Flow Modeling & Transport in Fiber Mats used as Reinforcement in Polymer Composites

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1 Flow Modeling & Transport in Fiber Mats used as Reinforcement in Polymer Composites Krishna M. Pillai Associate Professor Department of Mechanical Engineering

2 A few facts about Milwaukee and Wisconsin: Home of Harley-Davidson Motorcycles Home of Millers Beer Main base of the football team Greenbay Packers Famous for cheese Home of world famous Summer-Fest Award winning Arts Museum next to the lake

3 Outline. Introduction. Modeling Flow Variables during Unsaturated Flow 3. Single-Scale and Dual-Scale Porous Media 4. Modeling Bubble/Void Creation & Migration during Unsaturated Flow 5. Challenge of Unified theory for flow and Bubble/Void during Unsaturated Flow 6. Summary

Assisted Resin Transfer Molding (VARTM),

4 Liquid Composite Molding (LCM) Polymer Composites: Polymers + Reinforcing Fibers Properties of Polymer Composites: light weight, strong, stiff, corrosion resistant LCM: a technology to make polymer composites Examples of LCM: Resin Transfer Molding (RTM), Structural Reaction Injection Molding (SRIM),Vacuum Assisted Resin Transfer Molding (VARTM), and Seemann Composites Resin Infusion Molding Process (SCRIMP).

5 Process steps in Resin Transfer Molding (RTM). Preform Manufacturing 5. Demolding and Final Processing. Lay-up and Draping 4. Resin Injection and Cure 3. Mold Closure

6 Mold-Filling Simulation in RTM Flow-front progress Temperature distribution

7 Mold-Filling Simulation in RTM Mold fill-time Resin Velocity Distribution

8 Advantages of mold-filling simulation in RTM Passenger mini-van cross-member Optimize location of inlet gates and vents Monitor mold fill-time and cure Predict pressure and temperature buildup in the mold Study the effect of different fibrous reinforcements on filling

9 Flow through a porous medium: averaging of flow variables fiber averaging volume Volume average: r q = V V f T = V r v V f dv T dv Pore average: p = c = V f V f p V f c V f dv dv

10 Effects of averaging on balance equations in a porous medium Microscopic Energy Balance: ρ C p T t + ρ C p r v T = ( k T ) + ρ ( c,t ) + τ : D H R f c where D = r r [ ] T ( v ) + ( v) τ = μ D (Incompressible, Newtonian fluid)

11 Effects of averaging on balance equations in a porous medium Macroscopic Energy balance T t { ε ρ + ( ε ρ )} + ρ q T = C p C p s C p r μ r r K {( + ) T} + ε ρ ( c,t ) + q q k e K Effective thermal conductivity tensor k e D H r r = ε k δ + n b ds + fs εsks δ ε V S fs s ε V S R f c fs r n fs r b ds Dispersive thermal conductivity tensor K D r r = ε ρcp vˆ bdv V f V f where Tˆ = r b T

12 Conventional mold-filling simulation physics for RTM Real flow Simulation (FE/CV scheme) Mass balance v r = 0 Darcy s Law Energy balance T r { ρ ( ε ρ C p) } + ρ C p v T = ( + ) e D C p c r ε + v c = + t { k T} ε ρ H R f ( c, T ) c ε + + s t K Chemical reaction Elliptic pressure equation v r K K = p p = 0 μ μ f c, { ε ( Df + DD ) c} ε ( c T ) Finite Element/Control Volume (FE/CV): ) most widely used approach because its simplicity, efficiency and robustness. ) circumvents the front-tracking related problems associated with the adaptive mesh regeneration method by using an Eulerian fixed mesh.

13 Typical experiments to measure in-plane fiber-mat permeability radial flow mold -D flow mold constant pressure fluid supply constant injection-rate fluid supply -D flow: r K v = p μ K = Q A μ Δ L P

14 Experimental Investigation of the -D D unsaturated flow x

15 Mold and Flow Details Direction of Flow

16 Modeling Flow Variables during Unsaturated Flow

17 Inlet pressure history for -D D constant injection-rate experiment Pin Pin x f x Continuity: d u d x = 0 t Darcy s law: μ K d P d x = u Pin u K μ = ε t Front speed: d x f ε d t = u

18 Flow-front and Inlet-Pressure prediction for -D Flow: Random Fiber Mats Theoretical Pressure /Pa Experimental Time/sec

Flow-front and Inlet-Pressure prediction for -D Flow: Woven or Stitched Mat 80000 60000 40000

19 Flow-front and Inlet-Pressure prediction for -D Flow: Woven or Stitched Mat Pressure /Pa Experimental Theoretical Time/sec

20 Inlet pressure history: constant injection-rate -D D experiment random mat Pin P in woven or stitched mat Droop t

21 Random fiber mat Single-scale Porous Media One single fiber tow

22 Woven and stitched fiber mats Dual-scale or dual-porosity media Bi-axial woven mat Flow direction Fiber L i tows (Intra-tow Inter-tow space) spaces are absent

23 Travel of a dark colored test liquid in a unidirectional stitched mat injected first with a lighter color liquid Later injected dark-colored liquid Initially injected light-colored liquid Dry fiber mat Courtesy: Prof. Lee, Ohio State University

24 Radial Injection Pattern Random Mat (Single-scale Porous Medium) Biaxial Stitched Mat (Dual-Scale Porous Medium)

25 Two-Color Experiment: Random mat

Molding'', T. Roy, C. Dulmes, and K.M. Pillai, Proceedings of

26 Two-Color Experiment: Stitched mat ``Experimental investigations of the unsaturated flow in Liquid Composite Molding'', T. Roy, C. Dulmes, and K.M. Pillai, Proceedings of the 5th Canadian-International Conference in Vancouver, Canada, August 6-9, 005

27 Role of fiber bundles in creation of unsaturated flow and sink effect in woven/stitched fiber mats Typical micrograph of an RTM part made with stitched/woven fiber mat Schematic: absorption of resin by fiber bundles and creation of sink effect

28 Unsaturated flow in a simple, two-layer model

29 Details of the Mesh and the Control Volumes Outer CVs Inner CVs Finite difference scheme based on the control volume formulation

30 Evolution of temperature distribution y* T y* T y* 3 4 T x* x* x* t = 0. t ch t = t ch t = 4 t ch A Numerical Study of Non-Isothermal Reactive Flow in a Dual- Scale Porous Medium under Partial Saturation, K.M. Pillai and R.S. Jadhav, Numerical Heat Transfer, Part A: Applications, 46: -8, 004.

31 Evolution of cure distribution y* al E E E E E E-05 y* al E-05 y* 3 al x* x* x* t = 0. t ch t = t ch t = 4 t ch

32 Comparison of temperature predictions by the two-layer dual-scale model and the conventional single-scale scale model θ * for dual scale model T* for dual scale model τ for single scale model T*, θ*, τ x x*

33 Volume-averaging in a dual-scale porous media < Β g > = V Vg Β g dv < Β g > g = V g Vg Β g dv Averaging Theorems < Β g > = < Β g > + V Agt Βg < > = < Βg > t t V Agt Β g Β g n gt dv u n gt dv (Pillai & Murthy, 004)

34 Mass Balance: Governing equations for reactive, nonisothermal, unsaturated flow in dual-scale porous media Momentum Balance: K < v g > = S vg > = < Pg > < g Energy Balance: g T g g g C < > ( ρ p) g + < g > T g = th T g + g g H R f c + Qconv Q l ε v < > K < > ε ρ t μ cond Cure Balance: ε g < c g t > g + < v g > < c g > g = D < c g > g + ε g f c + M conv M diff Various source and sink terms: S = V A vg ngt gt da Q Q cond g = ρ g C p g S < T g> < T conv, = qg n V A gt gt da g > gt M M conv diff g = S < cg > < cg > = V A J g ngt gt da gt (Pillai & Murthy, 004)

35 Coupled macro-micro micro approach for modeling the unsaturated flow Governing equations for unsaturated flow through woven fiber mats, Part : Nonisothermal reactive flows, K.M. Pillai and M.S. Munagavalsa, Composites Part A: Applied Science and Manufacturing, v 35, 004, p

36 Governing equations for Governing equations for -D unsaturated isothermal flow D unsaturated isothermal flow Continuity: S o = x u S x u d d o = Darcy s law: u x d P d o o = μ K u x P o o d d = Front speed: u x o f o t d d = ε u t x o f d d = x x f

37 Inlet pressure history for the constant sink case P in = e S S t

38 Constant injection-rate radial flow Previous Experiments Governing Equations d d u r d or d + P r d d o r t u r f = or = = u u or or S

39 Inlet pressure history for the radial flow (analytical solution) s

40 Modeling Bubble Creation & Migration during Unsaturated Flow

41 Bubble Creation and Migration Numerous studies on Bubbles or Voids in RTM Bubbles created due to mechanical trapping of air pockets in porous media during mold filling Bubble creation due to evaporation of volatile compounds in resin is insignificant Capillary number plays an important role in deciding the type of bubble created

42 Void trapping mechanisms & its dependence on Capillary Number Patel, Lee et al. Macrovoids Microvoids % volume of voids % volume of voids Ca Ca = μ V σ Cosθ

43 Bucklet-Leverett Model for Bubble Migration Two-phase flow in porous media Phases are incompressible Capillary Pressure (P c = P nw P w ) is neglected Gravity neglected Fast Flooding regime (large Ca)

44 Mass Balance: Bucklet-Leverett Formulation r ( φ S ) + q = 0 t t r r ( φ S ) + q = 0 a r a S + S a r = Generalized Darcy s Law: r q r = k rel, r K μ r P r q a = k rel, a K μ a P

45 Bucklet-Leverett Equation S t r Sr + U = 0 x st order Quasi-linear Hyperbolic Equation -D Flow where U q + q ε d r d S r a = = signal speed r r r = = fractional flow rate q r q + q a

46 Bucklet-Leverett Model for Bubble Migration t (Sr) br 0.8 t 3 > t > t t t3 > t > t Sr t t 3 t t x(nodes) Typical Pattern of Bubble Migration (Lundstrom & Gebart) Buckley-Leverett Saturation Fronts (Pillai & Advani )

47 Buclet-Leverett Model: Void Distribution from Numerical Simulation If P > P critical, then air bubble moves S a, red = S a S S a, resid a, resid ( P) ( P) (Chui & Glimm et al.)

48 Bucklet-Leverett Model for Bubble Migration Lundstrom, T.S. & Gebart, B.R., Influence of Process Parameters on Void Formation in Resin Transfer Molding, Polymer Composites, 5():5-33, Feb. 994 Pillai, K.M. & Advani, S.G., Modeling of Void Migration in Resin Transfer Molding, Proceedings of 996 IMECE (ASME), page 4, 996. Chui, W.K., Glimm, J., Tangerman, F.M., Jardine, A.P., Madsen, J.S., Donnellan, T.M., and Leek, R., Process Modeling in Resin Transfer Molding as a Method to Enhance Product Quality, SIAM Rev., 39(4):74-77, Dec 997.

49 Conventional -Phase Flow Models to Predict Void Distribution/Saturation during Slow (small Ca) -D Flows Experimentally Measured Saturation Distribution Numerically predicted saturation Breard et al. Numerical Simulation of void Formation in LCM, Composites: Part A 34 (003) 57-53

50 Challenge of Unified theory for flow and Bubble/Void during Unsaturated Flow

51 Bucklet-Leverett Flow: Inlet Pressure history μ r q(t) K random mat V=0000 V=00 V=0 V = woven or stitched mat Single Phase Darcy s Law P inj P in V = μ r μ a t 0 x f (Experimental Observation) B-L Flow is unable to recreate the inlet pressure droop.

52 Unification of Flow Variable & Bubble Creation/Migration Predictions during RTM mold filling Simulation involving `sink model for dual-scale porous media recreate the pressure droop during high Ca flow, but is unable to predict bubble creation/migration. Buclet-Leverett type formulations can predict bubble creation/migration during high Ca flow, but are unable to recreate the pressure droop Conventional -phase flow formulations have been tried to model low Ca flows, but are unable to recreate the pressure droop.

53 Unification of Flow Variable & Bubble Creation/Migration Predictions during RTM mold filling (Cont d) Numerical, algorithmic approach: FE/CV formulation for modeling flow; use of line elements attached to FE nodes to model `sink like disappearance of fluid; saturation = as P > P crit. Example: LIMS of U. of Delaware Several problems: possible inaccuracy in physics; ad-hoc approach; temperature and cure modeling absent; weak experimental validation. Challenge of developing a comprehensive continuum model still unmet.

54 Summary Unsaturated flow fundamentally different in singlescale and dual-scale fiber mats. `Sink model developed to model the unsaturated flow in dual-scale porous media. Both low and high Ca flows can be modeled using - phase approach to predict saturation (bubble) distribution. However recreation of inlet pressure droop not achieved for dual-scale media. Challenge: a need for a comprehensive continuum model for predicting pressure/velocity and saturation/bubbles

55 Thank you for your attention Questions?

56 Permeability Estimation in Fibrous Porous Media Polymer Processing Laboratory

57 p K u =. η = ZZ ZY ZX YZ YY yx XZ XY xx k k k k k k k k k K = z p y p x p k k k k k k k k k u u u ZZ ZY ZX YZ YY yx XZ XY xx z y x η If the selected coordinate directions are along the principal directions, we have = x p x p x p k k k u u u η Tensor nature of the Permeability

58 Permeability of a Flat Fiber Mat Assumption: out-of-plane z direction is a principal direction K = k k k xx yx zx k k k xy yy zy k k k zz xz yz K = k k 0 xx yx k k 0 xy yy 0 0 k zz y x

59 Typical experiments to measure in-plane fiber-mat permeability radial flow mold -D flow mold constant pressure fluid supply constant injection-rate fluid supply

60 Permeability measurement through -D Channel Flow Resin inlet Isotropic Medium: r K v = p μ K = Q A μ Δ L P

61 In-plane Permeability Tensor for Anisotropic Fiber Mats III, y II K = K K xx yx K K xy yy K = K 0 0 K K = K I A D A D / Cos θ where 45 o θ I, x K = K θ = III tan A + D A+ D / Cos θ A D K D II A D K I + K A= III K I K D = III

62 -D Flow Permeability Measuring Process Advantages Simpler physics Easier data analysis Low mold deflection Disadvantages Sliding of Fiber Mats Increased flow on the side edges (racetracking) Three experiments for anisotropic fiber mats

63 Permeability Measurement by Radial flow

64 Permeability tensor measured through transient radial flow In a constant flow-rate experiment, the flow front positions and inlet pressure are functions of time [,]. The in-plane permeability tensor can be expressed as K K = 0 0 K Based on the Darcy s law and continuity equation, the effective permeability can be expressed as where K eff μq R f = KK = ln( ) + ln f πhpin r0 f = f r R K K K K + R R f f Injection port r0 K K y Flow front position and are two principal directions where Rf and Rf are the radial flow front along the principal directions, respectively; r0 is the inlet port radius, K and K are principal permeabilities.. Adams, K.L.,etc. Forced in-plane flow of an epoxy resin in fibrous networks. Polym. Eng. Sci. 986, 6(0),434.. Chan, A.W., Hwang Sun-Tak, Anisotropic in-plane permeability of fabric media. Polym. Eng. Sci. 99, 3(6),33. = x

65 Permeability tensor measured through transient radial flow (Cont.) Data analysis procedure. the ratio α(k/k) can be determined by flow visualization. While the flow pattern forms a fully developed ellipse, the degree of anisotropy α of the flow pattern can be expressed as length of α = length of major minor elliptic elliptic axis axis R = R f f. Since Q, μ, and h are known, we have the following equation μq K eff = KK = πh m where m can be obtained from the slope of the curve Pin versus (ln(rf /r0)+lnf ). 3. The values of K and K are then calculated from the relations: K = αk eff K = K eff α

66 Permeability tensor measured through steady-state radial flow a steady-state radial flow experiment is developed to measure permeability[3]. In this method, pressures at four locations in the flow field, instead of flow front locations, are measured and used for determining the permeabilities.two equations are used to find the permeability tensor the preform. P P P 3 X Y R=3 P 0 θ=0 0 β positions of pressure transducers ( ) ( ) 0 log log = + + K eff Q p p h x x x x μ π α α α α ( ) ( ) 0 log log = K eff Q p p h y y y y μ π α α α α α α where (x0, y0), (x, y) and (x, y) are the coordinates of points P0, P, and P respectively. Q is the flow rate; h is the mold gap thickness. 3. K. Ken Han, etc. (000) Measurements of the permeability of fiber preforms and applications, Composites Science and Technology, 60 (-3):

67 Permeability tensor measured through steady-state radial flow (Cont.) After plugging the steady-state pressure values into these two nonlinear equation, one can solve for the degree of anisotropy α and Keff. The principal permeabilities K and K through relations α = K/K and β is the angle between the principal axis and lab coordinates X-Y as shown in the figure and can be estimated by plugging the third pressure P3 into a coordinate transformation equation.

68 Radial-Flow Permeability Measuring Process Advantages Disadvantages K estimation in a single experiment No race-tracking No sliding of fiber mats Larger Mold Deflection Complex Physics Involved Data Analysis

Pillai, to appear in Journal of Composite Materials.

69 Calibration Devices for Permeability Measuring Setup Radial Flow -D Flow ) A method to estimate the accuracy of radial flow based permeability measuring devices, Hua Tan and Krishna M. Pillai, to appear in Journal of Composite Materials. ) A method to estimate the accuracy of -D flow based permeability measuring devices Hua Tan, Tonmoy Roy and Krishna. M. Pillai, Journal of Composite Materials, 007.

70 Theoretical Models for Permeability of Fiber Beds

71 Permeability Models There are many permeability models that has been proposed so far. The simplest model is to consider the porous medium as a bundle of capillaries [4]. The model takes the general form of φ τ A 3 K = v where Ø is porosity, is tortuosity, and Av is surface area per unit volume. Since the determination of tortuosity is arbitrary, this makes the model difficult to apply. Another capillary model is the network model in which a multitude of capillaries are arranged in the form of a regular network [5]. The wellknown Kozeny-Carman equation is based on this approach where C is Kozeny-Carman equation τ K = CA v 3 φ ( φ) 4. L. Skartsis,etc. Resin flow through fiber beds during composite manufacturing processes, Polymer Engineering and Science, 3,,99 5. Lenormand, etc. Mechanisms of the displacement of one fluid by another in a network of capillary ducts. Journal of Fluid Mechanics, 35, 337, 983.

72 Permeability Models Basing on the Kozeny-Carman equation, many researchers propose the following permeability model for flow along the fiber direction [6] K x = r f ( V ) 4C where Kx is the permeability in the fiber direction, rf is the fiber radius, C is the Kozeny constant to be determined experimentally, and Vf is the fiber volume fraction. For flow transverse to the fiber bundle, Gutowski [7] proposed relation is K z rf = 4C Where Kz is the permeability along the transverse direction, Va is the available fiber volume fraction at which the transverse flow stops. The constant C was measured experimentally and was found to be 0., and Va was determined to be around V f f 3 ( V ) 3 a V f ( V V + ) a 6. Williams, J.G., etc., Liquid flow through aligned fibre beds. Polymer Engineering and Science, 4, 43, Gutowski, T.G. etc, Consolidation experiments for laminate composites. Journal of composite materials,, 7, 650, 987. f

73 Permeability Models Gebart [8] assumed that most of the flow resistance is concentrated in the narrow gaps between adjacent fibers, therefore developed an analytical model to estimate the permeability. The permeability for quadratic and hexagonal arrangement of fibers can be written as K x = 8r ( V ) f B V f f 3 V f,max K z = B rf V f The parameters B, B, and V f,max depend on the fiber arrangement Gebart, B.R. Permeability of unidirectional reinforcements for RTM. Journal of Composite Materials, 6(8), 00, 99

74 Permeability-estimating model (Cont.) Another approach is the self-consistent method. This method assumes that a unit cell of a heterogeneous medium can be considered as being embedded in an equivalent homogeneous media whose properties are unknown and to be determined [9]. The flow inside the unit cell satisfies Navier-Stokes equation, while the flow outside of the unit cell follows Darcy s law. The consistency conditions are that the total amount of the flow and the dissipation energy remain the same with and without this insertion of unit cell. r f K x ln f 8V f V f ( 3 V )( V ) = f K z ln V f rf V f = 8V f + V f 9. Berdichevsky, A. and Cai, Z. Preform permeability predictions by self-consistent method and finite elements simulation, Polymer composites, 4(), 3, 993.

75 Thank you for your attention. Questions?

76 Wicking of Liquids into Fibrous Porous Media Polymer Processing Laboratory

77 Physics Behind Wicking G S L θ L θ L Young s Equation o θ <90 o θ >90 σ Cosθ = σ GL SG σ SL Wetting Non-Wetting

78 Wicking across a Fiber-bank ``Wicking across a Fiber-bank'', K. M. Pillai and S. G. Advani, Journal for Colloid and Interface Science, v 83, no., 996, p00-0.

79 Modeling Micro-Flow inside the Fiber Bank h

80 Suction Pressure Non-dimensional suction pressure P s = P s / 4γ cos( θ ) d f Energy model P s = v f Ahn et. al. model P s = v f 4( v f ) Williams et. al. model v f Ps = v f

Porous media parameters: Porosity Permeability

81 Measuring Wicking Parameters Liquid parameters: Density Viscosity Surface Tension Rheometer Porous media parameters: Porosity Permeability Fiber diameter Liquid-solid interaction Contact angle DCA

82 Washburn Equation ) ) cos( ( μ τ θ hγ o R h = ) = ( L L e τ π = L L e f f f h v v d R = ) ( 8 ) ) / cos( ( ~ f f f cs v v d A M M = = τ μ θ γ ρ t h h o f = t M t M o ) = ( / 3/ 0 ) ( ) cos( 8 f f f cs v v d A M = μ θ γ τ ρ Mass absorbed Liquid Height

83 Darcy s Law based Model for Wicking M ( t) = M o t M 0 = A cs ρ ( v μ f ) KP s

84 Liquid Mass wicked into Fiber Bank Motor Oil

85 Experiments with different Liquids M ( t) = M o t

86 Effectiveness of Models Mineral oil Silicon oil Glycerine Motor oil

87 Wicking into polymer wicks Experimental setup Sintered Polymer Wicks

88 Comparing Models with Test Data Test Results Capillary Model E.B. Model with gravity Capillary Model with gravity E.B. Model Washburn Model Test Results Capillary Model E.B.Model with gravity Capillary Model with gravity E.B. Model Washburn Model m [g] m [g] t [s] Polypropylene Wick and Liquid Hexadecane t [s] Polycarbonate Wick and Liquid Decane Darcy s Law based Models for Liquid Absorption in Polymer Wicks, by Reza Masoodi, Krishna M. Pillai, P. Varanasi, to appear in AIChE Journal.

89 Thank you for your attention. Questions?

COMPARISON OF WETTABILITY AND CAPILLARY EFFECT EVALUATED BY DIFFERENT CHARACTERIZING METHODS

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS COMPARISON OF WETTABILITY AND CAPILLARY EFFECT EVALUATED BY DIFFERENT CHARACTERIZING METHODS S.K. Wang*, M. Li*, Y.Z. Gu, Y.X. Li and Z.G. Zhang Key