Flow Modeling & Transport in Fiber Mats used as ... · Fiber Mats used as Reinforcement in Polymer...
Transcript of Flow Modeling & Transport in Fiber Mats used as ... · Fiber Mats used as Reinforcement in Polymer...
Flow Modeling & Transport in Fiber Mats used as
Reinforcement in Polymer Composites
Krishna M. PillaiAssociate Professor
Department of Mechanical Engineering
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
Outline
1. Introduction
2. 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
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).
1. Preform Manufacturing
2. Lay-up and Draping
3. Mold Closure
4. Resin Injection and Cure
5. Demolding and Final Processing
Process steps in Resin Transfer Molding (RTM)Process steps in Resin Transfer Molding (RTM)
Passenger mini-van cross-member
Advantages of moldAdvantages of mold--filling simulation in RTMfilling simulation in RTM
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
Flow through a porous medium: averaging of flow variablesFlow through a porous medium: averaging of flow variables
fiber
averaging volume
∫=V f
dVV
vq rr 1∫=V f
V fdVpp 1
Volume average: Pore average:
∫=V f
V fdVcc 1∫=
V f
dVTV
T1
( ) ( ) Dτ :++∇⋅∇=∇⋅+∂∂ T,cTkTvtT fHCC cRpp ρρρ r
where
( ) ( )[ ]vv Trr∇∇ +=
21D
Microscopic Energy Balance:
Dτ μ2=
Effects of averaging on balance equations in a porous mediuEffects of averaging on balance equations in a porous mediumm
(Incompressible, Newtonian fluid)
TbT̂wheredVbv̂V fV
Cf
pD ∇⋅=⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧−= ∫
rrr1ρεK
Macroscopic Energy balance
( ){ } =∇⋅+∂∂
+ TqtT
CCC psp prρρε ρε
( ){ } ( ) qqK
T,cT fH cRDerr⋅++∇⋅+⋅∇
μρεKk
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+= ∫∫ dSbdSbV
kSS fsfs
nV
kn fss
ssfserr rr
εεε
ε 11 δδk
Effective thermal conductivity tensor
Dispersive thermal conductivity tensor
Effects of averaging on balance equations in a porous mediuEffects of averaging on balance equations in a porous mediumm
Conventional moldConventional mold--filling simulation physics for RTMfilling simulation physics for RTM
( ){ } ( )Tcccvtc f
c,εεε +∇⋅+⋅∇=∇⋅+
∂∂
DD Dfr
Mass balance 0=⋅∇ vr
Energy balance
( ){ } ( ){ } ( )TcfHTTvCtT
C p cRDepspC ,ρερρερε +∇⋅+⋅∇=∇⋅+∂∂
+ Kkr
Chemical reaction
Darcy’s Law pv ∇−=μKr 0=∇⋅∇ p
μK
Elliptic pressure equation
Real flow Simulation (FE/CV scheme)
Finite Element/Control Volume (FE/CV): 1) most widely used approach because its simplicity, efficiency and robustness. 2) circumvents the front-tracking related problems associated with the adaptive mesh
regeneration method by using an Eulerian fixed mesh.
Typical experiments to measure inTypical experiments to measure in--plane fiberplane fiber--mat permeabilitymat permeability
radial flow mold1-D flow mold
constant pressurefluid supply
constant injection-ratefluid supply
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ
μ⎟⎠⎞
⎜⎝⎛=
PL
AQKpv ∇−=
μKr
1-D flow:
Experimental Investigation of the 1Experimental Investigation of the 1--D unsaturated flowD unsaturated flow
x
Inlet pressure history for 1Inlet pressure history for 1--D constant injectionD constant injection--rate experimentrate experiment
Continuity:
Darcy’s law:
Front speed:
xx f
Pin
0x=
dud
uxdPd=−
μK
utd
d x f =ε
tKuPin ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
εμ2
Pin
t
Flow-front and Inlet-Pressure prediction
for 1-D Flow: Random Fiber Mats
0 10 20 30 400
20000
40000
60000
80000
100000
120000
140000
Pre
ssur
e /P
a
Time/sec
Experimental
Theoretical
Flow-front and Inlet-Pressure predictionfor 1-D Flow: Woven or Stitched Mat
0 10 20 30 40 50 600
20000
40000
60000
80000
100000
120000
140000
160000
180000
Experimental Theoretical
Pres
sure
/Pa
Time/sec
Pin
t
rando
m mat
woven or stitched mat
Inlet pressure history: constant injectionInlet pressure history: constant injection--rate 1rate 1--D experimentD experiment
Pin
Droop
Woven and stitched fiber matsWoven and stitched fiber mats
Fibertows
Flow direction
Inter-tow spaces are absent
Li(Intra-tow
space)
Bi-axial woven mat
Dual-scale or dual-porosity media
Travel of a dark colored test liquid in a unidirectional stitched mat injected first with a
lighter color liquid
Courtesy: Prof. Lee, Ohio State University
Dry fiber matInitially injected light-colored liquid
Later injected dark-colored liquid
Radial Injection Pattern
Random Mat (Single-scale Porous Medium)
Biaxial Stitched Mat (Dual-Scale Porous Medium)
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 16-19, 2005
Role of fiber bundles in creation of unsaturated flow and sink effect in woven/stitched fiber mats
Typical micrograph of an RTMpart made with stitched/woven
fiber mat
Schematic: absorption of resin by fiberbundles and creation of sink effect
Details of the Mesh and the Control Volumes Details of the Mesh and the Control Volumes
Inner CVs
Outer CVs
Finite difference scheme based on the control volumeformulation
x*
y*
0 0.1 0.2 0.3 0.4 0.5 0.6
0
1
2
3
4
T60.208457.693555.178652.663750.148947.63445.119142.604240.089337.574435.059532.544730.029827.514925
x*
y*
0 0.025 0.05 0.075 0.1 0.125
0
0.2
0.4
0.6
0.8
T48.378746.708845.038943.36941.699140.029238.359336.689435.019533.349531.679630.009728.339826.669925
Evolution of temperature distributionEvolution of temperature distribution
x*
y*
0 0.1 0.2 0.3
0
0.5
1
1.5
2
T56.252254.019951.787649.555347.32345.090742.858440.626138.393836.161533.929231.696929.464627.232325
t = 0.2 tch t = 4 tcht = tch
•“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: 1-28, 2004.
x*
y*
0 0.025 0.05 0.075 0.1 0.125
0
0.2
0.4
0.6
0.8
al0.0002268710.0002117460.0001966220.0001814970.0001663720.0001512470.0001361230.0001209980.0001058739.07485E-057.56237E-056.0499E-054.53742E-053.02495E-051.51247E-05
Evolution of cure distributionEvolution of cure distribution
x*
y*
0 0.1 0.2 0.3
0
0.5
1
1.5
2
al0.0009852540.0009195710.0008538870.0007882040.000722520.0006568360.0005911530.0005254690.0004597850.0003941020.0003284180.0002627350.0001970510.0001313676.56836E-05
x*
y*
0 0.1 0.2 0.3 0.4 0.5 0.6
0
1
2
3
4
al0.003983060.003717530.003451990.003186450.002920910.002655380.002389840.00212430.001858760.001593230.001327690.001062150.0007966130.0005310750.000265538
t = 0.2 tch t = tch t = 4 tch
Comparison of temperature predictions by the twoComparison of temperature predictions by the two--layer layer dualdual--scale model and the conventional singlescale model and the conventional single--scale modelscale model
x*
T*,θ
*,τ
0 .0 5 0 .10
0 .0 5
0 .1
0 .1 5
0 .2
0 .2 5
0 .3
0 .3 5
0 .4
θ * fo r d u a l sca le m o d e lT * fo r d u a l sca le m o d e lτ fo r s in g le sca le m o d e l
x
VolumeVolume--averaging in a dualaveraging in a dual--scale porous mediascale porous media
dVΒVΒ gVg
g ∫=><1
dVΒV
Β gVgg
gg ∫=><
1
dVnΒVΒΒ gtgAgt
gg ∫+><∇=>∇<1
dVΒVΒttΒ
gtgAgt
gg nu ⋅−><
∂∂
=>∂∂
< ∫1
Averaging Theorems
(Pillai & Murthy, 2004)
Governing equations for reactive, Governing equations for reactive, nonisothermalnonisothermal,,unsaturated flow in dualunsaturated flow in dual--scale porous mediascale porous media
Sg −=><⋅∇ v ><∇−=>< ggg P
Kμ
v
QQfHTTtT
C condconvcRggg
gthg
gg
gg
gp l−++∇⋅⋅∇=
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡∇⋅><+
∂
∂><><><
ρεερ Kv)(
MMfcctc
diffconvcgg
gg
gg
gg
g −++∇⋅⋅∇=∇⋅><+∂
∂><><><
εε Dv
Mass Balance: Momentum Balance:
Energy Balance:
Cure Balance:
Various source and sink terms:dA
AVS gtg
gt
nv ⋅= ∫1
dAAV
Q gtgcondgt
nq ⋅= ∫1
⎥⎦⎤
⎢⎣⎡ ><−><= TTSCQ g
gtg
ggpgconv ,ρ ⎥⎦
⎤⎢⎣⎡ ><−><= ccSM g
gtg
gconv
dAAVM gtgdiffgt
nJ ⋅= ∫1
(Pillai & Murthy, 2004)
Coupled macroCoupled macro--micro approach for modelingmicro approach for modelingthe unsaturated flowthe unsaturated flow
•“Governing equations for unsaturated flow through woven fiber mats, Part 2: Nonisothermal reactive flows”, K.M. Pillai and M.S. Munagavalsa, Composites Part A: Applied Science and Manufacturing, v 35, 2004, p403-415.
Governing equations for 1Governing equations for 1--D unsaturated isothermal flowD unsaturated isothermal flow
Continuity: So −=∂∂xu Sx
ud
do ∗
∗
∗
−=
Darcy’s law: uxdPd
oo =−
μK
uxP
oo
d
d ∗
∗
∗
−=
Front speed: uxo
fo td
d=ε ut
xo
f
d
d ∗
∗
∗
=
xx f
Inlet pressure history for the constant sink caseInlet pressure history for the constant sink case
SeP
tSin 2
21∗
−∗
∗∗−
=
Previous ExperimentsPrevious Experiments
Governing EquationsGoverning EquationsConstant injectionConstant injection--rate radial flowrate radial flow
Sru
ru oror
d
d ∗
∗
∗
∗
∗
−=+
urP
oro
d
d ∗
∗
∗
−=
utr
orf
d
d ∗
∗
∗
=
Inlet pressure history for the radial flow (analytical sInlet pressure history for the radial flow (analytical solution)olution)
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
0.1 1.00.010.0010.0001
% v
olum
e of
voi
ds
% v
olum
e of
voi
ds
Macrovoids Microvoids
5
10 10
5
Void trapping mechanisms & its dependence on Capillary Number
CaCa V
Cos= μσ θ
Patel, Lee et al.
Bucklet-Leverett Model for Bubble Migration
• Two-phase flow in porous media• Phases are incompressible• Capillary Pressure (Pc = Pnw – Pw) is
neglected• Gravity neglected• ‘Fast Flooding’ regime (large Ca)
Bucklet-Leverett Formulation0)( =⋅∇+∂ rrt qS rφ
0)( =⋅∇+∂ aat qS rφ
1=+ ra SS
Mass Balance:
Generalized Darcy’s Law:PKkq
rrrelr ∇=μ,
r
PKkqa
arela ∇=μ,
r
Bucklet-Leverett Equation
where
1-D Flow
ar
r
qqqr+
= = fractional flow rate
1st order Quasi-linear Hyperbolic Equation
0=∂∂
+∂∂
xSU
tS rr
r
ar
SdrdqqU
ε+
= = signal speed
Bucklet-Leverett Model for Bubble Migration
t1
t2
t3
t3 > t2 > t1
(Sr)
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40 45 50
br1t> 2 >
2
t3t
t 31
Sr
x(nodes)
t t
Typical Pattern of Bubble Migration(Lundstrom & Gebart)
Buckley-Leverett Saturation Fronts (Pillai & Advani )
Buclet-Leverett Model: Void Distribution from Numerical Simulation
)(1)(
,
,, PS
PSSS
resida
residaareda −
−=If P > Pcritical, then air bubble moves
(Chui & Glimm et al.)
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, 15(1):25-33, Feb. 1994
• Pillai, K.M. & Advani, S.G., Modeling of Void Migration in Resin Transfer Molding, Proceedings of 1996 IMECE(ASME), page 14, 1996.
• 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):714-727, Dec 1997.
Conventional 2-Phase Flow Models to Predict Void Distribution/Saturation during Slow (small
Ca) 1-D Flows
Experimentally Measured Saturation Distribution
Numerically predicted saturation
• Breard et al. “Numerical Simulation of void Formation in LCM”, Composites: Part A 34 (2003) 517-523
Bucklet-Leverett Flow: Inlet Pressure history
Pinj
μ r q(t)
K
x f
V=1000
0
V=100
V=1
0
V =
1
Single
Phase
Darc
y’s L
aw
0 1
a
r
μμ
=V
P in
t
rando
m mat
woven or
stitched mat
B-L Flow is unable to recreate the inlet pressure droop.
(Experimental Observation)
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 2-phase flow formulations have been tried to model low Ca flows, but are unable to recreate the pressure droop.
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 = 1 as P > Pcrit. 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.
Summary
Unsaturated flow fundamentally different in single-scale 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 2-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
pKu ∇−= .η
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=
ZZZYZX
YZYYyx
XZXYxx
kkk
kkkkkk
K
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
∂∂∂∂∂∂
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
zpypxp
kkk
kkkkkk
u
uu
ZZZYZX
YZYYyx
XZXYxx
z
y
x
η1
If the selected coordinate directions are along the principal directions, we have
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
∂∂∂∂∂∂
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
3
2
1
33
22
11
3
2
1
0000
001
xpxpxp
kk
k
uuu
η
Tensor nature of the Permeability
Permeability of a Flat Fiber Mat
• Assumption: out-of-plane z direction is a principal direction
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
zzzyzx
yzyyyx
xzxyxx
kkk
kkk
kkk
K⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=
zz
yyyx
xyxx
k
kk
kk
00
0
0
K
x
y
Typical experiments to measure inTypical experiments to measure in--plane fiberplane fiber--mat permeabilitymat permeability
radial flow mold1-D flow mold
constant pressurefluid supply
constant injection-ratefluid supply
Permeability measurement through 1-D Channel Flow
Resin inlet
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ
μ⎟⎠⎞
⎜⎝⎛=
PL
AQKpv ∇−=
μKr
Isotropic Medium:
In-plane Permeability Tensor for Anisotropic Fiber Mats
⎥⎦
⎤⎢⎣
⎡=
yyyx
xyxx
KKKK
K ⎥⎦
⎤⎢⎣
⎡=
2
1
00K
KK
θ2/1 CosDADAKK I −
−=
, y
45o
III
I , x
1
2 II
θ
θ2/2 CosDADAKK III +
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−= −
DA
DKDA
II
221tan
21θ
2IIII KKA +
=
2IIII KKD −
=
where
1-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 (race-tracking)
• Three experiments for anisotropic fiber mats
Permeability tensor measured through transient radial flow
x
y
12
Flow front position
1 and 2 are two principal directions
In a constant flow-rate experiment, the flow front positions and inlet pressure are functions of time [1,2]. The in-plane permeability tensor can be expressed as
⎥⎦
⎤⎢⎣
⎡=
2
1
00K
KK
Based on the Darcy’s law and continuity equation, the effective permeability can be expressed as
⎥⎦
⎤⎢⎣
⎡+== f
rR
hPQKKK f
ineff ln)ln(
2 0
121 π
μ
Injection port r0
where
1
111
2
1
2
12
1
20
+
⎟⎟⎠
⎞⎜⎜⎝
⎛−++
=
KK
KK
Rr
f f 2
1
2
1
KK
RR
f
f =
where R1f and R2f are the radial flow front along the principal directions, respectively; r0 is the inlet port radius, K1 and K2 are principal permeabilities. 1. Adams, K.L.,etc. Forced in-plane flow of an epoxy resin in fibrous networks. Polym. Eng. Sci. 1986, 26(20),1434.
2. Chan, A.W., Hwang Sun-Tak, Anisotropic in-plane permeability of fabric media. Polym. Eng. Sci. 1991, 31(16),1233.
Permeability tensor measured through transient radial flow (Cont.)
Data analysis procedure1. the ratio α(K1/K2) 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
2. Since Q, μ, and h are known, we have the following equation
where m can be obtained from the slope of the curve Pin versus (ln(R1f /r0)+lnf ).
3. The values of K1 and K2 are then calculated from the relations:
2
2
12
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
f
f
RR
axisellipticminoroflengthaxisellipticmajoroflengthα
mhQKKKeff
1221 πμ
==
effKK α=1α
effKK =2
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.
P1
P2
P3
X
Y
R=3”
P0θ=1200
1
2 β
positions of pressure transducers
( )( ) 02
1
1log111
11log 10
21
21
0
1
2
0
1 =−
−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
+−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛
−effK
Qpph
xx
xx
μπ
α
ααα
( )( ) 02
1
1log1
11
log 20
21
21
0
2
2
0
2 =−
−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−
+−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−++⎟⎟
⎠
⎞⎜⎜⎝
⎛
−effK
Qpph
yy
yy
μπ
α
αα
αα
α
where (x0, y0), (x1, y1) and (x2, y2) are the coordinates of points P0, P1, and P2 respectively. Q is the flow rate; h is the mold gap thickness.
3. K. Ken Han, etc. (2000) Measurements of the permeability of fiber preforms and applications, Composites Science and Technology, 60 (12-13): 2435-2441.
Permeability tensor measured through steady-state radial flow (Cont.)
• After plugging the steady-state pressure values into these two non-linear equation, one can solve for the degree of anisotropy α and Keff . The principal permeabilities K1 and K2 through relations α = K1/K2 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.
Radial-Flow Permeability Measuring Process
Advantages
• K estimation in a single experiment
• No race-tracking
• No sliding of fiber mats
Disadvantages
• Larger Mold Deflection
• Complex Physics
• Involved Data Analysis
Calibration Devices for Permeability Measuring Setup
1) 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.
2) A method to estimate the accuracy of 1-D flow based permeability measuring devicesHua Tan, Tonmoy Roy and Krishna. M. Pillai, Journal of Composite Materials, 2007.
Radial Flow 1-D Flow
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
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 well-known Kozeny-Carman equation is based on this approach
where C is Kozeny-Carman equation
vAK 22
3
τφ
=
τ
22
3
)1( φφ−
=vCA
K
4. L. Skartsis,etc. Resin flow through fiber beds during composite manufacturing processes, Polymer Engineering and Science, 32, 221,1992
5. Lenormand, etc. Mechanisms of the displacement of one fluid by another in a network of capillary ducts. Journal of Fluid Mechanics, 135, 337, 1983.
Permeability Models• Basing on the Kozeny-Carman equation, many researchers propose the
following permeability model for flow along the fiber direction [6]
where Kx is the permeability in the fiber direction, rf is the fiber radius, Cis 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
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 0.2, and Va was determined to be around 0.8-0.85.
( )2
32
41
f
ffx VC
VrK
−=
( )( )14
132
+′
−′=
fa
fafz VVC
VVrK
6. Williams, J.G., etc., Liquid flow through aligned fibre beds. Polymer Engineering and Science, 14, 413, 1974.
7. Gutowski, T.G. etc, Consolidation experiments for laminate composites. Journal of composite materials, 21, 7, 650, 1987.
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
The parameters B1, B2, and Vf,max depend on the fiber arrangement.
( )2
1
32 18
f
ffx VB
VrK
−=
2
25
max,2 1 f
f
fz r
VV
BK ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
8. Gebart, B.R. Permeability of unidirectional reinforcements for RTM. Journal of Composite Materials, 26(8), 1100, 1992
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]. Theflow 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.
( )( )⎥⎥⎦
⎤
⎢⎢⎣
⎡−−−= ff
ff
fx VV
VVr
K 131ln8 2
2
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+
−−= 2
22
111ln
8 f
f
ff
fz V
VVV
rK
9. Berdichevsky, A. and Cai, Z. Preform permeability predictions by self-consistent method and finite elements simulation, Polymer composites, 14(2), 132, 1993.
Physics Behind Wicking
SLSGGL Cos σσθσ −=
G
L
S
Young’s Equation
θL
o90<θ
Wetting
θL
o90>θ
Non-Wetting
Wicking across a Fiber-bank
•``Wicking across a Fiber-bank'', K. M. Pillai and S. G. Advani, Journal for Colloid and Interface Science, v 183, no.1, 1996, p100-110.
Suction PressureNon-dimensional suction pressure
• Energy model
• Ahn et. al. model
• Williams et. al. model
⎟⎟⎠
⎞⎜⎜⎝
⎛=
fss dPP )cos(4/ θγ
fs vP =
)1(4 f
fs v
vP
−=
f
fs v
vP
−=
1
Measuring Wicking ParametersLiquid parameters:• Density• Viscosity • Surface Tension
Porous media parameters:• Porosity• Permeability • Fiber diameter
Liquid-solid interaction• Contact angle
Rheometer
DCA
Washburn Equation
21
2 )2
)cos((μτθγh
oRh =
2)(LLe=τ
2π
=LLe
f
ffh v
vdR
−=
142
1
23
00
)1(8
1)/)cos((
~
f
f
fcs v
vdA
MM−
==τμθγρ
thh of =tMtM o=)(
2/1
2/321
0
)1()cos(8 f
ffcs
vvdAM
−⎟⎟⎠
⎞⎜⎜⎝
⎛=
μθγ
τρ
Mass absorbedLiquid Height
Comparing Models with Test Data
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 10 20 30 40 50 60 70 80
t [s]
m [g
]
Test ResultsCapillary ModelE.B. Model with gravityCapillary Model with gravityE.B. ModelWashburn Model
Polycarbonate Wick
and Liquid Decane
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 20 40 60 80 100 120
t [s]
m [g
]
Test ResultsCapillary ModelE.B.Model with gravityCapillary Model with gravityE.B. ModelWashburn Model
Polypropylene Wick
and Liquid Hexadecane
• “Darcy’s Law based Models for Liquid Absorption in Polymer Wicks,” by Reza Masoodi, Krishna M. Pillai, P. Varanasi, to appear in AIChE Journal.