICFD and DoIfunc:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PM PL 2 i¼1 j¼1 ðDoICFD;ij DoIfunc ðP i ; t j ÞÞ RMSD ¼ ML
ð11Þ
where M and L are the number of sampling points for pressure and time, respectively. The rate of impregnation (RoI) can be obtained by the partial differentiation of DoIfunc with respect to time (t) at a constant pressure (P):
qffiffiffiffiffiffiffiffiffiffi
P a
@ðDoIfunc Þ
1106 p ffiffi RoIðP; tÞ ¼ ¼
@t 2 t P
t¼
ð13Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P x P þb þc 1 106 uT 1 106
DoIfunc ðP; xÞ ¼ a
DoIcum ¼ 2ðDoI1 ðP1 ; x1 Þ þ
R1 X ½DoIfunc ðPi ; xiþ1 Þ DoIfunc ðPi ; xi ÞÞ
ð14Þ
ð15Þ
i¼2
where DoI1 is the initial value caused by the surface coverage and capillary effects (see Section 4.1), R is the number of computational cells located in the transverse middle of each tow (see Fig. 1b) along P þP
the tow moving direction, and Pð¼ i 2 iþ1 Þ is the mean pressure for two adjacent cells. It is noted that the resin seeps into the middle of the tow from both the top and bottom sides (see Fig. 1b). Therefore, DoIacc is twice the one-sided DoI in the micro-scale CFD model. Finally, the DoI of the 15 CF tows with respect to the axial length of impregnation die is obtained by combining the micro- and macroscale CFD.
ð12Þ
Since the 15 tows are considered as a porous medium, the degree of impregnation cannot be obtained in the macro-scale CFD. However, the pressure distribution inside the 15 tows is identified. The position (x) of the tow is converted into time (t), using the constant tow speed (uT).
4. Results and discussions The continuum micro- and macro-scale CFD models were solved using a finite volume-based solver, ANSYS Fluent R15.0 (ANSYS Inc., USA, 2016). The calculation was performed on a workstation with 24 cores (2.7 GHz CPU) and 128 GB RAM. An adaptive
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time step size was applied for the micro-scale CFD model. The minimum and maximum time step sizes are 1 107 and 1 103 (s), respectively. 4.1. Micro-scale CFD results The transient micro-scale CFD simulation was carried out at eight pressures from 2 104 Pa to 9 104 Pa. As mentioned earlier, the pressure range is obtained from the macro-scale CFD (see Section 4.2). The average viscosity of the resin (0.5 103 Pa s g 1100 103 Pa s) was 8.07 10-3 Pa s with a mean shear rate of 182 s-1 (0.4 s-1 c 1150 s-1) in the micro-scale domain at T = 260 °C. The impregnation mechanism is identified based on the DoI calculated from the micro-scale CFD. Fig. 5 shows the DoI profile at pffiffi P = 2 104 Pa with respect to t within an initial impregnation time (0.0 s t 1.0 s). This initial profile can be divided into three parts: (i) resin coverage on the CF surface, (ii) capillary permeation to CF (Carleton and Nelson, 1994), and (iii) penetration through the inter-CF filaments (Klinkmüller et al., 1995). The capillary impreg-
nation is caused by the attractive force between carbon fiber and Nylon 6 resin, while the penetration to CF filaments is caused by the pressure difference between the inlet and outlet of the resin. The capillary effect dominates at the beginning of impregnation, while the penetration effect plays a major role after the capillary effect terminates. The resin coverage on CF surface takes place in pffiffi a short time of about 4.0 10-4 s ( t ¼ 0:02 s0:5 ), as shown in Fig. 5. The capillary permeation of resin is observed until t = pffiffi 0.013 s ( t ¼ 0:114 s0:5 ). After the capillary permeation, the CF pffiffi tow is almost linearly impregnated with t . Fig. 6 shows five snapshots of the resin volume fraction (ar) at t = 0.0001, 0.01, 1, 10, and 100 s for an inlet pressure of P = 9 104 Pa. The resin coverage on the CF surface is observed at t = 0.0001 s (see Fig. 6a), which is faster than that at P = 2 104 Pa. The capillary permeation is also seen at t = 0.001 s in Fig. 6b. The resin flow reaches the border of the CF tow at about t = 100 s (see Fig. 6e). Fig. 7 illustrates the effect of operating pressure on the penetration depth of the resin at a flow time of 20 s. For the linear increment of operating pressures at 3 104, 6 104, and 9 104 Pa, the ratio of the penetration depths is h1 : h2 : h3 ¼ 1 : 1:175 : 1:65. Kim et al. experimentally reported an increase of DoI with increasing pressure (Kim et al., 1989). The DoI versus time obtained from CFD (DoICFD) for the eight pressures was regressed with DoIfunc in Eq. (10), and its model parameters were found to be a = 0.1497, b = 0.6137, and c = 0.0704. Fig. 8a compares the two DoIs and the correlation coefficient (R2) between the two DoIs is 0.9673 (see Fig. 8b). The DoIfunc is applicable at pressures lower than 9 104 Pa. It is worth noting that the first, second, and third terms of the DoIfunc in Eq. (10) correspond to the penetration, capillary permeation, and surface coverage, respectively. Thus, the DoI of surface coverage is estimated to about 0.07, which is the third parameter (c) of Eq. (10). 4.2. Macro-scale CFD results
Fig. 5. Impregnation mechanism from micro-scale CFD at P = 2 104 Pa.
(a) t=0.0001 s
(b) t=0.001 s
The macro-scale CFD provides the micro-scale CFD and the process-scale simulation with the resin pressure range (from P1 to PM) and the resin pressure (Pi) applied to the tow in the axial direction (xi), respectively (see Fig. 4). Fig. 9 shows the pressure profiles of 15 tows according to the impregnation die length. Here, the gauge pressure was collected from an upper cell located in the
(c) t=1 s
(d) t=10 s
Fig. 6. Contours of resin volume fraction (ar) at five flow times (P = 9 104 Pa).
(e) t=100 s
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(a) P = 3×104 Pa
(b) P = 6 ×104 Pa h1
(c) P = 9 ×104 Pa
h2
h3
Fig. 7. Contours of resin volume fraction (ar) at t ¼ 20 s for three inlet pressures (P).
Fig. 8. DoI with respect to P and
pffiffi t , and correlation between DoICFD and DoIfunc.
transverse middle of each tow (see Fig. 1b). Point ➀ is the resin injection area from the bottom to the tow domain. Point ➁ is the outlet of both tow and resin. All 15 tows have a similar pressure profile ranging from 0 to 8 104 Pa. A maximum pressure of 7.5 104 Pa is observed near the bottle neck of x = 3 mm for every tow. RoI in Eq. (12) is estimated to 0.21 min-1 at P = 6 104 Pa, and x = 0.015 m (t = 0.45 s). Note that the cumulative DoI (DoIcum) in Eq. (15) is needed because the pressure (P) varies with position (x). 4.3. Process-scale simulation results The final stage of the present multi-scale model is the prediction of DoI at the process-scale with the data collected from the
micro- and macro-scale CFD simulations. Eq. (15) is applied to predict the DoI of each tow with respect to the die length. DoI1(P1,x1) was 0.105, which included the resin coverage on the surface and the capillary permeation at P = 6 104 Pa (see Fig. 9). Fig. 10a shows 15 DoIcum profiles along the impregnation die length (x). DoI at the end of the impregnation die (DoIcum(xend)) is seen in Fig. 10b; it is not symmetric and uniform along the die width. In fact, DoIcum(xend) is strongly related to the quality of the final UD-CFP product. The impregnation die should be designed to have a symmetric and uniform DoIcum(xend). Fig. 10 demonstrates that the multi-scale CFD model is able to reproduce the resin uniformity of the impregnation process. Ten segments of the final UD-CFP tape produced in this impregnation die were sintered in a muffle furnace (AJ-MB8, Ajeon
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73
Fig. 9. Pressure (P) profiles of 15 tows along impregnation die length (x) in macro-scale CFD simulation.
Fig. 10. Cumulative degree of impregnation (DoIcum) along impregnation die length (x) for 15 tows.
Heating Industrial Co., Ltd., Korea), and the resin weight of each segment was measured (Ngo et al., 2017). Since the resin weight and DoIcum(xend) cannot be compared directly, a relative resin
amount and a relative DoIcum(xend) were obtained from the ratio of original values to their average. Fig. 11 compares the three relative values scaled in the range of 0.942 y 1.047, which is the
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scale simulation approach provided a promising tool to investigate the effect of process geometry on the quality of the final UD-CFP product. This micro-scale CFD under the assumption of rigid CF filaments did not consider deformation of the microscopic CF structure by mechanical forces. DoI would be more accurately predicted, if the clustering and spreading of CF filaments during impregnation were taken into account. Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science ICT and Future Planning (Grant number: NRF-2016R1A2B4010423) in South Korea.
Fig. 11. Comparison between CFD results for relative DoI and experimental data for relative resin amount of final UD-CFP.
lower and upper limits of the relative resin amount obtained by experiment, with respect to the UD-tape width (or die width). The relative resin amount is asymmetric in the UD-tape width, and is higher in the center than at the two edges. The relative resin velocity obtained over the entire cross-sectional area at xend (Ngo et al., 2017) has a similar shape to the experimental data. The relative DoIcum(xend) is relatively lower in the center than at the edges unlike the experimental data. The UD-CFP passes through a grooved area for further resin impregnation and spreading. Then, it is compressed by the pressing roller. After cooling to room temperature, the final UD-tape is produced (Ngo et al., 2017). Since the relative resin amount was measured for the final UD-tape, the DoI at xend obtained from the multi-scale CFD simulation has some discrepancy with the experimental data. Further investigation including the grooved area and the pressing roller would be helpful to correctly predict the distribution of resin along the UD-tape width.
5. Conclusions A multi-scale simulation approach was presented to investigate thermoplastic resin impregnation into carbon fiber (CF) tows. The approach included (i) micro-scale transient computational fluid dynamics (CFD) of the two-dimensional (2D) representative elementary volume (REV) of a CF tow, (ii) macro-scale steady-state CFD of an entire 3D impregnation die with 15 porous media moving tows, and (iii) process-scale simulation to predict cumulative degree of impregnation (DoI) for unidirectional carbon fiber prepreg (UD-CFP) prepreg. Of particular interest were the impregnation mechanisms of resin coverage on the CF surface, capillary permeation, and penetration through CF filaments in the micro-scale CFD simulation. Several micro-scale volume of fluid (VOF) CFD simulations were performed within a range of resin pressures identified from the macro-scale CFD simulation. A global DoI function with respect to pressure and time, inspired by the three impregnation mechanisms, was obtained from the regression of the micro-scale CFD results. Using the data collected from the micro- and macro-scale CFD simulations, a cumulative DoI of the 15 tows with respect to the impregnation die length was predicted in the process-scale simulation. The cumulative DoI at the end of the impregnation die was compared to the relative resin amount measured experimentally. Though there was deviation between the multi-scale CFD prediction and the experimental resin amount, the multi-
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Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Prediction of degree of impregnation in thermoplastic unidirectional carbon fiber prepreg by multi-scale computational fluid dynamics Son Ich Ngo a, Young-Il Lim a,⇑, Moon-Heui Hahn b, Jaeho Jung b a Center of Sustainable Process Engineering (CoSPE), Department of Chemical Engineering, Hankyong National University, 327 Jungang-ro, Anseong-si, Gyeonggi-do 17579, Republic of Korea b Hyosung R&D Business Lab., Hyosung Corporation, 74 Simin-daero, Dongan-gu, Anyang-si, Gyeonggi-do 14080, Republic of Korea
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Degree of impregnation (DoI) for
unidirectional carbon fiber prepreg (UD-CFP). Pressure profile of impregnation die obtained using macro-scale 3D Eulerian CFD model. DoI with time and pressure obtained from micro-scale 2D volume of fluid CFD model. Prediction of cumulative DoI for UDCFP impregnation die by multi-scale CFD model.
a r t i c l e
i n f o
Article history: Received 2 February 2017 Received in revised form 6 March 2018 Accepted 5 April 2018 Available online 7 April 2018 Keywords: Unidirectional carbon fiber prepreg (UD-CFP) Thermoplastic resin Resin impregnation die Degree of impregnation (DoI) Computational fluid dynamics (CFD) Multi-scale simulation
a b s t r a c t A multi-scale simulation approach was proposed to predict the degree of impregnation (DoI) in thermoplastic unidirectional carbon fiber prepreg (UD-CFP). The multi-scale approach included a twodimensional (2D) micro-scale computational fluid dynamics (CFD) in a representative elementary volume (REV) of carbon fiber (CF) tow, a 3D macro-scale CFD of an entire impregnation die with 15 sliding CF tows, and a process-scale simulation assembling data from the micro- and macro-scale CFDs. In the macro-scale steady-state CFD, thermoplastic resin injection and CF tow insertion were considered for an impregnation die 10 cm in width. In the micro-scale transient CFD, impregnation mechanisms of resin into CF filaments 7 lm in diameter were identified in terms of surface coverage, capillary permeation, and penetration through CF filaments. The DoI as a function of pressure and time was obtained from the micro-scale CFD within a range of pressures found in the macro-scale CFD. In the process-scale simulation, the cumulative DoI of the 15 tows was predicted along the impregnation die length with the aid
Abbreviations: 2D, two-dimensional; 3D, three-dimensional; CF, carbon fiber; CFD, computational fluid dynamics; CFRC, carbon fiber reinforced composite; CFP, carbon fiber prepreg; CSF, continuum surface force; DMSO, dimethyl sulfoxide; DoI, degree of impregnation; PA6, polyamide 6; REV, representative elementary volume; RMSD, root mean squared deviation; RoI, rate of impregnation; UD, uni-directional; VOF, volume of fluid. ⇑ Corresponding author. E-mail address: [email protected] (Y.-I. Lim). https://doi.org/10.1016/j.ces.2018.04.010 0009-2509/Ó 2018 Elsevier Ltd. All rights reserved.
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65
Nomenclature a, b, c A C2 Ca cp dH DoICFD DoICum DoIfunc E F ! g h H ! i I ! j k K L LP M n ! n N P R R2 Re t T T0
constants of DoI function cross sectional area (m2) inertial resistance coefficient (–) capillary number (–) heat capacity (J/kg/K) hydraulic diameter (m) DoI obtained from micro-scale CFD cumulative DoI DoI function obtained by regression of DoICFD energy (J) source term force (N) gravity vector (=9.81 m/s2) penetration depth (m) temperature dependence term of viscosity (–) unit direction vector of x-axis identity tensor (–) unit direction vector of y-axis thermal conductivity (W/m/K) permeability (m2) number index of times length of wetted perimeter (m) number index of pressures power-law index normal vector (–) number index of computational cells in 2D CFD domain gauge pressure (Pa) number index of computational cells in tow domain correlation coefficient (–) Reynolds number (–) time (s) temperature (K or °C) zero temperature (K)
Tref u V x xend
reference temperature (K) velocity (m/s) molar volume (cm3/mol) axial length (m) end of impregnation die length (m)
Greek letters a volume fraction (–) b slope limiter value (–) c shear rate (1/s) e voidage (–) g viscosity (kg/m/s) g0 zero shear viscosity (kg/m/s) g1 infinite shear viscosity (kg/m/s) h contact angle (rad or °) j surface curvature (1/m) s time constant (s) q density (kg/m3) r surface tension (N/m) s stress tensor (Pa) U molar volume constant (–) Subscripts a activation D DMSO eff effective r resin surf surface tension T tow w wall
of the micro- and macro-scale CFD results. Combining the multi-scale models gives a potential to predict the uniformity of the transverse resin amount in the final UD-CFP product. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction High strength fiber-reinforced composite materials have been widely applied in many structural and mechanical systems such as aerospace, automotive, sporting goods, biomedical, building, and infrastructure (Chang and Lees, 1988; Han et al., 2015; Ngo et al., 2017; Rodríguez-Tembleque and Aliabadi, 2016). Carbon fiber-reinforced composites (CFRCs) with thermoplastic matrices show excellent properties including light weight, high strength and modulus, good electrical and thermal conductivity, environmental resistance, and recyclability (Chang and Lees, 1988; Luo et al., 2014; Sakaguchi et al., 2000). However, the use of CFRCs with thermoplastic matrices suffers from drawbacks regarding their manufacture in industrial applications. Since the molten viscosity of thermoplastic resins is higher than that of thermoset resins, there are difficulties in impregnating thermoplastic resins into carbon fiber (CF) tows (Sakaguchi et al., 2000; Ye et al., 1995), where the tow is a bundle of strands (or filaments) having a diameter of a few micrometers. Understanding the impregnation mechanisms at the micro-scale is important to achieve a good quality of CFRCs (Hou et al., 1998; Ye et al., 1995). The prepreg used to fabricate CFRC intermediates (Ngo et al., 2017) is ready to lay into the mold without the addition of any resin,
because of pre-impregnation with a resin. Many researchers have presented impregnation and compaction behaviors of intermediate thermoplastic composites for commingled yarn (Hamada et al., 1993; Klinkmüller et al., 1995; Van West et al., 1991), powderimpregnated unidirectional fiber bundles (Kim et al., 1989), and prepreg tapes (Lee and Springer, 1987). The axial and transverse permeabilities were dependent on the microscopic characteristics and orientation of the fiber network (Gutowski et al., 1987). A comprehensive model describing the impregnation and consolidation mechanisms was developed for thermoplastic CFRCs under the assumption of an isothermal Newtonian fluid, porous media tow, and absence of capillary effect between the fluid and CF tow (Hou et al., 1998; Ye et al., 1995). Bijsterbosch and Gaymans (1993) studied the degree of impregnation (DoI) of glass fiber with polyamide 6 (PA6) in an impregnation bath, where the DoI experimentally obtained for various process and material parameters was expressed as a function of the square roots of time and pressure (Bijsterbosch and Gaymans, 1993). A resin permeability to CF filaments obtained at the micro-scale was used to calculate the resin velocity inside CFRC tows at the macro-scale, assuming that the CF filaments were entirely impregnated (Ngo et al., 2017). However, few studies have addressed the influence of the microscopic impregnation mechanism on the quality of CFRCs at the macro-scale.
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The fiber impregnation process by melting the polymer matrix is intrinsically heterogeneous at all length scales. When modeling the fluid flow at macro-scale, it is generally not possible to account for all pertinent scales in a single simulation. The heterogeneous effects of smaller scales are introduced into the homogeneous macro-scale by constitutive relations (Weinan, 2011). By simultaneously considering models at different scales, multi-scale modeling approach has the potential to share the efficiency of macroscale models as well as the accuracy of micro-scale models (Weinan, 2011; Li et al., 2013). A central issue that invariably arises in the literature is how the understanding of mechanisms at the micro-scale can enable predictions of functional behavior at the macro-scale (Yip and Short, 2013). With the increase of computational power nowadays, computational fluid dynamics (CFD) has become a viable tool for simulating multi-physics and complex geometry problems at a wide range of fluid scales (Tu et al., 2012). In small scale flow, CFD is seen to yield very detailed flow field solutions of all relevant variables such as velocities, pressure, temperature, concentrations, and volume fraction. Such detailed solutions are expected to achieve a better understanding of phenomena occurring in a micro-scale impregnation process. For larger scales in industrial applications, process and design optimization is sped up at low cost through the introduction of CFD tools (Joshi and Ranade, 2003). A multi-scale steady-state CFD simulation sequentially approaching from small to large scales of a structured-packing column was presented to examine the liquid holdup and pressure drop (Raynal and Royon-Lebeaud, 2007). A structure-based drag correlation obtained at meso-scale was incorporated into the gas-liquid Eulerian CFD model of bubble columns (Yang et al., 2011). The volume of fluid CFD at the micro- and meso-scales, and porous media Eulerian CFD at the column-scale were combined to investigate hydrodynamics of three different types of structured-packings (Li et al., 2016). A dual-scale porous media model for microscopic and macroscopic resin flows was proposed to investigate the DoI of a thermoplastic matrix composite during the pultrusion process (Kim et al., 2001). A multi-scale CFD model including both the full 3D impregnation die and the microscopic creeping flow in a representative element volume (REV) was used to optimize operating conditions such as tow speed and resin flow rate (Ngo et al., 2017). Unfortunately, these multi-scale CFD approaches focused only on evaluating a physical property (i.e., pressure drop, film thickness, or permeability) at micro-scale CFD for use in macro-scale CFD. There remains a need for an efficient method that can predict functional behaviors at the macro-scale from microscopic mechanisms. The product quality in the process-scale has to be obtained by an extensive coupling between the micro- and macro-scale CFD simulations. This article aims to develop a multi-scale CFD model to predict the DoI of an impregnation die for the production of a thermoplastic unidirectional carbon fiber prepreg (UD-CFP). In the micro-scale transient CFD within a REV of CF tow, impregnation mechanisms are found and the microscopic DoI is calculated at various resin inlet pressures. The macro-scale 3D CFD covers the entire impregnation die and identifies the pressure distribution along the die length. Combining the micro-scale CFD and macro-scale CFD simulation results, a cumulative DoI at the end of the impregnation die is obtained and compared to experimental data for the relative resin amount of the final UD-CFP tape. This article reveals how to connect the micro- and macro-scale CFDs to obtain process-scale performance required for industrial applications. 2. Material properties and geometry Fifteen CF tows were impregnated with resin in a prepreg die. One tow consisted of 12,000 CF filaments, and polyamide 6 (PA6)
was used as the thermoplastic resin. The micro- and macro-scale CFDs took into account 128 filaments of CF and the entire impregnation die, respectively. Before the tow entered the impregnation die, the void area of the tow was filled with dimethyl sulfoxide (DMSO), which is an organic solvent used in the upstream process of UD-CFP. DMSO in the liquid phase was replaced by the resin (PA6) during impregnation and volatiles at a die operating temperature over 220 °C. 2.1. Material properties The material properties of PA6, DMSO, and the CF tow are listed in Table 1. The density, heat capacity, and thermal conductivity of the resin and DMSO are given at 260 °C, and 189 °C. The nonNewtonian resin viscosity (g) was modeled by Carreau’s formula (Tanner, 2000) as a function of shear rate (c) and temperature (T):
n1
g ¼ HðTÞ g1 þ ðg0 g1 Þ½1 þ c2 s2 2
ð1Þ
where g0, g1, s, and n are the zero shear viscosity, infinite shear viscosity, time constant, and power-law index, respectively. The temperature dependence term, H(T), is expressed as:
1 1 HðTÞ ¼ exp T a T T 0 T ref T 0
ð2Þ
where Ta, T0, and Tref are the activation, zero, and reference temperatures, respectively. The model parameters (Table 2) were obtained from the regression of Carreau’s model with the viscosity experimentally measured at four temperatures (T = 220, 240, 260, and 280 °C) and shear rate values ranging from 10 to 1000 s1 (Ngo et al., 2017). 2.2. Geometry and meshing Two CFD domains are considered in the multi-scale simulation. The macro-scale 3D domain is the impregnation die (see Fig. 1) including a porous media moving tow domain and a resin fluid domain. The geometry is asymmetric because the resin is injected through a pipe introduced from the left. The micro-scale domain is
Table 1 Material properties of PA6, DMSO, and CF tow.
a b
Resin (PA6)
DMSO
CF tow
Density (q, kg/m3) Heat capacity (cp, J/kg/K) Thermal conductivity (k, W/m/K) Viscosity (g, kg/m/s)
932a 3728a 0.264a Carreau’s model
1800 1075 0.357 –
Voidage (e)
–
974b 2177b 0.0454b 0.55 103b –
0.408
Values at 260 °C. Values at 189 °C.
Table 2 Parameter values of Carreau’s viscosity model as function of shear rate (c) and temperature (T) for PA6 (Ngo et al., 2017). Parameter
Value
Time constant (s, s) Power-law index (n) Zero shear viscosity (g0 , kg/m/s) Infinite shear viscosity (g1 , kg/m/s) Reference temperature (T ref , K) Zero temperature (T 0 , K) Activation temperature (Ta, K)
0.0058 3.556 330.9 1.89 493 0 5148
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Fig. 1. Geometry and meshing of impregnation die in macro-scale 3D CFD.
a 2D microscopic tow structure with randomly-distributed CF filaments 7 lm in diameter. The micro-scale 2D domain illustrates a representative elementary volume (REV) of the tow domain (see Fig. 2). The voidage of the REV (e = 0.408) is the same as that of the CF tow. It is assumed that the micro-scale REV is periodically repeated in the macroscopic tow domain. One tow has a width of 6 mm and a thickness of 0.13 mm. The die width is 100 mm. In the macro-scale CFD, the mesh independent test was performed to find the optimal mesh number in the range from 1 to 11 million cells. A mesh structure of 4.2 million cells was chosen to save computational time while retaining the accuracy of the
CFD simulation (Ngo et al., 2017). The resin and tow domains have 2.8 million polyhedral cells and 1.4 million hexahedral cells, respectively. In the micro-scale 2D CFD domain (see Fig. 2), the triangular and rectangular mixed-mesh structure was constructed. The REV has about 27,800 meshes. It is assumed that the resin passes through in one direction from left to right.
3. CFD modeling The macro-scale CFD model is solved in one liquid phase at steady state under the assumption that the impregnation die is fully filled with resin. An unsteady-state liquid-liquid volume of fluid (VOF) model with surface tension force is used in the micro-scale CFD simulation, where the two liquid phases are DMSO and PA6.
3.1. Macro-scale CFD model
Fig. 2. Geometry and meshing of representative elementary volume (REV) in micro-scale 2D CFD.
There are two domains in the continuum macro-scale CFD model: tow and resin domains. The former is assumed to be a porous medium moving at constant speed, uT, called the tow speed. The porous tow domain is filled 40.8% with resin (e = 0.408, see Table 1). The solid phase of the tow is not modeled, and the microscopic creeping-flow inside the porous medium (Whitaker, 1986) is neglected in this study. However, the resistance force between the liquid (resin) and the solid (porous media) was considered using Darcy’s law (Ngo et al., 2017). The continuity, momentum, and energy equations are applied to the continuous resin phase in both the tow and resin domains. Considering a moving coordinate system of the tow domain with ! a linear velocity ( uT ), the fluid phase inside the tow is transformed from a stationary frame to a moving frame. The governing equations of the macro-scale steady state CFD model are listed in Table 3. The continuity equations of the resin and tow domains are shown in Eqs. (T1) and (T2). The fluid is assumed to be incompressible (i.e., constant density, q).
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S.I. Ngo et al. / Chemical Engineering Science 185 (2018) 64–75 Table 3 Macro-scale CFD model in resin and tow domains. ! ! Continuity equation (resin domain): r ðq u Þ ¼ 0 ! ! ! (tow domain): r ðqð u uT ÞÞ ¼ 0 ! ! ! !! ! Momentum equation (resin domain): r ðq u u Þ ¼ r P þ r ðsÞ þ q g ! ! ! ! ! ! ! ! ! (tow domain): r ðqð u uT Þð u uT ÞÞ ¼ r P þ r ðsÞ þ q g þ F ! where F in the tow domain includes the forces of porous resistance and inertia: ! ! ! ! ! ! ! F ¼ Kg ð u uT Þ C 2 12 qj u uT jð u uT Þ
(T1) (T2) (T3) (T4) (T5)
s is defined as ! !! ! ! u þ r u T Þ 23 r u I s ¼ g½ð r !
(T6)
! ! ! ! ! ! ! Energy equation (resin domain): qcp u r T þ u r P ¼ r ðkr r T þ s u Þ ! ! ! ! ! ! ! ! ! ! (tow domain): qcp ð u uT Þ r T T þ ð u uT Þ r P ¼ r ðkT;eff r T T þ ðs ð u uT ÞÞÞ where kT;eff ¼ ekr þ ð1 eÞkT
The momentum conservation equations for the resin and tow domains are expressed in Eqs. (T3) and (T4), respectively. P is the ! static pressure, q g is the gravitational force, and s is the stress ! tensor. F is the source term including resistance forces in Eq. (T5), where K is the permeability, C 2 is the inertial resistance coef! ! ficient, and j u uT j is the magnitude of the relative velocity. The first term on the right-hand side of Eq. (T5) refers to the viscous loss defined by Darcy’s law. The second term is the inertial loss. The viscous resistance factor is the inverse of the permeability (1/K). In this study, the inertial resistance coefficient is neglected in the thin tow domain (C2 = 0) because the resin flow rate inside the tow is very low (Ngo et al., 2017). The stress tensor (s) is expressed by Eq. (T6), where I is the identity tensor and ! !T ðr u þ ðr u Þ Þ is the strain rate tensor for the resin flow. The energy transport equation is introduced to the resin phase of the two domains. Under the assumption of thermal equilibrium inside the porous tow, the steady-state energy conservation equations of the resin and tow domains are given in Eqs. (T7) and (T8), respectively. Here, TT and T are the resin temperatures in the tow and resin domains, respectively, cp is the heat capacity of the resin, and kr is the thermal conductivity of the resin (see Table 1). The effective thermal conductivity (kT,eff) in the tow domain is calculated by Eq. (T9), where kT is the thermal conductivity of the tow. The second terms on the right-hand sides of Eqs. (T7) and (T8) represent the viscous energy dissipation. 3.2. Micro-scale 2D VOF model The volume of fluid (VOF) model relies on the assumption that two phases do not interpenetrate and is applied to the micro-scale CFD. A liquid-liquid system including DMSO and PA6 is considered in the VOF model at unsteady-state. Since the boiling point of DMSO is 189 °C, DMSO evaporates at the operating temperature (TT 260 °C). However, it is assumed that the microscopic tow is initially filled with liquid DMSO at a slightly elevated pressure (about 1.5 atm). The resin (PA6) is impregnated by extruding DMSO. The 128 CF filaments randomly distributed are supposed to be a rigid body without deformation. Thus, the applied pressure, length and width of REV during impregnation do not change in this micro-scale CFD. The surface tension effect becomes important when Reynolds number (Re) and Capillary number (Ca) are less than unity.
qu dH g u g Ca ¼ r
Re ¼
(T7) (T8) (T9)
dH = 4A/Lp, where A is the area and Lp is the wetted perimeter. Re and Ca of the resin inside the tow are about 0.044 and 0.32, respectively. In this study, the continuum surface force model (Brackbill et al., 1992) is used as a source term in the momentum equation. The governing equations are summarized in Table 4. Tracking of the interface between phases is accomplished by solving the continuity equation for the incompressible liquid. Assuming there is no mass transfer between the two phases and no source term exists in each phase, the continuity equation for DMSO (denoted as subscript D) is shown in Eq. (T10), where a is the volume frac! tion, and u is the velocity. The volume fraction will not be solved for the primary resin phase (denoted as subscript r), but ar is computed based on the relation of Eq. (T11). In this study, a special interpolation treatment of the volume fraction to the cells that contain the interface between the two phases is used, which is based on the slope limiter second-order reconstruction compressive scheme (ANSYS Fluent Theory Guide, 2016). Here, the slope limiter value (b) is set to b ¼ 2. In the VOF model, a single momentum equation shown in Eq. (T12) is solved throughout the domain, and the resulting velocity field is shared in both two phases. The left hand side of Eq. (T12) includes accumulation and convection of momentum per unit volume. In the right-hand side of Eq. (T12), P is the pressure shared in !! !! ! both phases, q g is the gravity force, gð r u þ r u T Þ is the stress tensor, and F surf is the surface tension force on the interface between two phases as an additional source term. The continuum surface force (CSF) model (Brackbill et al., 1992) is expressed by Eq. (T13), where rrD is the interfacial surface tension between the resin and DMSO, and jr is the surface curvature of the resin as defined in Eq. (T14). jr is a divergence of the unit ! ! normal vector for the gradient of resin volume fraction ( n r ¼ r ar ). By combining Young’s equation of surface tension and contact angle (Young, 1805), Fokews’ theory of adsorptive adhesion
Table 4 Micro-scale two-phase volume of fluid (VOF) model. ! ! aD Continuity equation (DMSO): @@t þ r ðaD u D Þ ¼ 0 (resin): aD þ ar ¼ 1 Single momentum equation: ! ! ! !! !!T ! !! ! @ @t ðq u Þ þ r ðq u u Þ ¼ r P þ r ½gð r u þ r u Þ þ q g þ F surf ! ar where surface tension force (F surf ): F surf ¼ rrD 1qðjqr r þqr Þ 2 D ! ! r ar Þ surface curvature (jr): jr ¼ r ð !
(T10) (T11) (T12) (T13) (T14)
j r ar j
ð3Þ
; and r are the mean velocity, mean viscosity, and sur; g where u face tension, respectively. dH is the hydraulic diameter defined as
! ! ! ! @ ðqEÞ þ r ð u ðqE þ PÞÞ ¼ r ðk r TÞ Single energy equation: @t Single properties (density): q ¼ ar qr þ aD qD (viscosity): g ¼ ar gr þ aD gD (energy): E ¼ ar qr Er þqaD qD ED (mass-averaged energy)
(conductivity): k ¼ ar kr þ aD kD
(T15) (T16) (T17) (T18) (T19)
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(Fowkes, 1964), and the Girifalco-Good correlation (Girifalco and Good, 1957), it is possible to calculate the interfacial surface tension (rrD) using the following equation:
rrD ¼ rr
ð1 þ coshÞ2 4U2
!
cosh
ð4Þ
where U is a constant defined as follows (Girifalco and Good, 1957):
U¼
4ðV r V D Þ1=3
ð5Þ
1=3 2
½V r1=3 þ V D
V r and V D are the molar volumes of the resin and DMSO, respectively. The contact angle (h) between the resin and DMSO is calculated from (Girifalco and Good, 1957):
12 rr cosh 2U 1
ð6Þ
rD
Table 5 reports the physical properties of the resin and DMSO, and the contact angle and interfacial surface tension between the two liquid phases estimated from Eqs. (4)–(6). The surface tensions of PA6 and DMSO are calculated by the group contribution method (Baum, 1997). Since the difference between the surface tensions of PA6 and DMSO is small, rrD is small. The contact angle between the two liquid phases is estimated to h = 33.5°. The energy equation is shared for the two phases in a given control volume, as shown in Eq. (T15). Single properties such as density (q), viscosity (g), energy (E), and thermal conductivity (k) are employed to solve the VOF model, as shown in Eqs. (T16–T19). Energy (E) is treated as a mass-averaged value in Eq. (T18), where Er and ED are calculated with the specific heat capacity (cp) and the shared temperature (T). In order to include the resin adhesion effect on the wall of the CF filament, a dynamic boundary condition is applied to adjust the surface curvature (jw) on the wall (Brackbill et al., 1992).
!
n wÞ jw ¼ r ð!
ð7Þ
die. The temperature of the wall is the same as the inlet resin temperature. In the micro-scale 2D CFD, the resin enters at 260 °C and at eight different pressures from 2 to 9 104 Pa, which are identified from the macro-scale CFD in the entire impregnation domain (see Section 4.2). It is assumed that the temperature of the CF surface (250 °C) is a little lower than that of the resin, because the tow comes in at 220 °C in the macro-scale domain (see Table 6). 3.4. Multi-scale coupling strategy The present multi-scale model is divided into four main parts: (i) macro-scale CFD, (ii) micro-scale CFD, (iii) regression with the micro-scale CFD data, and (iv) process-scale simulation, as illustrated in Fig. 4. The multi-scale model aims to obtain the degree of impregnation (DoI) and rate of impregnation (RoI) as a function of pressure (P) and time (t) in the production of UD-CFP. First, the macro-scale 3D CFD is performed for the entire impregnation die considering the tow as a porous media with a transverse permeability of 5.61 1013 m2 (Ngo et al., 2017), that is about twice higher than that of a laminate graphite composite (Gutowski et al., 1987). The range of resin pressures induced in the moving tow is found in the macro-scale CFD simulation. Then, the microscale 2D CFD simulation at unsteady-state is repeated at various pressure values (P 1 P M ). Each simulation provides DoI with respect to t, and DoI data at M pressures are obtained in the micro-scale CFD. The DoI at a given time is calculated by the ratio of the area occupied by resin (Ar) to the total area (A) (Kim et al., 1989):
DoICFD ¼
Ar ¼ A
PN
a
i¼1 Ai r;i PN i¼1 Ai
ð9Þ
where DoICFD is the DoI obtained from CFD, N is the number of computational cells, Ai is the area of cell i, and ar,i is the resin volume fraction within cell i. ar,i = 1.0 for a cell fully filled with resin and ar,i has a value between 0 and 1 for a cell lying at the interface of the resin and DMSO.
! where n w is the unit normal vector to the tangent to the surface that makes with the CF solid surface. When the contact angle between the resin and CF surface (hw) is known, the unit normal vector is expressed as follows:
! ! ! n w ¼ sin hw i w þ cos hw j w
ð8Þ
! ! where i w and j w are the unit vectors tangential and normal to the CF surface, respectively. Fig. 3 illustrates the unit normal vector of resin on the CF wall. The contact angle (hw ) of PA6 on the CF surface is around 42° (Scott, 1997). 3.3. Boundary conditions In the macro-scale 3D CFD, the resin enters at 260 °C and 26.65 g/min. The tow (220 °C) at the inlet moves with a speed of 2 m/min. The gauge pressure (P) is set to 0 Pa at the exit of the impregnation
Fig. 3. Contact angle and normal vector of resin on CF wall.
Table 5 Physical properties of PA6 and DMSO. Parameters
Symbol (unit)
Value
Reference
Surface tension of PA6 with air Surface tension of DMSO with air Molar volume of PA6 Molar volume of DMSO Contact angle between PA6 and DMSO Molar volume constant Interfacial surface tension
rr ðN=mÞ rD ðN=mÞ
55.7 10-3 65.7 10-3 104.39 71.03 0.83 0.996 0.769 10-3
Baum (1997)
V r ðcm3 =molÞ V D ðcm3 =molÞ cosh
U
rrD ðN=mÞ
Fowkes (1964), Girifalco and Good (1957), Young (1805)
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Table 6 Boundary conditions of macro- and micro-scale CFD models. Macro-scale CFD model Boundary area
T (°C)
P (Pa)
! j uT j (m/min)
Mass (g/min)
Resin inlet Tow inlet Resin/tow outlet Wall
260 220 260 260
– – 0 –
– 2 – –
26.65 – – –
Micro-scale CFD model Boundary area
T (°C)
P (Pa)
Resin inlet
260
Resin outlet CF surface
260 250
ð2 9Þ 104 0 –
–
Resin pressure range (P1 ~ PM) DoIcum for 15 tows
Micro-scale 2D CFD (Unsteady-state)
4
1
Micro-scale CFD at P1
Macro-scale 3D CFD (Steady-state)
Micro-scale CFD at P2
Process-scale Simulation
Positions (xi)
Micro-scale CFD at P3 …
Pressures (Pi)
Micro-scale CFD at PM 2
DoICFD
3
Data fitting of micro-scale CFD
DoIfunc and RoI
Fig. 4. Coupling strategy of multi-scale CFD models.
x uT
Once a set of DoICFD data is acquired from the micro-scale CFD, a DoI function (DoIfunc) with respect to pressure P and t is obtained by means of regression with this data set. Since the DoI of fiberreinforced composites was expressed as a function of the square pffiffiffiffiffi root of both time and pressure ( Pt) (Bijsterbosch and Gaymans, 1993; Carleton and Nelson, 1994; Kim et al., 1989), the global funcpffiffiffi pffiffi tion is proposed in terms of t and P:
Substituting Eq. (13) in Eq. (10), the final DoI function is obtained as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P DoIfunc ðP; tÞ ¼ a tþb þc 1 106 1 106
The cumulative DoI (DoIcum) for each tow from 1 to 15 is calculated by the summation of DoI in each cell:
ð10Þ
where a, b, and c are the model constants, which are found by minimizing the root mean squared deviation (RMSD) between DoICFD and DoIfunc:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PM PL 2 i¼1 j¼1 ðDoICFD;ij DoIfunc ðP i ; t j ÞÞ RMSD ¼ ML
ð11Þ
where M and L are the number of sampling points for pressure and time, respectively. The rate of impregnation (RoI) can be obtained by the partial differentiation of DoIfunc with respect to time (t) at a constant pressure (P):
qffiffiffiffiffiffiffiffiffiffi
P a
@ðDoIfunc Þ
1106 p ffiffi RoIðP; tÞ ¼ ¼
@t 2 t P
t¼
ð13Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P x P þb þc 1 106 uT 1 106
DoIfunc ðP; xÞ ¼ a
DoIcum ¼ 2ðDoI1 ðP1 ; x1 Þ þ
R1 X ½DoIfunc ðPi ; xiþ1 Þ DoIfunc ðPi ; xi ÞÞ
ð14Þ
ð15Þ
i¼2
where DoI1 is the initial value caused by the surface coverage and capillary effects (see Section 4.1), R is the number of computational cells located in the transverse middle of each tow (see Fig. 1b) along P þP
the tow moving direction, and Pð¼ i 2 iþ1 Þ is the mean pressure for two adjacent cells. It is noted that the resin seeps into the middle of the tow from both the top and bottom sides (see Fig. 1b). Therefore, DoIacc is twice the one-sided DoI in the micro-scale CFD model. Finally, the DoI of the 15 CF tows with respect to the axial length of impregnation die is obtained by combining the micro- and macroscale CFD.
ð12Þ
Since the 15 tows are considered as a porous medium, the degree of impregnation cannot be obtained in the macro-scale CFD. However, the pressure distribution inside the 15 tows is identified. The position (x) of the tow is converted into time (t), using the constant tow speed (uT).
4. Results and discussions The continuum micro- and macro-scale CFD models were solved using a finite volume-based solver, ANSYS Fluent R15.0 (ANSYS Inc., USA, 2016). The calculation was performed on a workstation with 24 cores (2.7 GHz CPU) and 128 GB RAM. An adaptive
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time step size was applied for the micro-scale CFD model. The minimum and maximum time step sizes are 1 107 and 1 103 (s), respectively. 4.1. Micro-scale CFD results The transient micro-scale CFD simulation was carried out at eight pressures from 2 104 Pa to 9 104 Pa. As mentioned earlier, the pressure range is obtained from the macro-scale CFD (see Section 4.2). The average viscosity of the resin (0.5 103 Pa s g 1100 103 Pa s) was 8.07 10-3 Pa s with a mean shear rate of 182 s-1 (0.4 s-1 c 1150 s-1) in the micro-scale domain at T = 260 °C. The impregnation mechanism is identified based on the DoI calculated from the micro-scale CFD. Fig. 5 shows the DoI profile at pffiffi P = 2 104 Pa with respect to t within an initial impregnation time (0.0 s t 1.0 s). This initial profile can be divided into three parts: (i) resin coverage on the CF surface, (ii) capillary permeation to CF (Carleton and Nelson, 1994), and (iii) penetration through the inter-CF filaments (Klinkmüller et al., 1995). The capillary impreg-
nation is caused by the attractive force between carbon fiber and Nylon 6 resin, while the penetration to CF filaments is caused by the pressure difference between the inlet and outlet of the resin. The capillary effect dominates at the beginning of impregnation, while the penetration effect plays a major role after the capillary effect terminates. The resin coverage on CF surface takes place in pffiffi a short time of about 4.0 10-4 s ( t ¼ 0:02 s0:5 ), as shown in Fig. 5. The capillary permeation of resin is observed until t = pffiffi 0.013 s ( t ¼ 0:114 s0:5 ). After the capillary permeation, the CF pffiffi tow is almost linearly impregnated with t . Fig. 6 shows five snapshots of the resin volume fraction (ar) at t = 0.0001, 0.01, 1, 10, and 100 s for an inlet pressure of P = 9 104 Pa. The resin coverage on the CF surface is observed at t = 0.0001 s (see Fig. 6a), which is faster than that at P = 2 104 Pa. The capillary permeation is also seen at t = 0.001 s in Fig. 6b. The resin flow reaches the border of the CF tow at about t = 100 s (see Fig. 6e). Fig. 7 illustrates the effect of operating pressure on the penetration depth of the resin at a flow time of 20 s. For the linear increment of operating pressures at 3 104, 6 104, and 9 104 Pa, the ratio of the penetration depths is h1 : h2 : h3 ¼ 1 : 1:175 : 1:65. Kim et al. experimentally reported an increase of DoI with increasing pressure (Kim et al., 1989). The DoI versus time obtained from CFD (DoICFD) for the eight pressures was regressed with DoIfunc in Eq. (10), and its model parameters were found to be a = 0.1497, b = 0.6137, and c = 0.0704. Fig. 8a compares the two DoIs and the correlation coefficient (R2) between the two DoIs is 0.9673 (see Fig. 8b). The DoIfunc is applicable at pressures lower than 9 104 Pa. It is worth noting that the first, second, and third terms of the DoIfunc in Eq. (10) correspond to the penetration, capillary permeation, and surface coverage, respectively. Thus, the DoI of surface coverage is estimated to about 0.07, which is the third parameter (c) of Eq. (10). 4.2. Macro-scale CFD results
Fig. 5. Impregnation mechanism from micro-scale CFD at P = 2 104 Pa.
(a) t=0.0001 s
(b) t=0.001 s
The macro-scale CFD provides the micro-scale CFD and the process-scale simulation with the resin pressure range (from P1 to PM) and the resin pressure (Pi) applied to the tow in the axial direction (xi), respectively (see Fig. 4). Fig. 9 shows the pressure profiles of 15 tows according to the impregnation die length. Here, the gauge pressure was collected from an upper cell located in the
(c) t=1 s
(d) t=10 s
Fig. 6. Contours of resin volume fraction (ar) at five flow times (P = 9 104 Pa).
(e) t=100 s
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(a) P = 3×104 Pa
(b) P = 6 ×104 Pa h1
(c) P = 9 ×104 Pa
h2
h3
Fig. 7. Contours of resin volume fraction (ar) at t ¼ 20 s for three inlet pressures (P).
Fig. 8. DoI with respect to P and
pffiffi t , and correlation between DoICFD and DoIfunc.
transverse middle of each tow (see Fig. 1b). Point ➀ is the resin injection area from the bottom to the tow domain. Point ➁ is the outlet of both tow and resin. All 15 tows have a similar pressure profile ranging from 0 to 8 104 Pa. A maximum pressure of 7.5 104 Pa is observed near the bottle neck of x = 3 mm for every tow. RoI in Eq. (12) is estimated to 0.21 min-1 at P = 6 104 Pa, and x = 0.015 m (t = 0.45 s). Note that the cumulative DoI (DoIcum) in Eq. (15) is needed because the pressure (P) varies with position (x). 4.3. Process-scale simulation results The final stage of the present multi-scale model is the prediction of DoI at the process-scale with the data collected from the
micro- and macro-scale CFD simulations. Eq. (15) is applied to predict the DoI of each tow with respect to the die length. DoI1(P1,x1) was 0.105, which included the resin coverage on the surface and the capillary permeation at P = 6 104 Pa (see Fig. 9). Fig. 10a shows 15 DoIcum profiles along the impregnation die length (x). DoI at the end of the impregnation die (DoIcum(xend)) is seen in Fig. 10b; it is not symmetric and uniform along the die width. In fact, DoIcum(xend) is strongly related to the quality of the final UD-CFP product. The impregnation die should be designed to have a symmetric and uniform DoIcum(xend). Fig. 10 demonstrates that the multi-scale CFD model is able to reproduce the resin uniformity of the impregnation process. Ten segments of the final UD-CFP tape produced in this impregnation die were sintered in a muffle furnace (AJ-MB8, Ajeon
S.I. Ngo et al. / Chemical Engineering Science 185 (2018) 64–75
73
Fig. 9. Pressure (P) profiles of 15 tows along impregnation die length (x) in macro-scale CFD simulation.
Fig. 10. Cumulative degree of impregnation (DoIcum) along impregnation die length (x) for 15 tows.
Heating Industrial Co., Ltd., Korea), and the resin weight of each segment was measured (Ngo et al., 2017). Since the resin weight and DoIcum(xend) cannot be compared directly, a relative resin
amount and a relative DoIcum(xend) were obtained from the ratio of original values to their average. Fig. 11 compares the three relative values scaled in the range of 0.942 y 1.047, which is the
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S.I. Ngo et al. / Chemical Engineering Science 185 (2018) 64–75
scale simulation approach provided a promising tool to investigate the effect of process geometry on the quality of the final UD-CFP product. This micro-scale CFD under the assumption of rigid CF filaments did not consider deformation of the microscopic CF structure by mechanical forces. DoI would be more accurately predicted, if the clustering and spreading of CF filaments during impregnation were taken into account. Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science ICT and Future Planning (Grant number: NRF-2016R1A2B4010423) in South Korea.
Fig. 11. Comparison between CFD results for relative DoI and experimental data for relative resin amount of final UD-CFP.
lower and upper limits of the relative resin amount obtained by experiment, with respect to the UD-tape width (or die width). The relative resin amount is asymmetric in the UD-tape width, and is higher in the center than at the two edges. The relative resin velocity obtained over the entire cross-sectional area at xend (Ngo et al., 2017) has a similar shape to the experimental data. The relative DoIcum(xend) is relatively lower in the center than at the edges unlike the experimental data. The UD-CFP passes through a grooved area for further resin impregnation and spreading. Then, it is compressed by the pressing roller. After cooling to room temperature, the final UD-tape is produced (Ngo et al., 2017). Since the relative resin amount was measured for the final UD-tape, the DoI at xend obtained from the multi-scale CFD simulation has some discrepancy with the experimental data. Further investigation including the grooved area and the pressing roller would be helpful to correctly predict the distribution of resin along the UD-tape width.
5. Conclusions A multi-scale simulation approach was presented to investigate thermoplastic resin impregnation into carbon fiber (CF) tows. The approach included (i) micro-scale transient computational fluid dynamics (CFD) of the two-dimensional (2D) representative elementary volume (REV) of a CF tow, (ii) macro-scale steady-state CFD of an entire 3D impregnation die with 15 porous media moving tows, and (iii) process-scale simulation to predict cumulative degree of impregnation (DoI) for unidirectional carbon fiber prepreg (UD-CFP) prepreg. Of particular interest were the impregnation mechanisms of resin coverage on the CF surface, capillary permeation, and penetration through CF filaments in the micro-scale CFD simulation. Several micro-scale volume of fluid (VOF) CFD simulations were performed within a range of resin pressures identified from the macro-scale CFD simulation. A global DoI function with respect to pressure and time, inspired by the three impregnation mechanisms, was obtained from the regression of the micro-scale CFD results. Using the data collected from the micro- and macro-scale CFD simulations, a cumulative DoI of the 15 tows with respect to the impregnation die length was predicted in the process-scale simulation. The cumulative DoI at the end of the impregnation die was compared to the relative resin amount measured experimentally. Though there was deviation between the multi-scale CFD prediction and the experimental resin amount, the multi-
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