Nbm
Ya
b
c
a
ARRAA
KSTFCVF
1
enpbn
pm&oa
2
0h
Computers and Chemical Engineering 46 (2012) 190– 204
Contents lists available at SciVerse ScienceDirect
Computers and Chemical Engineering
jou rn al h om epa ge: w ww.elsev ier .com/ locate /compchemeng
umerical investigation of the thermally and flow induced crystallizationehavior of semi-crystalline polymers by using finite element–finite differenceethod
ue Mua,b,c,∗, Guoqun Zhaoa,b,∗, Anbiao Chenb, Xianghong Wua,b
Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandong 250061, PR ChinaEngineering Research Center for Mould & Die Technologies, Shandong University, Jinan, Shandong 250061, PR ChinaKey Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China
r t i c l e i n f o
rticle history:eceived 8 September 2011eceived in revised form 20 June 2012ccepted 21 June 2012vailable online 30 June 2012
eywords:emi-crystalline polymershermally inducedlow inducedrystallizationiscoelasticinite element–finite difference simulation
a b s t r a c t
The thermally and flow induced crystallization behavior of semi-crystalline polymer in processing cansignificantly influence the quality of final products. The investigation of its mechanism has both scien-tific and industrial interest. A mathematical model in three dimensions for thermally and flow inducedcrystallization of polymer melts obeying differential Phan-Thien and Tanner (PTT) constitutive modelhas been developed and solved by using the finite element–finite difference method. A penalty methodis introduced to solve the nonlinear governing equations with a decoupled algorithm. The correspondingfinite element–finite difference model is derived by using the discrete elastic viscous split stress algo-rithm incorporating the streamline upwind scheme. A modified Schneider’s approach is employed todiscriminate the relative roles of the thermal state and the flow state on the crystallization phenomenon.The thermally and flow induced crystallization characteristics of polypropylene is investigated based onthe proposed mathematical model and numerical scheme. The half crystallization time of polypropylenein a cooled couette flow configuration obtained by simulation are compared with Koscher’s experimen-tal results, which show that they agree well with each other. Two reasons to cause crystallization ofpolypropylene in pipe extrusion process including the thermal state and the flow state are investigated.Both the crystalline distribution and crystalline size of polypropylene are obtained by using the finiteelement–finite difference simulation of three-dimensional thermally and flow induced crystallization.The effects of processing conditions including the volume flow rate and the temperature boundary onthe crystallization kinetics process are further discussed.
© 2012 Published by Elsevier Ltd.
. Introduction
The final properties of semi-crystalline polymer products are to a great extent determined by the internal crystalline structures thatstablished during processing. The flow and thermo-mechanical history experienced by polymer melts during processing can enhanceucleation and crystallization, and hence to accelerate the process and lead to different types of crystalline structure. Therefore, accuraterediction of properties of the final products by computer simulation requires not only appropriate crystallization model that consideringoth the thermal and the flow effects on crystalline structure development but also robust numerical solution methods for the complicatedonlinear mathematical model.
Many researchers have been working to investigate and explain the thermally and flow induced crystallization behavior in polymerrocessing by theoretical research. The main idea is firstly to choose appropriate rheological constitutive equation, such as Doi–Edwards
odel (Acierno, Coppola, & Grizzuti, 2008), upper convective Maxwell model (Koscher & Fulchiron, 2002), Leonov model (Zuidema, Peters,Meijer, 2001) and conformation tensor model (Bushman & McHugh, 1997). The variables like stress tensor, strain tensor, shear rater supercooling degree are then assumed to be the driving force of nucleation. The simultaneous equations of the constitutive equationnd the crystallization kinetic equations are finally solved to predict the behavior of thermally and flow induced crystallization. The
∗ Corresponding authors at: Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandong50061, PR China. Tel.: +86 531 88393238; fax: +86 531 88392811.
E-mail addresses: [email protected] (Y. Mu), [email protected] (G. Zhao).
098-1354/$ – see front matter © 2012 Published by Elsevier Ltd.ttp://dx.doi.org/10.1016/j.compchemeng.2012.06.026
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 191
Nomenclature
u velocity vectorp hydrostatic pressure� extra stress tensord deformation rate tensorS polymer-contribution stress�v Newtonian-contribution viscosity
retardation ratio�p polymer-contribution viscosity�t total viscosity� relaxation time� reference viscosityε elasticity parameter� slip parameter�p penalty factor� material densityCp specific heatk thermal conductivity˛∞ absolute crystallinity
relative crystallinity�Hc latent heat of crystallizationa0 shape factor corresponding to crystallization growthT0
m equilibrium melting temperatureU* activation energyR gas constant
mflftettcsa(anpwZu
iphsp&f
esTfivTcpdh
Tg glass transition temperature
icro-rheological model that can depict the microscopic morphology of polymer melts was adopted in Coppola’s work to establish theow induced crystallization model where the variation of conformational structure caused by the flow field should change the system
ree energy (Acierno et al., 2008). The variation of free energy was thought to affect the crystallization kinetics process and a free energyerm was hence added into the nucleation equation. Doufas (Doufas, Dairanieh, & McHugh, 1999; Doufas, 2006) deduced a constitutivequation for the flow induced crystallization behavior of polymer melts in uniform flow field based on the Hamiltonian/Poisson theory andhe Avrami equation. The presented constitutive equation can not only describe the viscosity variation of polymer melts but also predicthe experimental results of flow birefringence. Lin, Wang, and Zheng (2008) adopted cellular automaton model to investigate isothermalrystallization of monomer casting nylon 6 based on the Kim and Kim’s rate theory for spherulite expansion. Tanner (2002) proposed auspension model for polymer solidification at low shear rate. The spherulites were assumed to be hard spheres that suspended in themorphous phase and the stress distribution was calculated based on the hypothesis of concentrated suspension system. Zuidema et al.2001) considered the recovery strain as the driving force of flow induced crystallization by using the Leonov constitutive model. Zhengnd Kennedy (2004) presented an idea that to insert the free energy term calculated with the conformation tensor model into the Ederucleation rate equation (Stadlbauer, Janeschitz-Kriegl, Eder, & Ratajski, 2004) to predict the flow induced crystallization behavior ofolypropylene. Ouyang (Ruan, Ouyang, Liu, & Zhang, 2011) investigated the isothermal crystallization of short fiber reinforced compositesith athermal nucleation and the morphology evolution was captured by using the “pixel coloring” technique. Yu, Zhang, Zheng, Yu, and
hou (2008) developed a modified two-phase microstructure rheological model that can predict the evolution of viscosity and modulusnder the flow state to investigate the flow induced crystallization behavior of the supercooling polymer melts.
Although many theories have been proposed to model the crystallization kinetics process of polymeric material, the difficulty in solv-ng the complicated governing equations for the thermally and flow induced crystallization hinders its application to practical polymerrocessing. Numerical simulation as a highly effective method can well predict such complex phenomenon as in polymer processing andence to explain relevant material forming mechanism (Keunings, 1995). However, there is few research work performed aiming at how toolve the nonlinear thermally and flow induced crystallization problem by using numerical methods. It is still a difficult task to improve theroperties of final products through obtaining desired crystalline structure under rational control of processing conditions (Costa, Maciel,
Filho, 2007). This may hinder the application of the proposed theoretical model in practical polymer processing and hence to hinderurther development of corresponding crystallization kinetics theory.
In the present study, the thermally and flow induced crystallization behavior of polymeric material is investigated by means of finitelement–finite difference method. The viscoelastic rheological properties are described by using differential Phan-Thien–Tanner (PTT) con-titutive model. The mathematical model of three-dimensional thermally and flow induced crystallization of polymer melts is established.he corresponding finite element–finite difference model is derived and the details of numerical schemes are introduced. The penaltynite element method and a decoupled solving method are adopted to reduce the computation memory requirement. The discrete elasticiscous split stress (DEVSS) algorithm incorporating the streamline upwind scheme is employed to improve the computation stability.he ellipticity of the momentum equation is improved by the addition of a stabilization factor and the influence of convection term in the
onstitutive equation and the energy equation is controlled by an asymmetric weighting function. The evolution of crystallization kineticsrocess is determined by using two sets of Schneider equations with the thermal state and the flow state assumed to be two distinctriving forces. The feasibility of the proposed mathematical model and numerical solving method is verified by comparing the calculatedalf crystallization time in cooled couette flow configuration with those of Koscher’s experimental results. The thermally and flow induced
1
cnc
2
2
cc
wpl
s2P
wd
w
wP
G
2
aSd
92 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
rystallization behavior of polypropylene in pipe extrusion process are investigated based on the established mathematical model andumerical algorithm. The crystalline distribution and crystalline size within the flow channel are predicted and the effects of processingonditions are further discussed.
. Mathematical modeling
.1. Balance equations
According to the theory of computational fluid dynamics, the governing equations to solve the flow problem of semi-crystalline polymersan be obtained from the continuity equation, Eq. (1), momentum equation, Eq. (2) and energy equation, Eq. (3), respectively, based on theonservation of mass, momentum and energy.
∇ · u = 0 (1)
�
(∂u
∂t+ u · ∇u
)= ∇ · � − ∇p (2)
�Cp
(∂T
∂t+ u · ∇T
)= k∇ · (∇T) + � : ∇u + �˛∞�Hc
∂˛
∂t(3)
here � is the material density, u is the flow velocity, � is the Hamilton differential operator, t means time, � is the additional stress tensor, is the hydrostatic pressure, Cp is the specific heat, T is temperature, k is the thermal conductivity, ˛∞ is absolute crystallinity, �Hc is theatent heat of crystallization, and is the relative crystallinity.
A general Maxwell-type differential constitutive equation is adopted to model the viscoelastic properties of polymer melts, namely,hear thinning, non-zero second normal stress coefficient, and stress overshoot in transient shear flows (Favero, Secchi, Cardozo, & Jasak,010). The rheology of the polymer solutions are here represented as the sum of a Newtonian solvent contribution with a full multi-modeTT polymer contribution as follows
� = 2�vd + S (4)
here �v is the Newtonian-contribution viscosity, S =∑n
k=1Sk is the sum of polymer-contribution stress of each different mode k, d is theeformation rate tensor which obeys
d = ∇u + ∇uT
2(5)
By introducing the retardation ratio ˇ, defined as
= �p
�t(6)
here �t = �p + �v is the total viscosity and �p is the polymer-contribution viscosity.The polymer-contribution stress Sk of each mode obeys the following equation
�k
�Sk + g(Sk)Sk = 2ˇ�td (7)
here �k is each single relaxation time and g(Sk) is a stress function following the exponential form proposed in the original work ofhan-Thien and Tanner (1977)
g(Sk) = exp(
�kεk
ˇ�ttr(Sk)
)(8)
The full PTT constitutive model based on Gordon–Schowalter (GS) convective derivative is adopted and�Sk denotes the following
ordon–Schowalter convective derivative operator
�Sk = ∂Sk
∂t+ u · ∇Sk − (∇u − �kd) · Sk − Sk · (∇u − �kd)T (9)
Eq. (9) can be simplified into the upper convective derivative when the parameter � of the GS derivative is set to zero.
.2. Crystallization kinetics model
Crystallization of semi-crystalline polymers can be described as a nucleation and growth process of crystallites. The nuclei are activatednd grow into crystallites at rates depending on the flow and the thermo-mechanical state of the molten polymers in processing. Thechneider equations are adopted in the study to model the thermally and flow induced crystallization by means of a set of first orderifferential equations (Schneider, Koppl, & Berger, 1988). In order to discriminate the thermal and the flow effects, the Schneider equations,
Ea
witT
wv
d
w
s
wr
d
wi
rsa
w
drs
wt
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 193
q. (10) are modified by expressing the total number of nuclei as the sum of two kind nuclei that one induced by the thermal state NT andnother one induced by the flow state Nf (Boutaous, Bourgin, & Zinet, 2010)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂ϕT,f3 (t)
∂t+ u · ∇ϕT,f
3 (t) = 8NT,f (t)
∂ϕT,f2 (t)∂t
+ u · ∇ϕT,f2 (t) = G(t)ϕT,f
3 (t)
∂ϕT,f1 (t)∂t
+ u · ∇ϕT,f1 (t) = G(t)ϕT,f
2 (t)
∂ϕT,f0 (t)
∂t+ u · ∇ϕT,f
0 (t) = G(t)ϕT,f1 (t)
(10)
here N(t) is the nucleation rate, G(t) is the spherulitic growth rate, ϕ0 is the total extended volume of the spherulite Vtot per unit volume ifmpingement is disregarded, ϕ1 is the total surface area of the spherulite Stot per unit volume, ϕ2(ϕ2 = 4Rtot) is proportional to the sum ofhe spherulite radii Rtot per unit volume, ϕ3(ϕ3 = 8N) is proportional to the number of the spherulites N per unit volume, the superscripts
and f, respectively, denote the nuclei induced by the thermal state and the flow state.The correction for the impingements is then introduced by using the Avrami equation and lead to the following relative crystallinity
˛(t) = 1 − exp(−ϕg0(t)) (11)
here ϕg0(t) is the extended global crystalline volume fraction that can be obtained by adding the thermally induced extended crystalline
olume and the flow induced extended crystalline volume as follows
ϕg0(t) = ϕT
0(t) + ϕf0(t) (12)
According to the experimental study of Koscher, the thermally induced nuclei number of the molten polymer NT in quiescent stateepends on the degree of supercooling �T (Koscher & Fulchiron, 2002)
NT (t) = exp(a�T(t) + b) (13)
here �T(t) = T0m − T(t), T0
m is the equilibrium melting temperature, a and b are two material parameters to be identified with experiments.Considering the flow effect on crystallization kinetics, the number of flow induced nuclei is linked to both the shear rate and the shear
train as proposed by Tanner and Qi (2005). The number of flow induced nuclei Nf can be estimated from the following differential equation
dNf ( �, t)dt
= Nf ( �, t) = A| �(t)|p( �t) (14)
here | �(t)|p represents the effect of the chain relaxation, � is the shear rate, A and p are two material parameters to be identified and Nf
epresents the flow induced nucleation rate.The following Hoffman–Lauritzen expression is adopted to describe the radial growth rate of the spherulites G(t) that is considered to
epend on the thermal state (Hoffman & Miller, 1997)
G(t) = G0exp(
− U∗
R (T(t) − T∞)
)exp
(− Kg
T(t)�T
)(15)
here U* is the activation energy, R is the gas constant, T∞ = Tg − 30 ◦C is the temperature below which molecular motion becomesmpossible, Tg is the glass transition temperature, parameters G0 and Kg can be determined by experiment.
Due to the effect of crystallization behavior on the viscosity of molten polymers, sudden increase of viscosity has been found in manyesearches when the relative crystallinity reaches a critical value (Titomanlio, Speranza, & Brucato, 1997). According to the concentrateduspension theory as proposed by Tanner, the viscosity of polymer melts in crystallization process can be linked to the relative crystallinitys follows (Tanner, 2003)
�˛ = �t
(1 − ˛
a0
)−2(16)
here �˛ is the polymer viscosity for a given relative crystallinity, a0 is a shape factor that corresponding to crystalline growth.The thermophysical properties of semi-crystalline polymers can also be influenced by the crystallization behaviors and they can be
escribed with a simple mixing rule between the properties of the amorphous phase and the semi-crystalline phase weighted by theelative crystallinity (Zinet, Otmani, Boutaous, & Chantrenne, 2010). The variation of heat capacity Cp and thermal conductivity k of theemi-crystalline polymers in the crystallization process can be written as follows⎧⎪⎨
⎪Cp = ˛Cpc + (1 − ˛)Cpa
Cpa(T) = c1T + c2
⎧⎪⎨⎪
k = ˛kc + (1 − ˛)ka
ka(T) = c5T + c6 (17)
⎩Cpc(T) = c3T + c4⎩kc(T) = c7T + c8
here Cpa and Cpc are, respectively, the specific heat of the amorphous state and the semi-crystalline state, ka and kc are, respectively, thehermal conductivity of the amorphous state and the semi-crystalline state.
194 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
Table 1Model analysis.
Equation Variables calculated Variables specified
Momentum and continuity equation, Eq. (25) u, p �t , ˇ, �p
3
3
mif
1o
w
tiT
tZr
wi
t
w�
f
wh
is
W
Constitutive equation, Eq. (30) S �, ε, �Energy equation, Eq. (37) T Cp , ˛∞ , �Hc , kSchneider equation, Eq. (41) ϕ0, ϕ1, ϕ2, ϕ3 NT , Nf , G
. Numerical algorithm
.1. Solution method
In the study, the penalty function method is employed to solve the nonlinear system consisting of the continuity equation and theomentum equation (Kihara, Gouda, Matsunaga, & Funatsu, 1999). The penalty factor as a function obeying �p = − (p/(∇ · u)) is firstly
ntroduced to avoid a direct calculation of the hydrostatic pressure and hence to reduce the computation memory requirement. Theollowing momentum governing equation can be obtained
�
(∂u
∂t+ u · ∇u
)− ∇(�p∇ · u) − 2(1 − ˇ)�t∇ · d − ∇ · S = 0 (18)
To improve the convergence property of the numerical scheme, the discrete elastic viscous split stress formulation (Guenette & Fortin,995) is adopted and an ellipticity factor 2 �[˙(u) − d] is introduced into the momentum equation, Eq. (18). The following equation is thusbtained
�
(∂u
∂t+ u · ∇u
)− �p∇(∇ · u) + 2 �∇ · ˙(u) = ∇ · S + 2( � + (1 − ˇ)�t)∇ · d (19)
here ˙( u) = (∇ u + ∇ T u)/2, � is a reference viscosity.The nonlinear system comprised by the flow balance equations and the crystallization kinetics equations can be used to predict the
hermally and flow induced crystallization behavior of semi-crystalline polymers. The governing equations as shown in Table 1 are numer-cally discretized in the time and the spatial domain respectively by using the finite differential method and the finite element method.he simulation flow chart of the thermally and flow induced crystallization can be seen from Fig. 1.
As it is shown in Fig. 1, a decoupled iteration method is employed to solve the governing equations including the continuity equation,he momentum equation, the energy equation, the constitutive equation and the Schneider equation in current time step j (Mu, Zhao, &hang, 2010). The velocity vector is firstly solved from Eq. (19) by treating the polymer-contribution stress tensor and the deformationate tensor as known terms
�
(∂ui
∂t+ ui · ∇ui
)− �p∇(∇ · ui) + 2 �∇ · ˙(ui) = ∇ · Si−1 + 2( � + (1 − ˇ)�t)∇ · di−1 (20)
here di−1 and Si−1 are, respectively, the deformation rate tensor and the stress tensor calculated with the flow field at the previousteration step i−1 and their initial values are derived from a viscous flow field, i is the current iteration step of flow field calculation.
The polymer-contribution stress tensor of each different mode is then calculated through the constitutive equation, Eq. (21) based onhe velocity solution obtained from the calculation of the momentum equation, Eq. (20).
�k
(∂Si
k
∂t+ ui · ∇Si
k − (∇ui) · Sik − Si
k · (∇ui)T + �k
2
((∇ui + (∇ui)
T) · Si
k + Sik · (∇ui + (∇ui)
T)))
+(
exp(
�kεk
ˇ�ttr(Si−1
k )))
Sik = ˇ�t(∇ui + (∇ui)
T) (21)
here the non-linear term is derived from the stress tensor obtained from last iteration step i−1. The polymer-contribution stress tensor can then be obtained by adding the polymer-contribution stress tensor of each different mode Sk.
Based on the velocity vector and stress tensor, respectively, obtained from Eqs. (20) and (21), the temperature field is then calculatedrom the energy equation (22)
�Cp
(∂Ti
∂t+ ui · ∇Ti
)= k∇ · (∇Ti) + �i : ∇ui + �˛∞�Hc
∂˛i−1
∂t(22)
here the terms on the right hand side, respectively, correspond to the heat exchange, the viscous heat dissipation and the release of latenteat due to the crystallization behavior.
According to the calculated results of the flow and temperature field, the number of flow induced nuclei Nf, the number of thermallynduced nuclei NT and the growth rate of crystallites G(t) can be obtained. Both the relative crystallinity and the information on crystallinetructure can then be solved based on the thermally and flow induced crystallization kinetics model as depicted in Section 2.2.
The convergence criteria of calculation within the current flow field are defined upon the following requirements as shown in Eq. (23).
hen the average relative error of each variable is less than a set value, the convergence is considered to be achieved.1N
∑∣∣∣∣ui − ui−1
ui
∣∣∣∣ < ıu,1N
∑∣∣∣∣Si − Si−1
Si
∣∣∣∣ < ıS,1N
∑∣∣∣∣Ti − Ti−1
Ti
∣∣∣∣ < ıT ,1N
∑∣∣∣∣˛i − ˛i−1
˛i
∣∣∣∣ < ı˛ (23)
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 195
NO
YES
START
Input material parameters and boundary conditions
Establishment of the FE/FD model of velocity & pressure field (Momentum & Continuity equation Eq.25)
Initialization of the flow field, t = 0
Calculation of velocity and pressure ( )u, p using Penalty function & DEVSS algorithm
Establishment of the FE/FD model of viscoelastic stress field (Constitutive equation Eq.30)
END
Calculation of stress tensor ( )τ using SU algorithm
Output results of the simulation
YES
NO Is iteration convergent?
Solution iteration, i = 0
i = i +1
Update of material parameters
Is iteration convergent?t = t + dt
Establishment of the FE/FD model of temperature field (Energy equation Eq.37)
Calculation of temperature ( )T using SUPG algorithm
Establishment of the FE/FD model of crystallization field (Schneider equation Eq.41)
Calculation of relative crystallinity ( )α and crystalline morphology
wwcc
3
if
wa
Fig. 1. Simulation flow chart of the thermally and flow induced crystallization.
here N is the total number of finite element nodes in the computational domain, ıu is the convergence criteria of the velocity vectorhich is set to 10−4, ıS is the convergence criteria of the polymer-contribution stress tensor which is set to 10−3, ıT is the convergence
riteria of the temperature which is set to 10−4 and ı˛ is the convergence criteria of the relative crystallinity which is set to 10−4 in theurrent study.
.2. Finite element formulations
The Galerkin weighting residual method is adopted for the discretization of the momentum equation where the weighting functions taken as the same form as the interpretation function. The discretized momentum equation at the elemental level can be expressed asollows∫
˝e
N
(�
(∂u
∂t+ u · ∇u
)− �p∇(∇ · u) + 2 �∇ · ˙(u)
)d =
∫˝e
N(∇ · S + 2( � + (1 − ˇ)�t)∇ · d)d (24)
After Green–Gauss transformation, the following weak form of Eq. (24) can be obtained∫
N
(�
(∂u + u · ∇u
))d +
∫∇N( �(∇u + ∇uT ) − �p(∇ · u))d =
∫∇N(S + 2( � + (1 − ˇ)�t)d) +
∫N∗(S − pI) · ndS (25)
˝e ∂t˝e ˝e Se
here N* is equal to the interpolation function on the boundary force, ˝e and Se are, respectively, the element region and element boundary,nd n is the outer normal unit vector.
1
e
e
we
cow
w(
e
f
a
we
w
w
96 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
The discretized momentum equation (25) is then expanded along three directions of coordinate components (x, y, z) and the followinglemental stiffness matrix equation can be obtained.⎡
⎢⎢⎣[kij
xx] [kijxy] [kij
xz]
[kijyx] [kij
yy] [kijyz]
[kijzx] [kij
zy] [kijzz]
⎤⎥⎥⎦
⎧⎪⎪⎨⎪⎪⎩
{ujx}
{ujy}
{ujz}
⎫⎪⎪⎬⎪⎪⎭
+
⎡⎢⎢⎣
[nijxx] [nij
xy] [nijxz]
[nijyx] [nij
yy] [nijyz]
[nijzx] [nij
zy] [nijzz]
⎤⎥⎥⎦
⎧⎪⎪⎨⎪⎪⎩
{∂ujx/∂t}
{∂ujy/∂t}
{∂ujz/∂t}
⎫⎪⎪⎬⎪⎪⎭
=
⎧⎪⎨⎪⎩
{f ix}
{f iy}
{f iz }
⎫⎪⎬⎪⎭
(i, j = 1, 2, . . . , 8)
(26)
According to the nodes superposition principle and element connectivity, the global stiffness matrix equation of the momentumquation, Eq. (27) for all the elements can be assembled after evaluating the elemental stiffness matrix at elemental level using Eq. (26).
[K]{u}t + [N]
{∂u
∂t
}t
= {f }t (27)
here [K] is the global stiffness matrix of the unknown velocity vectors {u}, [N] is the transient diffusion matrix, {f} corresponds to thequation residuals and the subscript t denotes the current time step.
As for the discretization of the constitutive equation, the effect of convective term u · ∇ Sk should be well considered. When theonvective term becomes dominant as the Weissenberg number increases, the classical Galerkin weighting residual method will lose itsptimal approximation. The inconsistent streamline upwind (SU) method is adopted here to control the convective effect and an asymmetriceighting function W is introduced
W = N + ku
(u · u)· ∇N (28)
here N is equal to the classical Galerkin weighting function, the coefficient k is defined by the velocity components at the element centeru� , u�, u�) as proposed by Marchal and Crochet (1987)
k =(u2
�+ u2
� + u2� )
1/2
2(29)
The additional term of the weighting function (ku/(u · u)) · (∇N) is only imposed on the purely advective term in the constitutivequation (21) and the following elemental constitutive equation can be obtained∫
˝e
N
(�k
(∂Sk
∂t− (∇ui) · Sk − Sk · (∇u)T + �k
2
((∇u + (∇u)T ) · Sk + Sk · (∇u + (∇u)T )
)))d +
∫˝e
W (�k(u · ∇Sk))d˝
+∫
˝e
N((
exp(
�kεk
ˇ�ttr(Sk)
))Sk
)d =
∫˝e
N(ˇ�t(∇u + (∇u)T ))d (30)
Eq. (30) is then expanded along three directions of coordinate components and the elemental stiffness matrix equation is obtained asollows⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
[kij11] [kij
12] [kij13] 0 0 0
[kij21] [kij
22] [kij23] [kij
24] [kij25] 0
[kij31] [kij
32] [kij33] 0 [kij
35] [kij36]
0 [kij42] 0 [kij
44][
kij45
]0
0 [kij52] [kij
53] [kij54] [kij
55] [kij56]
0 0 [kij63] 0 [kij
65] [kij66]
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
{Sjxx}
{Sjxy}
{Sjxz}
{Sjyy}
{Sjyz}
{Sjzz}
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
+
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
[nij11] [nij
12] [nij13] 0 0 0
[nij21] [nij
22] [nij23] [nij
24] [nij25] 0
[nij31] [nij
32] [nij33] 0 [nij
35] [nij36]
0 [nij42] 0 [nij
44] [nij45] 0
0 [nij52] [nij
53] [nij54] [nij
55][
nij56
]0 0 [nij
63] 0 [nij65] [nij
66]
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
{∂Sjxx/∂t}
{∂Sjxy/∂t}
{∂Sjxz/∂t}
{∂Sjyy/∂t}
{∂Sjyz/∂t}
{∂Sjzz/∂t}
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
=
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
{f i1}
{f i2}
{f i3}
{f i4}
{f i5}
{f i6}
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(i, j = 1, 2, . . . , 8)
(31)
The global stiffness matrix equation of the constitutive equation, Eq. (32) for all the elements is assembled after the elemental calculationccording to the nodes superposition principle.
[K]{S}t + [N]
{∂S
∂t
}t
= {f }t (32)
here [K] is the global stiffness matrix of the unknown stress tensors {S}, [N] is the transient diffusion matrix, {f} corresponds to thequation residuals and the subscript t means the current time step.
The energy equation (22) is a typical convection–diffusion problem and calculation oscillation of temperature field probably happensith the increase of the Peclet number Pe that defined as (Brooks & Hughes, 1982)
Pe = �Cpu
k(33)
here u is the mean flow velocity in the downstream of the flow channel.
t
w
w
e
w
b
wr
f
a
wr
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 197
The streamline upwind Petrov–Galerkin (SUPG) formulation is employed to control the undesirable oscillations in the calculation ofemperature field. Another asymmetric weighting function � is introduced in the study
� = N + ku
||u||2 · ∇N (34)
here the coefficient k is defined as
k = �u�h� + �u�h� + �u�h�
2(35)
here the coefficient (�, �, �) is related to the local Peclet number in each element as proposed by Brooks and Hughes (1982).The weighted function � is applied to all terms of the energy equation (22) at the elemental level and the following elemental energy
quation is obtained.∫˝e
�
(�Cp
(∂T
∂t+ u · ∇T
))d =
∫˝e
� (k∇2T)d +∫
˝e
�
(� : ∇u + �˛∞�Hc
∂˛
∂t
)d (36)
After Green–Gauss transformation, the weak form of Eq. (36) is obtained∫˝e
�
(�Cp
(∂T
∂t+ u · ∇T
))d +
∫˝e
∇� (k∇T)d =∫
˝e
�
(� : ∇u + �˛∞�Hc
∂˛
∂t
)d +
∫Se
� ∗(k∇T) · ndS (37)
here �* is equal to the interpolation function of the boundary force.The discretized energy equation (37) can be written as the following elemental stiffness matrix equation.
⎡⎢⎢⎢⎢⎣
k11 k12 · · · k18
k21 k22 · · · k28
......
. . ....
k81 k82 · · · k88
⎤⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
T1
T2
...
T8
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
+
⎡⎢⎢⎢⎢⎣
n11 n12 · · · n18
n21 n22 · · · n28
......
. . ....
n81 n82 · · · n88
⎤⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
∂T1/∂t
∂T2/∂t
...
∂T8/∂t
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
f1
f2
...
f8
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
(38)
According to the nodes superposition principle, the global stiffness matrix equation of the energy equation (39) for all the elements cane assembled after evaluating the elemental stiffness matrix at elemental level by using Eq. (38).
[K]{T}t + [N]
{∂T
∂t
}t
= {f }t (39)
here [K] is the global stiffness matrix of the unknown variable {T}, [N] is the transient diffusion matrix, {f} corresponds to the equationesiduals and the subscript t means the current time step.
As for the Schneider equation (10), each differential equation can be written as the following convective diffusion equation
∂ϕ
∂t+ u · ∇ϕ = Q (40)
According to classical Galerkin weighting residual method where the weighting function is taken as the same form as the interpretationunction, the discretization form of the Schneider equation can be obtained∫
˝e
N
(∂ϕ
∂t
)d +
∫˝e
N(u · ∇ϕ)d =∫
˝e
NQd˝ (41)
The discretized Schneider equation (41) can be written as the following elemental stiffness matrix equation.
⎡⎢⎢⎢⎢⎣
k11 k11 · · · k18
k21 k22 · · · k28
......
. . ....
k81 k82 · · · k88
⎤⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
ϕ1
ϕ2
...
ϕ8
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
+
⎡⎢⎢⎢⎢⎣
n11 n11 · · · n18
n21 n22 · · · n28
......
. . ....
n81 n82 · · · n88
⎤⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
∂ϕ1/∂t
∂ϕ2/∂t
...
∂ϕ8/∂t
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
f1
f2
...
f8
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
(42)
The global stiffness matrix equation of the Schneider equation Eq. (43) for all the elements is assembled after the elemental calculationccording to the nodes superposition principle.{
∂ϕ}
[K]{ϕ}t + [N]∂t
t
= {f }t (43)
here [K] is the global stiffness matrix of the unknown variable {ϕ}, [N] is the transient diffusion matrix, {f} corresponds to the equationesiduals and the subscript t means the current time step.
198 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
3
w
dos
c
w
o
4
d(ripboK
cmwiscw1t
Fig. 2. Geometric model of the plane couette flow.
.3. Finite element–finite difference formulations
According to the definition of finite difference method, two-point difference scheme of a typical convective diffusion equation can beritten as follows
a
{∂C
∂t
}t
+ (1 − a)
{∂C
∂t
}t−�t
= 1�t
({C}t − {C}t−�t) (44)
When different coefficient value a (a = 1, a = 0, a = 1/2 and a = 2/3) is adopted in Eq. (44), different difference schemes like the backwardifference scheme, the forward difference scheme, the Crank–Nicolson difference scheme and the Galerkin difference scheme can bebtained. Among the above difference schemes, Crank–Nicolson difference scheme and Galerkin difference scheme are unconditionallytable and show higher calculation accuracy (Li, Zhao, & He, 2007).
After discretized in the spatial domain by using finite element method, the basic governing equations of the thermally and flow inducedrystallization can all be written in the following format
[K]{C}t + [N]
{∂C
∂t
}t
= {f }t (45)
here the variable {C} are used to denote the unknown variables like {u}, {S}, {T} or {ϕ} here.The basic governing equation (45) on the current time step t and the last time step t − �t can be respectively written as⎧⎪⎪⎪⎨
⎪⎪⎪⎩
[K]{C}t + [N]
{∂C
∂t
}t
= {f }t
[K]{C}t−�t + [N]
{∂C
∂t
}t−�t
= {f }t−�t
(46)
Substituting Eq. (46) into Eq. (44), the following finite element–finite difference formulations with Crank–Nicolson difference schemef the governing equations can thus be obtained(
[K] + 2[N]�t
){C}t = ({f }t + {f }t−�t) +
(2[N]�t
− [K])
{C}t−�t (47)
. Experimental verification
Crystallization experiment after shear treatment is performed on both the shearing hot stage with polarizing microscope and theynamic rheometer in Koscher’s study to investigate the effect of shear on the crystallization kinetics and morphology of polypropyleneKoscher & Fulchiron, 2002). The half crystallization time was defined using the transmitted intensity between crossed polarizers and theheological measurement. According to Koscher’s experiment, the flow behavior of polymer melts in the experimental cell can be simplifiednto a plane couette flow as shown in Fig. 2. Polymer melts lie between two parallel plates with a constant gap H(H = 0.001 m). The lowerlate is fixed (Ulower = 0) and the upper plate is imposed a constant velocity Uupper(Uupper = H · �), where � is the shear rate. Thermaloundary conditions can be imposed on both the lower plate and the upper plate with temperature or heating flux. The physical propertiesf polypropylene as shown in Table 2 are adopted in the study according to the shear induced crystallization experiment executed byoscher on the dynamic rheometer, the differential scanning calorimeter and the shearing hot stage with polarizing microscope.
In order to verify the rationality of the proposed mathematical model and numerical method for the thermally and flow inducedrystallization, the simulated half crystallization time of polypropylene in the plane couette flow cell is compared with the correspondingeasured results in Koscher’s shear induced crystallization experiment as shown in Fig. 3. It is found that the simulated results agree wellith the experimental results at lower shear rate. The deviation from experimental results is found to become larger as the shear rate is
ncreasing. This is because the material parameters adopted in the study are mainly obtained based on the experiments performed at lowhear rate. When the shear rate is smaller than 0.5 s−1, the half crystallization time keeps constant. As the shear rate keeps increasing, the
rystallization kinetics process is obviously accelerated by the flow phenomenon. The simulated and experimental results can also agreeell with each other even if the crystallization temperature is changed. When the crystallization temperature is decreased from 140 ◦C to30 ◦C at constant shear rate, the increase of supercooling degree of polymer melts enhances the process of nucleation and growth rate, andhe half crystallization time is hence decreased. However, it is still difficult to quantitatively predict the field variables in the thermally and
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 199
Table 2Physical property parameters of polypropylene (Koscher & Fulchiron, 2002).
Parameter Unit Value
c1 J kg K−2 2124.0c2 J kg K−1 3.10c3 J kg K−2 1451.0c4 J kg K−1 10.68c5 W m−1 K−2 −6.25 × 10−5
c6 W m−1 K−1 0.189c7 W m−1 K−2 −4.96 × 10−4
c8 W m−1 K−1 0.31˛∞ – 0.7�Hc J kg−1 5 × 104
a K−1 1.56 × 10−1
b – 1.51 × 101
T0m
◦C 210.0Tg
◦C −4.0A – 7.4 × 1013
p – −0.38U* J mol−1 6270.0R J K−1 mol−1 8.31Kg K2 5.5 × 105
G0 m s−1 2.83 × 102
ε – 0.1� – 0.0ˇ – 1.0�t Pa s 3411
flaa
5
soi
5
tm
p
�p – 1.0 × 109
a0 – 0.54
ow induced crystallization process directly through experimental measurement. The proposed numerical simulation technology can be reasonable alternative for the investigation of complex crystallization behaviors in polymer processing. It can be adopted to qualitativelynd quantitatively predict melts flow and crystallization patterns and hence to reveal material forming mechanism.
. Application to the crystallization in plastic pipe extrusion
In plastic pipe extrusion process, the flow and thermo-mechanical history experienced within the flow channel of the extrusion die ashown in Fig. 4 can significantly affect the crystallization behavior of polymeric materials in the shaped mold where the internal structuref the products is finally determined. The thermally and flow induced crystallization behavior of polypropylene in pipe extrusion processs investigated based on the proposed mathematical model and numerical solving method in the study.
.1. Modeling of the pipe crystallization
The geometric model of the flow channel in the shaped mold of pipe extrusion is shown in Fig. 5(a) where the outer radius R is 0.02 m,he inner radius r is 0.01 m and the length L is 0.08 m. The solved domain is divided into 1248 tri-linear brick elements by means of mapping
esh generation technology as shown in Fig. 5(b).When real boundary conditions are introduced into the governing equations, the corresponding calculation can reflect practical physical
henomenon. The crystallization of polymer melts in the shaped mold of pipe extrusion has the following characteristic boundaries.
10-3
10-1
100
101
102
103
104
105
10-2
100
10 1
Simulation Tc= 130oC
Simulation Tc= 134oC
Simulation Tc= 140oC
Half c
rysta
llization tim
e (
s)
Shear rate (s-1 )
Experiment Tc= 130oC
Experiment Tc= 134oC
Experiment Tc= 140oC
Fig. 3. Comparison of the simulated and experimental result of the half crystallization time.
200 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
z
wcb
ot
5
tmpdimaC
Fig. 4. Geometric model of the flow channel in pipe extrusion die.
On the inflow boundary, the flow is fully developed and the axial velocity can be imposed as Eq. (48) with radial velocity being set toero
uz = 2Qv
R2o
(1 − r2
R2o
+ 1 − k2
ln(1/k)ln
(r
Ro
))·[
(1 − k4) − (1 − k2)2
ln(1/k)
]−1
(48)
here r =√
x2 + y2, k = Ri/Ro is the ratio of inner to outer radius on the inlet face, Qv is the volume flow rate. The associate inlet stressan be calculated by submitting the velocity profiled into Eq. (30) along with all of the gradient components in the primary flow directioneing set to zero. A fixed temperature boundary T = Tinlet is also imposed on the inlet face.
On the die wall, the non-slip condition can be imposed on all the velocity components as follows
ux = uy = uz = 0 (49)
Due to the vanishing of convective derivatives in the constitutive Eq. (30), the stress components can be calculated iteratively in termsf the non-zero components of the velocity gradients. An isothermal contact is imposed on the die wall and the melts temperature nearhe die wall is assigned to be T = Twall .
.2. Crystallization characteristics in pipe extrusion
According to practical processing conditions in pipe extrusion, the inlet volume flow rate Qv is set to be 9.42 × 10−6 m3/s, the inletemperature of polymer melts Tinlet is set to be 150 ◦C and the temperature on the die wall Twall is set to be 100 ◦C. Based on the theoretical
odel and numerical algorithm proposed for the simulation of thermally and flow induced crystallization behavior of semi-crystallineolymers, the crystallization characteristics of polypropylene in pipe extrusion process including the variation of relative crystallinity, theistribution of crystalline structure are investigated in the study as respectively shown from Figs. 6–12. For each simulation, convergence
s met through prescribed tolerances that are detailed in Section 3.1. The CPU time spent in these computations mainly depends on theaterial parameters, the boundary conditions and the mesh density within the calculation domain. In the study, about thirty time iterations
re carried out per cam revolution. Average execution time of convergence requires about 4 h on the computer with a processor of Intelore i5 CPU [email protected] GHz.
Fig. 5. Model of the flow channel in the shaped mold of pipe extrusion: (a) geometric model and (b) mesh model.
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 201
141210864200.0
0.2
0.4
0.6
0.8
1.0
Rela
tive c
rysta
llin
ity
time (s)
core
wall-in
wall-out
Fig. 6. Variation of the relative crystallinity.
Fig. 7. Distribution of the relative crystallinity at half crystallization time: (a) x = 0.0 and (b) z = 0.04.
Fig. 8. Distribution of the thermally induced nucleation density: (a) x = 0.0 and (b) z = 0.04.
Fig. 9. Distribution of the flow induced nucleation density: (a) x = 0.0 and (b) z = 0.04.
202 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
Fig. 10. Distribution of the thermally induced average crystallite radius: (a) x = 0.0 and (b) z = 0.04.
ncaphm
rlcocctmF
Fig. 11. Distribution of the flow induced average crystallite radius: (a) x = 0.0 and (b) z = 0.04.
Fig. 6 shows the variation of relative crystallinity in three points respectively near the flow channel center (x = 0.0, y = 0.015, and z = 0.04),ear the outer die wall (x = 0.0, y = 0.02, and z = 0.04) and near the inner die wall (x = 0.0, y = 0.01, and z = 0.04). It can be found that therystallization rate near the die wall is relatively larger than that in the flow channel center. This is because both the supercooling degreend the shearing action near the die wall is greater than those in the flow channel center and hence to accelerate the crystallization kineticsrocess. Fig. 7 shows the distribution of relative crystallinity on the axial cross-section (x = 0.0) and the radial cross-section (z = 0.04) atalf crystallization time. It can be found that the relative crystallinity is larger in the downstream of the flow channel where polymericaterial experience longer shearing and cooling effects compared with that in the upstream of the flow channel.Figs. 8 and 9 show the distribution of nucleation density on the axial cross-section (x = 0.0) and the radial cross-section (z = 0.04),
espectively, induced by the thermal state and the flow state. It can be found that the nucleation density near the die wall is relativelyarger than that in the flow channel center and the flow induced nuclei is dominated compared with the thermally induced nuclei. In theenter of the flow channel, the effect of temperature on the nucleation density is relatively slight because of the small thermal conductivityf polymer melts. Figs. 10 and 11 show the distribution of the average crystallite radius on the axial cross-section (x = 0.0) and the radialross-section (z = 0.04), respectively, induced by the thermal state and the flow state. It can be found that the radius of flow inducedrystallite is smaller than that induced by the thermal state. The crystallite radius near the die wall is smaller than that in the center of
he flow channel for the reason of stronger shear effect and greater supercooling degree. With the effect of shear stress, the polymericolecular chain would be straightened along the direction of force and the inner “skin-score” structure of plastic products as shown inig. 12 can hence be obtained (Jerschow & Janeschitz-Kriegl, 1997).
Fig. 12. “Skin-core” structure of the plastic product.
Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204 203
3025201510500.0
0.2
0.4
0.6
0.8
1.0
Re
lati
ve
cry
sta
llin
ity
time (s)
Q1
Q2
Q3
Q4
Fig. 13. Variation of the relative crystallinity with different volume flow rate (Q1 = 9.42 × 10−6 m3/s, Q2 = 9.42 × 10−7 m3/s, Q3 = 9.42 × 10−8 m3/s, and Q4 = 9.42 × 10−9 m3/s).
201510500.0
0.2
0.4
0.6
0.8
1.0
Re
lati
ve
cry
sta
llin
ity
time (s)
T1
T2
T3
T4
ftFreieoTb
6
nstcrpmIcadp
Fig. 14. Variation of the relative crystallinity with different temperature boundary (T1 = 80 ◦C, T2 = 90 ◦C, T3 = 100 ◦C, and T4 = 110 ◦C).
The effects of the processing conditions on the crystallization behavior including the volume flow rate and the temperature state areurther investigated as respectively shown in Figs. 13 and 14. The volume flow rate is an important processing parameter to controlhe production efficiency in extrusion process, which can be adjusted with the extruder rotational speed and the product pulled speed.ig. 13 shows the variation of relative crystallinity of point (x = 0.0, y = 0.015, and z = 0.04) in the flow channel with different volume flowate (Q1 = 9.42 × 10−6 m3/s, Q2 = 9.42 × 10−7 m3/s, Q3 = 9.42 × 10−8 m3/s, and Q4 = 9.42 × 10−9 m3/s). It can be found that polymer meltsxperience even stronger shear effect in the flow channel with the increase of volume flow rate. The flow induced nuclei number is hencencreased and the crystallization kinetics process is accelerated. The temperature state is another important processing parameter in thextrusion process which can be adjusted by the cooling system in the shaped mold. Fig. 14 shows the variation of relative crystallinityf point (x = 0.0, y = 0.015, and z = 0.04) in the flow channel with different temperature boundary (T1 = 80 ◦C, T2 = 90 ◦C, T3 = 100 ◦C, and4 = 110 ◦C). It can be found that the crystallization kinetics process is accelerated with the decrease of the die wall temperature. This isecause of the increase of the supercooling degree of polymer melts.
. Conclusions
The three-dimensional thermally and flow induced crystallization behavior of semi-crystalline polymers obeying Phan-Thien and Tan-er constitutive model has been modeled and simulated by using a penalty finite element–finite difference method with a decoupledolving method. The evolution of crystallization kinetics process was determined by using two sets of modified Schneider equations withhe thermal state and the flow state assumed to be two distinct driving forces. The discrete elastic viscous stress splitting algorithm inooperating with streamline upwinding approach can serve as a relatively robust numerical scheme. A comparison between the numericalesults of half crystallization time and its corresponding experimental results in the thermally and flow induced crystallization process iserformed. The simulated results show satisfactory agreement with those of Koscher’s experimental observations. The proposed mathe-atical model and numerical solving method have been successfully applied in the crystallization process of polypropylene pipe extrusion.
t was found that when polymer melts are extruded from the extrusion die to the shaped mold, the flow and thermo-mechanical historyan significantly influence the crystallization kinetics process. Compared with that in the flow channel center, the stronger shearing effect
nd supercooling degree near the die wall can increase the nucleation density and decrease the crystallite size. The crystallite near theie wall is found to be smaller than that in the flow channel center and the “skin-core” structure of plastic product is hence obtained inractical processing conditions.
2
A
fF(
R
ABB
BC
DD
F
GH
J
KK
KL
LMM
PR
SS
TTT
T
Y
ZZ
Z
04 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190– 204
cknowledgements
This work is financially supported by China Postdoctoral Science Foundation Special Funded Project (no. 201104621), the Programor Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (no. IRT0931), the Natural Scienceoundation of Shandong Province (no. ZR2012EEQ001) and the Specialized Research Fund for the Doctoral Program of Higher Educationno. 20090131120028).
eferences
cierno, S., Coppola, S., & Grizzuti, N. (2008). Effects of molecular weight distribution on the flow-enhanced crystallization of poly(1-butene). Journal of Rheology, 52, 551–566.outaous, M., Bourgin, P., & Zinet, M. (2010). Thermally and flow induced crystallization of polymers at low shear rate. Journal of Non-Newtonian Fluid Mechanics, 165, 227–237.rooks, A. N., & Hughes, T. J. R. (1982). Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible
Navier–Stokes equations. Computation Methods in Applied Mechanics and Engineering, 32(2), 199–259.ushman, A. C., & McHugh, A. J. (1997). A continuum model for the dynamics of flow-induced crystallization. Journal of Polymer Science Part B: Polymer Physics, 35, 1649–1650.osta, C. B. B, Maciel, M. R. W., & Filho, R. M. (2007). Considerations on the crystallization modeling: Population balance solution. Computers and Chemical Engineering, 31,
206–218.oufas, A. K., Dairanieh, I. S., & McHugh, A. J. (1999). A continuum model for flow-induced crystallization of polymer melts. Journal of Rheology, 43, 85–109.oufas, A. K. (2006). Analysis of the rheotens experiment with viscoelastic constitutive equations for probing extensional rheology of polymer melts. Journal of Rheology, 50,
749–769.avero, J. L., Secchi, A. R., Cardozo, N. S. M., & Jasak, H. (2010). Viscoelastic fluid analysis in internal and in free surface flows using the software OpenFOAM. Computers and
Chemical Engineering, 34, 1984–1993.uenette, R., & Fortin, M. (1995). A new mixed finite method for computing viscoelastic flows. Journal of Non-Newtonian Fluid Mechanics, 60, 27–52.offman, J. D., & Miller, R. L. (1997). Kinetic of crystallization from the melt and chain folding in polyethylene fractions revisited: theory and experiment. Polymer, 38,
3151–3212.erschow, P., & Janeschitz-Kriegl, H. (1997). The role of long molecules and nucleating agents in shear induced crystallization of isotactic polypropylenes. International Polymer
Processing XII, 1, 72–77.eunings, R. (1995). Parallel finite element algorithms applied to computational rheology. Computers and Chemical Engineering, 19, 647–669.ihara, S., Gouda, T., Matsunaga, K., & Funatsu, K. (1999). Numerical simulation of three-dimensional viscoelastic flow within dies. Polymer Engineering and Science, 39,
152–163.oscher, E., & Fulchiron, R. (2002). Influence of shear on polypropylene crystallization: Morphology development and kinetics. Polymer, 43(25), 6931–6942.i, H., Zhao, G., & He, L. (2007). Research on key technologies of FEM simulation of temperature field in the process of quenching. Frontiers of Materials Science in China, 1(4),
359–365.in, J. X., Wang, C. Y., & Zheng, Y. Y. (2008). Prediction of isothermal crystallization parameters in monomer cast nylon 6. Computers and Chemical Engineering, 32, 3023–3029.archal, J. M., & Crochet, M. J. (1987). A new mixed finite element for calculating viscoelastic flows. Journal of Non-Newtonian Fluid Mechanics, 26, 77–114.u, Y., Zhao, G. Q., & Zhang, C. R. (2010). Three-dimensional simulation of planar contraction viscoelastic flow by penalty finite element method. International Journal for
Numerical Methods in Fluids, 63(7), 811–827.han-Thien, N., & Tanner, R. I. (1977). A new constitutive equation derived from network theory. Journal of Non-Newtonian Fluid Mechanics, 2, 353–365.uan, C. L., Ouyang, J., Liu, S. Q., & Zhang, L. (2011). Computer modeling of isothermal crystallization in short fiber reinforced composites. Computers and Chemical Engineering,
35, 2306–2317.chneider, W., Koppl, A., & Berger, J. (1988). Non-isothermal crystallization of polymers. International Polymer Processing, 2, 151–154.tadlbauer, M., Janeschitz-Kriegl, H., Eder, G., & Ratajski, E. (2004). New extensional rheometer for creep flow at high tensile stress. Part II. Flow induced nucleation for the
crystallization of iPP. Journal of Rheology, 48, 631–639.anner, R. I. (2002). A suspension model for low shear rate polymer solidification. Journal of Non-Newtonian Fluid Mechanics, 102, 397–408.anner, R. I. (2003). On the flow of crystallizing polymers. I. Linear regime. Journal of Non-Newtonian Fluid Mechanics, 30, 251–268.anner, R. I., & Qi, F. (2005). A comparison of some models for describing polymer crystallization at low deformation rates. Journal of Non-Newtonian Fluid Mechanics, 127,
131–141.itomanlio, G., Speranza, V., & Brucato, V. (1997). On the simulation of thermoplastic injection molding process. Part 2. Relevance of interaction between flow and crystal-
lization. International Polymer Processing, 12, 45–53.u, F. Y., Zhang, H. B., Zheng, H., Yu, W., & Zhou, C. X. (2008). Experimental study of flow-induced crystallization in the blends of isotactic polypropylene and poly(ethylene-
co-octene). European Polymer Journal, 44, 79–86.
heng, R., & Kennedy, P. K. (2004). A model for post-flow induced crystallization: General equations and predictions. Journal of Rheology, 48(4), 823–842.inet, M., Otmani, R. E., Boutaous, M., & Chantrenne, P. (2010). Numerical modeling of nonisothermal polymer crystallization kinetics and thermal effects. Polymer Engineeringand Science, 50(10), 2044–2059.uidema, H., Peters, G. W. M., & Meijer, H. E. H. (2001). Development and validation of a recoverable strain-based model for flow-induced crystallization of polymers.
Macromolecular Theory and Simulations, 10, 447–460.
