Numerical investigation of the thermally and flow induced crystallization behavior of semi-crystalline polymers by using finite element–finite difference method

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  • 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

    Numerbehavimethod

    Yue Mua

    a Key Laboratob Engineering Rc Key Laborator

    a r t i c l

    Article history:Received 8 SepReceived in reAccepted 21 JuAvailable onlin

    Keywords:Semi-crystalliThermally indFlow inducedCrystallizationViscoelasticFinite element

    1. Introdu

    The naestablishednucleation prediction oboth the thenonlinear m

    Many reprocessing model (Acie& Meijer, 2or supercooand the cry

    Correspon250061, PR Ch

    E-mail add

    0098-1354/$ http://dx.doi.oical investigation of the thermally and ow induced crystallizationor of semi-crystalline polymers by using nite elementnite difference

    ,b,c,, Guoqun Zhaoa,b,, Anbiao Chenb, Xianghong Wua,b

    ry for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandong 250061, PR Chinaesearch Center for Mould & Die Technologies, Shandong University, Jinan, Shandong 250061, PR Chinay of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China

    e i n f o

    tember 2011vised form 20 June 2012ne 2012e 30 June 2012

    ne polymersuced

    nite difference simulation

    a b s t r a c t

    The thermally and ow induced crystallization behavior of semi-crystalline polymer in processing cansignicantly inuence the quality of nal products. The investigation of its mechanism has both scien-tic and industrial interest. A mathematical model in three dimensions for thermally and ow inducedcrystallization of polymer melts obeying differential Phan-Thien and Tanner (PTT) constitutive modelhas been developed and solved by using the nite elementnite difference method. A penalty methodis introduced to solve the nonlinear governing equations with a decoupled algorithm. The correspondingnite elementnite difference model is derived by using the discrete elastic viscous split stress algo-rithm incorporating the streamline upwind scheme. A modied Schneiders approach is employed todiscriminate the relative roles of the thermal state and the ow state on the crystallization phenomenon.The thermally and ow induced crystallization characteristics of polypropylene is investigated based onthe proposed mathematical model and numerical scheme. The half crystallization time of polypropylenein a cooled couette ow conguration obtained by simulation are compared with Koschers 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 ow state are investigated.Both the crystalline distribution and crystalline size of polypropylene are obtained by using the niteelementnite difference simulation of three-dimensional thermally and ow induced crystallization.The effects of processing conditions including the volume ow rate and the temperature boundary onthe crystallization kinetics process are further discussed.

    2012 Published by Elsevier Ltd.

    ction

    l properties of semi-crystalline polymer products are to a great extent determined by the internal crystalline structures that during processing. The ow and thermo-mechanical history experienced by polymer melts during processing can enhanceand crystallization, and hence to accelerate the process and lead to different types of crystalline structure. Therefore, accuratef properties of the nal products by computer simulation requires not only appropriate crystallization model that consideringrmal and the ow effects on crystalline structure development but also robust numerical solution methods for the complicatedathematical model.searchers have been working to investigate and explain the thermally and ow induced crystallization behavior in polymerby theoretical research. The main idea is rstly to choose appropriate rheological constitutive equation, such as DoiEdwardsrno, Coppola, & Grizzuti, 2008), upper convective Maxwell model (Koscher & Fulchiron, 2002), Leonov model (Zuidema, Peters,001) and conformation tensor model (Bushman & McHugh, 1997). The variables like stress tensor, strain tensor, shear rateling degree are then assumed to be the driving force of nucleation. The simultaneous equations of the constitutive equationstallization kinetic equations are nally solved to predict the behavior of thermally and ow induced crystallization. The

    ding authors at: Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandongina. Tel.: +86 531 88393238; fax: +86 531 88392811.resses: ymu@sdu.edu.cn (Y. Mu), zhaogq@sdu.edu.cn (G. Zhao).

    see front matter 2012 Published by Elsevier Ltd.rg/10.1016/j.compchemeng.2012.06.026

  • Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204 191

    Nomenclature

    u velocity vectorp d S v

    pt p Cpk

    Hca0T0mU*

    R Tg

    micro-rheoow inducefree energyterm was hequation fothe Avramithe experimcrystallizatisuspensionamorphous(2001) consand Kennednucleation polypropylewith athermZhou (2008under the

    Althoughing the comprocessing.hence to exsolve the noproperties o& Filho, 200further dev

    In the prelementnstitutive moThe correspnite elemeviscous spliThe ellipticconstitutiveprocess is ddriving forchalf crystallhydrostatic pressureextra stress tensordeformation rate tensorpolymer-contribution stressNewtonian-contribution viscosityretardation ratiopolymer-contribution viscositytotal viscosityrelaxation timereference viscosityelasticity parameterslip parameterpenalty factormaterial densityspecic heatthermal conductivityabsolute crystallinityrelative crystallinitylatent heat of crystallizationshape factor corresponding to crystallization growthequilibrium melting temperatureactivation energygas constantglass transition temperature

    logical model that can depict the microscopic morphology of polymer melts was adopted in Coppolas work to establish thed crystallization model where the variation of conformational structure caused by the ow eld should change the system

    (Acierno et al., 2008). The variation of free energy was thought to affect the crystallization kinetics process and a free energyence added into the nucleation equation. Doufas (Doufas, Dairanieh, & McHugh, 1999; Doufas, 2006) deduced a constitutiver the ow induced crystallization behavior of polymer melts in uniform ow eld based on the Hamiltonian/Poisson theory and

    equation. The presented constitutive equation can not only describe the viscosity variation of polymer melts but also predictental results of ow birefringence. Lin, Wang, and Zheng (2008) adopted cellular automaton model to investigate isothermalon of monomer casting nylon 6 based on the Kim and Kims rate theory for spherulite expansion. Tanner (2002) proposed a

    model for polymer solidication at low shear rate. The spherulites were assumed to be hard spheres that suspended in the phase and the stress distribution was calculated based on the hypothesis of concentrated suspension system. Zuidema et al.

    idered the recovery strain as the driving force of ow induced crystallization by using the Leonov constitutive model. Zhengy (2004) presented an idea that to insert the free energy term calculated with the conformation tensor model into the Ederrate equation (Stadlbauer, Janeschitz-Kriegl, Eder, & Ratajski, 2004) to predict the ow induced crystallization behavior ofne. Ouyang (Ruan, Ouyang, Liu, & Zhang, 2011) investigated the isothermal crystallization of short ber reinforced compositesal nucleation and the morphology evolution was captured by using the pixel coloring technique. Yu, Zhang, Zheng, Yu, and

    ) developed a modied two-phase microstructure rheological model that can predict the evolution of viscosity and modulusow state to investigate the ow induced crystallization behavior of the supercooling polymer melts.

    many theories have been proposed to model the crystallization kinetics process of polymeric material, the difculty in solv-plicated governing equations for the thermally and ow induced crystallization hinders its application to practical polymer

    Numerical simulation as a highly effective method can well predict such complex phenomenon as in polymer processing andplain relevant material forming mechanism (Keunings, 1995). However, there is few research work performed aiming at how tonlinear thermally and ow induced crystallization problem by using numerical methods. It is still a difcult task to improve thef nal products through obtaining desired crystalline structure under rational control of processing conditions (Costa, Maciel,7). This may hinder the application of the proposed theoretical model in practical polymer processing and hence to hinderelopment of corresponding crystallization kinetics theory.esent study, the thermally and ow induced crystallization behavior of polymeric material is investigated by means of niteite difference method. The viscoelastic rheological properties are described by using differential Phan-ThienTanner (PTT) con-del. The mathematical model of three-dimensional thermally and ow induced crystallization of polymer melts is established.onding nite elementnite difference model is derived and the details of numerical schemes are introduced. The penaltynt method and a decoupled solving method are adopted to reduce the computation memory requirement. The discrete elastict stress (DEVSS) algorithm incorporating the streamline upwind scheme is employed to improve the computation stability.ity of the momentum equation is improved by the addition of a stabilization factor and the inuence of convection term in the

    equation and the energy equation is controlled by an asymmetric weighting function. The evolution of crystallization kineticsetermined by using two sets of Schneider equations with the thermal state and the ow state assumed to be two distinctes. The feasibility of the proposed mathematical model and numerical solving method is veried by comparing the calculatedization time in cooled couette ow conguration with those of Koschers experimental results. The thermally and ow induced

  • 192 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204

    crystallization behavior of polypropylene in pipe extrusion process are investigated based on the established mathematical model andnumerical algorithm. The crystalline distribution and crystalline size within the ow channel are predicted and the effects of processingconditions are further discussed.

    2. Mathem

    2.1. Balanc

    Accordincan be obtaconservatio

    u

    (

    Cp

    (

    where is tp is the hydlatent heat

    A generashear thinn2010). The rPTT polyme

    = 2

    where v isdeformatio

    d =

    By intro

    =

    where t =The poly

    kSk

    where k isPhan-Thien

    g(Sk)

    The full GordonSch

    Sk =

    Eq. (9) c

    2.2. Crystal

    Crystalliand grow iSchneider edifferentialatical modeling

    e equations

    g to the theory of computational uid dynamics, the governing equations to solve the ow problem of semi-crystalline polymersined from the continuity equation, Eq. (1), momentum equation, Eq. (2) and energy equation, Eq. (3), respectively, based on then of mass, momentum and energy.

    = 0 (1)

    u

    t+ u u

    )= p (2)

    T

    t+ u T

    )= k (T) + : u + Hc

    t(3)

    he material density, u is the ow velocity, is the Hamilton differential operator, t means time, is the additional stress tensor,rostatic pressure, Cp is the specic heat, T is temperature, k is the thermal conductivity, is absolute crystallinity, Hc is theof crystallization, and is the relative crystallinity.l Maxwell-type differential constitutive equation is adopted to model the viscoelastic properties of polymer melts, namely,ing, non-zero second normal stress coefcient, and stress overshoot in transient shear ows (Favero, Secchi, Cardozo, & Jasak,heology of the polymer solutions are here represented as the sum of a Newtonian solvent contribution with a full multi-moder contribution as follows

    vd + S (4)

    the Newtonian-contribution viscosity, S =n

    k=1Sk is the sum of polymer-contribution stress of each different mode k, d is then rate tensor which obeys

    u + uT2

    (5)

    ducing the retardation ratio , dened as

    p

    t(6)

    p + v is the total viscosity and p is the polymer-contribution viscosity.mer-contribution stress Sk of each mode obeys the following equation

    + g(Sk)Sk = 2td (7)

    each single relaxation time and g(Sk) is a stress function following the exponential form proposed in the original work of and Tanner (1977)

    = exp(

    kkt

    tr(Sk))

    (8)

    PTT constitutive model based on GordonSchowalter (GS) convective derivative is adopted andSk denotes the following

    owalter convective derivative operator

    Skt

    + u Sk (u kd) Sk Sk (u kd)T (9)

    an be simplied into the upper convective derivative when the parameter of the GS derivative is set to zero.

    lization kinetics model

    zation of semi-crystalline polymers can be described as a nucleation and growth process of crystallites. The nuclei are activatednto crystallites at rates depending on the ow and the thermo-mechanical state of the molten polymers in processing. Thequations are adopted in the study to model the thermally and ow induced crystallization by means of a set of rst order

    equations (Schneider, Koppl, & Berger, 1988). In order to discriminate the thermal and the ow effects, the Schneider equations,

  • Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204 193

    Eq. (10) are modied by expressing the total number of nuclei as the sum of two kind nuclei that one induced by the thermal state NT andanother one induced by the ow state Nf (Boutaous, Bourgin, & Zinet, 2010)

    T,f3 (t) T,f T,f

    where N(t) impingemethe spherulT and f, resp

    The corr

    (t) =

    where g0(t)volume and

    g0(t)

    Accordindepends on

    NT (t)

    where T(tConsider

    strain as pro

    dNf (d

    where | (t)represents

    The follodepend on

    G(t) =

    where U* isimpossible,

    Due to tresearches suspensionas follows (

    =

    where isThe ther

    described wrelative crysemi-crysta

    Cp

    Cp

    Cp

    where Cpa athermal cont+ u 3 (t) = 8N (t)

    T,f2 (t)t

    + u T,f2 (t) = G(t)T,f3 (t)T,f1 (t)t

    + u T,f1 (t) = G(t)T,f2 (t)T,f0 (t)

    t+ u T,f0 (t) = G(t)T,f1 (t)

    (10)

    is the nucleation rate, G(t) is the spherulitic growth rate, 0 is the total extended volume of the spherulite Vtot per unit volume ifnt is disregarded, 1 is the total surface area of the spherulite Stot per unit volume, 2(2 = 4Rtot) is proportional to the sum ofite radii Rtot per unit volume, 3(3 = 8N) is proportional to the number of the spherulites N per unit volume, the superscriptsectively, denote the nuclei induced by the thermal state and the ow state.ection for the impingements is then introduced by using the Avrami equation and lead to the following relative crystallinity

    1 exp(g0(t)) (11)

    is the extended global crystalline volume fraction that can be obtained by adding the thermally induced extended crystalline the ow induced extended crystalline volume as follows

    = T0(t) + f0(t) (12)

    g to the experimental study of Koscher, the thermally induced nuclei number of the molten polymer NT in quiescent state the degree of supercooling T (Koscher & Fulchiron, 2002)

    = exp(aT(t) + b) (13)

    ) = T0m T(t), T0m is the equilibrium melting temperature, a and b are two material parameters to be identied with experiments.ing the ow effect on crystallization kinetics, the number of ow induced nuclei is linked to both the shear rate and the shearposed by Tanner and Qi (2005). The number of ow induced nuclei Nf can be estimated from the following differential equation

    , t)t

    = Nf ( , t) = A| (t)|p( t) (14)

    |p represents the effect of the chain relaxation, is the shear rate, A and p are two material parameters to be identied and Nfthe ow induced nucleation rate.wing HoffmanLauritzen expression is adopted to describe the radial growth rate of the spherulites G(t) that is considered tothe thermal state (Hoffman & Miller, 1997)

    G0exp(

    U

    R (T(t) T))

    exp(

    KgT(t)T

    )(15)

    the activation energy, R is the gas constant, T = Tg 30 C is the temperature below which molecular motion becomes Tg is the glass transition temperature, parameters G0 and Kg can be determined by experiment.he effect of crystallization behavior on the viscosity of molten polymers, sudden increase of viscosity has been found in manywhen the relative crystallinity reaches a critical value (Titomanlio, Speranza, & Brucato, 1997). According to the concentrated

    theory as proposed by Tanner, the viscosity of polymer melts in crystallization process can be linked to the relative crystallinityTanner, 2003)

    t

    (1

    a0

    )2(16)

    the polymer viscosity for a given relative crystallinity, a0 is a shape factor that corresponding to crystalline growth.mophysical properties of semi-crystalline polymers can also be inuenced by the crystallization behaviors and they can beith a simple mixing rule between the properties of the amorphous phase and the semi-crystalline phase weighted by the

    stallinity (Zinet, Otmani, Boutaous, & Chantrenne, 2010). The variation of heat capacity Cp and thermal conductivity k of thelline polymers in the crystallization process can be written as follows

    = Cpc + (1 )Cpaa(T) = c1T + c2c(T) = c3T + c4

    k = kc + (1 )kaka(T) = c5T + c6kc(T) = c7T + c8

    (17)

    nd Cpc are, respectively, the specic heat of the amorphous state and the semi-crystalline state, ka and kc are, respectively, theductivity 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 specied

    Momentum and continuity equation, Eq. (25) u, p , , ConstitutiveEnergy equaSchneider eq

    3. Numeri

    3.1. Solutio

    In the stmomentumintroduced following m

    (

    To impro1995) is adoobtained

    (

    where ( uThe non

    thermally aically discreThe simulat

    As it is sthe momenZhang, 201rate tensor

    (

    where di1

    iteration steThe poly

    the velocity

    k

    (

    +(

    where the n can then b

    Based onfrom the en

    Cp

    (

    where the theat due to

    Accordininduced nustructure ca

    The convWhen the a

    1N

    t p

    equation, Eq. (30) S , , tion, Eq. (37) T Cp , , Hc , kuation, Eq. (41) 0, 1, 2, 3 NT , Nf , G

    cal algorithm

    n method

    udy, the penalty function method is employed to solve the nonlinear system consisting of the continuity equation and the equation (Kihara, Gouda, Matsunaga, & Funatsu, 1999). The penalty factor as a function obeying p = (p/( u)) is rstlyto avoid a direct calculation of the hydrostatic pressure and hence to reduce the computation memory requirement. Theomentum governing equation can be obtained

    u

    t+ u u

    ) (p u) 2(1 )t d S = 0 (18)

    ve the convergence property of the numerical scheme, the discrete elastic viscous split stress formulation (Guenette & Fortin,pted and an ellipticity factor 2 [(u) d] is introduced into the momentum equation, Eq. (18). The following equation is thus

    u

    t+ u u

    ) p( u) + 2 (u) = S + 2( + (1 )t) d (19)

    ) = ( u + T u)/2, is a reference viscosity.linear system comprised by the ow balance equations and the crystallization kinetics equations can be used to predict thend ow induced crystallization behavior of semi-crystalline polymers. The governing equations as shown in Table 1 are numer-tized in the time and the spatial domain respectively by using the nite differential method and the nite element method.ion ow chart of the thermally and ow induced crystallization can be seen from Fig. 1.hown in Fig. 1, a decoupled iteration method is employed to solve the governing equations including the continuity equation,tum equation, the energy equation, the constitutive equation and the Schneider equation in current time step j (Mu, Zhao, &0). The velocity vector is rstly solved from Eq. (19) by treating the polymer-contribution stress tensor and the deformationas known terms

    ui

    t+ ui ui

    ) p( ui) + 2 (ui) = Si1 + 2( + (1 )t) di1 (20)

    and Si1 are, respectively, the deformation rate tensor and the stress tensor calculated with the ow eld at the previousp i1 and their initial values are derived from a viscous ow eld, i is the current iteration step of ow eld calculation.mer-contribution stress tensor of each different mode is then calculated through the constitutive equation, Eq. (21) based on

    solution obtained from the calculation of the momentum equation, Eq. (20).

    Sikt

    + ui Sik (ui) Sik Sik (ui)T + k

    2

    ((ui + (ui)T ) Sik + Sik (ui + (ui)

    T)))

    exp(

    kkt

    tr(Si1k )))

    Sik = t(ui + (ui)T) (21)

    on-linear term is derived from the stress tensor obtained from last iteration step i1. The polymer-contribution stress tensore obtained by adding the polymer-contribution stress tensor of each different mode Sk.

    the velocity vector and stress tensor, respectively, obtained from Eqs. (20) and (21), the temperature eld is then calculatedergy equation (22)

    Ti

    t+ ui Ti

    )= k (Ti) + i : ui + Hc

    i1

    t(22)

    erms on the right hand side, respectively, correspond to the heat exchange, the viscous heat dissipation and the release of latent the crystallization behavior.g to the calculated results of the ow and temperature eld, the number of ow induced nuclei Nf, the number of thermallyclei NT and the growth rate of crystallites G(t) can be obtained. Both the relative crystallinity and the information on crystallinen then be solved based on the thermally and ow induced crystallization kinetics model as depicted in Section 2.2.

    ergence criteria of calculation within the current ow eld are dened upon the following requirements as shown in Eq. (23).verage relative error of each variable is less than a set value, the convergence is considered to be achieved.u

    i ui1ui

    < u, 1NS

    i Si1Si

    < S, 1NT

    i Ti1Ti

    < T , 1N

    i i1i

    < (23)

  • Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204 195

    START

    Input material parameters and boundary conditions

    where N is which is secriteria of tcurrent stud

    3.2. Finite e

    The Galeis taken as tfollows

    eN

    After Gr

    eN

    where N* isand n is theEstablishment 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)

    Calculation of stress tensor ( ) using SU algorithm

    Solution iteration, i = 0

    Establishment of the FE/FD model of temperature field (Energy equation Eq.37)

    Calculation of temperature ( )T using SUPG algorithm NO

    YES

    END

    Output results of the simulation

    YES

    NO Is iteration convergent?

    i = i +1

    Update of material parameters

    Is iteration convergent?t = t + dt

    Establishment of the FE/FD model of crystallization field (Schneider equation Eq.41)

    Calculation of relative crystallinity ( ) and crystalline morphology

    Fig. 1. Simulation ow chart of the thermally and ow induced crystallization.

    the total number of nite element nodes in the computational domain, u is the convergence criteria of the velocity vectort to 104, S is the convergence criteria of the polymer-contribution stress tensor which is set to 103, T is the convergencehe temperature which is set to 104 and is the convergence criteria of the relative crystallinity which is set to 104 in they.

    lement formulations

    rkin weighting residual method is adopted for the discretization of the momentum equation where the weighting functionhe same form as the interpretation function. The discretized momentum equation at the elemental level can be expressed as

    (

    (u

    t+ u u

    ) p( u) + 2 (u)

    )d =

    e

    N( S + 2( + (1 )t) d)d (24)

    eenGauss transformation, the following weak form of Eq. (24) can be obtained(

    (u

    t+ u u

    ))d +

    e

    N( (u + uT ) p( u))d =

    eN(S + 2( + (1 )t)d) +

    Se

    N(S pI) ndS (25)

    equal to the interpolation function on the boundary force, e and Se are, respectively, the element region and element boundary, outer normal unit vector.

  • 196 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 followingelemental stiffness matrix equation can be obtained.

    [kijxx] [k

    ijxy] [k

    ijxz]

    [kiy

    [kiz

    {ujx} [nijxx] [nijxy] [nijxz]

    {ujx/t} {f ix}

    Accordinequation, E

    [K]{u

    where [K] iequation re

    As for thconvective optimal appweighting f

    W =

    where N is e(u , u, u)

    k =(

    The addequation (2

    eN

    +

    Eq. (30) follows

    [kij11

    [kij21

    [kij31

    0

    0

    0

    The globaccording t

    [K]{S

    where [K] iequation re

    The enerwith the inc

    Pe =

    where u is tjx] [k

    ijyy] [k

    ijyz]

    jx] [k

    ijzy] [k

    ijzz]

    {ujy}

    {ujz}

    + [nijyx] [nijyy] [nijyz]

    [nijzx] [nijzy] [n

    ijzz]

    {ujy/t}

    {ujz/t}

    = {f

    iy}

    {f iz }

    (i, j = 1, 2, . . . , 8)

    (26)

    g to the nodes superposition principle and element connectivity, the global stiffness matrix equation of the momentumq. (27) for all the elements can be assembled after evaluating the elemental stiffness matrix at elemental level using Eq. (26).

    }t + [N]{

    u

    t

    }t

    = {f }t (27)

    s the global stiffness matrix of the unknown velocity vectors {u}, [N] is the transient diffusion matrix, {f} corresponds to thesiduals and the subscript t denotes the current time step.e discretization of the constitutive equation, the effect of convective term u Sk should be well considered. When theterm becomes dominant as the Weissenberg number increases, the classical Galerkin weighting residual method will lose itsroximation. The inconsistent streamline upwind (SU) method is adopted here to control the convective effect and an asymmetricunction W is introduced

    N + ku(u u) N (28)

    qual to the classical Galerkin weighting function, the coefcient k is dened by the velocity components at the element centeras proposed by Marchal and Crochet (1987)

    u2

    + u2 + u2 )1/2

    2(29)

    itional term of the weighting function (ku/(u u)) (N) is only imposed on the purely advective term in the constitutive1) and the following elemental constitutive equation can be obtained(

    k

    (Skt

    (ui) Sk Sk (u)T + k2((u + (u)T ) Sk + Sk (u + (u)T )

    )))d +

    e

    W (k(u Sk))d

    eN((

    exp(

    kkt

    tr(Sk)))

    Sk

    )d =

    e

    N(t(u + (u)T ))d (30)

    is then expanded along three directions of coordinate components and the elemental stiffness matrix equation is obtained as

    ] [kij12] [kij13] 0 0 0

    ] [kij22] [kij23] [k

    ij24] [k

    ij25] 0

    ] [kij32] [kij33] 0 [k

    ij35] [k

    ij36]

    [kij42] 0 [kij44]

    [kij45

    ]0

    [kij52] [kij53] [k

    ij54] [k

    ij55] [k

    ij56]

    0 [kij63] 0 [kij65] [k

    ij66]

    {Sjxx}

    {Sjxy}

    {Sjxz}

    {Sjyy}

    {Sjyz}

    {Sjzz}

    +

    [nij11] [nij12] [n

    ij13] 0 0 0

    [nij21] [nij22] [n

    ij23] [n

    ij24] [n

    ij25] 0

    [nij31] [nij32] [n

    ij33] 0 [n

    ij35] [n

    ij36]

    0 [nij42] 0 [nij44] [n

    ij45] 0

    0 [nij52] [nij53] [n

    ij54] [n

    ij55]

    [nij56

    ]0 0 [nij63] 0 [n

    ij65] [n

    ij66]

    {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)

    al stiffness matrix equation of the constitutive equation, Eq. (32) for all the elements is assembled after the elemental calculationo the nodes superposition principle.

    }t + [N]{

    S

    t

    }t

    = {f }t (32)

    s the global stiffness matrix of the unknown stress tensors {S}, [N] is the transient diffusion matrix, {f} corresponds to thesiduals and the subscript t means the current time step.gy equation (22) is a typical convectiondiffusion problem and calculation oscillation of temperature eld probably happensrease of the Peclet number Pe that dened as (Brooks & Hughes, 1982)

    Cpu

    k(33)

    he mean ow velocity in the downstream of the ow channel.

  • Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204 197

    The streamline upwind PetrovGalerkin (SUPG) formulation is employed to control the undesirable oscillations in the calculation oftemperature eld. Another asymmetric weighting function is introduced in the study

    = N + ku N (34)

    where the c

    k =

    where the cThe weig

    equation is e

    After Gre

    where * isThe disc

    k11

    k21

    ...

    k81

    Accordinbe assemble

    [K]{T

    where [K] isresiduals an

    As for th

    t+

    Accordinfunction, th

    eN

    The disc

    k11

    k21

    ...

    k81

    The globaccording t

    [K]{

    where [K] isresiduals an||u||2

    oefcient k is dened as

    uh + uh + uh2

    (35)

    oefcient (, , ) is related to the local Peclet number in each element as proposed by Brooks and Hughes (1982).hted function is applied to all terms of the energy equation (22) at the elemental level and the following elemental energyobtained.(

    Cp

    (T

    t+ u T

    ))d =

    e

    (k2T)d +

    e

    ( : u + Hc

    t

    )d (36)

    eenGauss transformation, the weak form of Eq. (36) is obtained(Cp

    (T

    t+ u T

    ))d +

    e

    (kT)d =

    e

    ( : u + Hc

    t

    )d +

    Se

    (kT) ndS (37)

    equal to the interpolation function of the boundary force.retized energy equation (37) can be written as the following elemental stiffness matrix equation.

    k12 k18k22 k28...

    . . ....

    k82 k88

    T1

    T2

    ...

    T8

    +

    n11 n12 n18n21 n22 n28...

    .... . .

    ...

    n81 n82 n88

    T1/t

    T2/t

    ...

    T8/t

    =

    f1

    f2

    ...

    f8

    (38)

    g to the nodes superposition principle, the global stiffness matrix equation of the energy equation (39) for all the elements cand after evaluating the elemental stiffness matrix at elemental level by using Eq. (38).

    }t + [N]{

    T

    t

    }t

    = {f }t (39)

    the global stiffness matrix of the unknown variable {T}, [N] is the transient diffusion matrix, {f} corresponds to the equationd the subscript t means the current time step.e Schneider equation (10), each differential equation can be written as the following convective diffusion equation

    u = Q (40)

    g to classical Galerkin weighting residual method where the weighting function is taken as the same form as the interpretatione discretization form of the Schneider equation can be obtained(

    t

    )d +

    e

    N(u )d =

    eNQd (41)

    retized Schneider equation (41) can be written as the following elemental stiffness matrix equation.

    k11 k18k22 k28...

    . . ....

    k82 k88

    1

    2

    ...

    8

    +

    n11 n11 n18n21 n22 n28...

    .... . .

    ...

    n81 n82 n88

    1/t

    2/t

    ...

    8/t

    =

    f1

    f2

    ...

    f8

    (42)

    al stiffness matrix equation of the Schneider equation Eq. (43) for all the elements is assembled after the elemental calculationo the nodes superposition principle.

    }t + [N]{

    t

    }t

    = {f }t (43)

    the global stiffness matrix of the unknown variable {}, [N] is the transient diffusion matrix, {f} corresponds to the equationd the subscript t means the current time step.

  • 198 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204

    3.3. Finite e

    Accordinwritten as f

    a

    {C

    When didifference sobtained. Astable and s

    After discrystallizati

    [K]{C

    where the vThe basi

    [K

    [K

    Substituof the gove(

    [K]

    4. Experim

    Crystallidynamic rh(Koscher & rheologicalinto a planeplate is xeboundary cof polypropKoscher on

    In ordercrystallizatimeasured rwith the exincreasing. shear rate. Wcrystallizatiwell with e130 C at cothe half cryFig. 2. Geometric model of the plane couette ow.

    lementnite difference formulations

    g to the denition of nite difference method, two-point difference scheme of a typical convective diffusion equation can beollows

    t

    }t

    + (1 a){

    C

    t

    }tt

    = 1t

    ({C}t {C}tt) (44)

    fferent coefcient value a (a = 1, a = 0, a = 1/2 and a = 2/3) is adopted in Eq. (44), different difference schemes like the backwardcheme, the forward difference scheme, the CrankNicolson difference scheme and the Galerkin difference scheme can bemong the above difference schemes, CrankNicolson difference scheme and Galerkin difference scheme are unconditionallyhow higher calculation accuracy (Li, Zhao, & He, 2007).cretized in the spatial domain by using nite element method, the basic governing equations of the thermally and ow inducedon can all be written in the following format

    }t + [N]{

    C

    t

    }t

    = {f }t (45)

    ariable {C} are used to denote the unknown variables like {u}, {S}, {T} or {} here.c governing equation (45) on the current time step t and the last time step t t can be respectively written as

    ]{C}t + [N]{

    C

    t

    }t

    = {f }t

    ]{C}tt + [N]{

    C

    t

    }tt

    = {f }tt(46)

    ting Eq. (46) into Eq. (44), the following nite elementnite difference formulations with CrankNicolson difference schemerning equations can thus be obtained

    + 2[N]t

    ){C}t = ({f }t + {f }tt) +

    (2[N]t

    [K])

    {C}tt (47)

    ental verication

    zation experiment after shear treatment is performed on both the shearing hot stage with polarizing microscope and theeometer in Koschers study to investigate the effect of shear on the crystallization kinetics and morphology of polypropyleneFulchiron, 2002). The half crystallization time was dened using the transmitted intensity between crossed polarizers and the

    measurement. According to Koschers experiment, the ow behavior of polymer melts in the experimental cell can be simplied couette ow as shown in Fig. 2. Polymer melts lie between two parallel plates with a constant gap H(H = 0.001 m). The lowerd (Ulower = 0) and the upper plate is imposed a constant velocity Uupper(Uupper = H ), where is the shear rate. Thermalonditions can be imposed on both the lower plate and the upper plate with temperature or heating ux. The physical propertiesylene as shown in Table 2 are adopted in the study according to the shear induced crystallization experiment executed by the dynamic rheometer, the differential scanning calorimeter and the shearing hot stage with polarizing microscope. to verify the rationality of the proposed mathematical model and numerical method for the thermally and ow inducedon, the simulated half crystallization time of polypropylene in the plane couette ow cell is compared with the correspondingesults in Koschers shear induced crystallization experiment as shown in Fig. 3. It is found that the simulated results agree wellperimental results at lower shear rate. The deviation from experimental results is found to become larger as the shear rate isThis is because the material parameters adopted in the study are mainly obtained based on the experiments performed at low

    hen the shear rate is smaller than 0.5 s1, the half crystallization time keeps constant. As the shear rate keeps increasing, theon kinetics process is obviously accelerated by the ow phenomenon. The simulated and experimental results can also agreeach other even if the crystallization temperature is changed. When the crystallization temperature is decreased from 140 C tonstant shear rate, the increase of supercooling degree of polymer melts enhances the process of nucleation and growth rate, andstallization time is hence decreased. However, it is still difcult to quantitatively predict the eld 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 K2 2124.0c2 J kg K1 3.10c3 J kg K2 1451.0c4 J kg K1 10.68c5 W m1 K2 6.25 105c6 W m1 K1 0.189c7 W m1 K2 4.96 104c8 W m1 K1 0.31 0.7Hc J kg1 5 104a K1 1.56 101b 1.51 101T0m

    C 210.0Tg C 4.0A 7.4 1013p 0.38U* J mol1 6270.0R J K1 mol1 8.31Kg K2 5.5 105G0 m s1 2.83 102 0.1 0.0 1.0t Pa s 3411p 1.0 109a0

    ow inducea reasonabland quantit

    5. Applicat

    In plastishown in Fiof the produis investiga

    5.1. Modeli

    The geomthe inner ramesh gener

    When rephenomeno 0.54

    d crystallization process directly through experimental measurement. The proposed numerical simulation technology can bee alternative for the investigation of complex crystallization behaviors in polymer processing. It can be adopted to qualitativelyatively predict melts ow and crystallization patterns and hence to reveal material forming mechanism.

    ion to the crystallization in plastic pipe extrusion

    c pipe extrusion process, the ow and thermo-mechanical history experienced within the ow channel of the extrusion die asg. 4 can signicantly affect the crystallization behavior of polymeric materials in the shaped mold where the internal structurects is nally determined. The thermally and ow induced crystallization behavior of polypropylene in pipe extrusion process

    ted based on the proposed mathematical model and numerical solving method in the study.

    ng of the pipe crystallization

    etric model of the ow channel in the shaped mold of pipe extrusion is shown in Fig. 5(a) where the outer radius R is 0.02 m,dius 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 mappingation technology as shown in Fig. 5(b).

    al boundary conditions are introduced into the governing equations, the corresponding calculation can reect practical physicaln. 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

    Fig. 4. Geometric model of the ow channel in pipe extrusion die.

    On the inow boundary, the ow is fully developed and the axial velocity can be imposed as Eq. (48) with radial velocity being set tozero

    uz = 2Qv(

    1 r2

    + 1 k2

    ln(

    r))

    [(1 k4) (1 k

    2)2 ]1

    (48)

    where r =

    can be calcubeing set to

    On the d

    ux = Due to t

    of the non-the die wal

    5.2. Crystal

    Accordintemperaturmodel and polymers, tdistributionis met thromaterial paare carried Core i5 CPU

    R2o R2o ln(1/k) Ro ln(1/k)

    x2 + y2, k = Ri/Ro is the ratio of inner to outer radius on the inlet face, Qv is the volume ow rate. The associate inlet stresslated by submitting the velocity proled into Eq. (30) along with all of the gradient components in the primary ow direction

    zero. A xed temperature boundary T = Tinlet is also imposed on the inlet face.ie wall, the non-slip condition can be imposed on all the velocity components as follows

    uy = uz = 0 (49)he vanishing of convective derivatives in the constitutive Eq. (30), the stress components can be calculated iteratively in termszero components of the velocity gradients. An isothermal contact is imposed on the die wall and the melts temperature nearl is assigned to be T = Twall .

    lization characteristics in pipe extrusion

    g to practical processing conditions in pipe extrusion, the inlet volume ow rate Qv is set to be 9.42 106 m3/s, the inlete 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 theoreticalnumerical algorithm proposed for the simulation of thermally and ow induced crystallization behavior of semi-crystallinehe crystallization characteristics of polypropylene in pipe extrusion process including the variation of relative crystallinity, the

    of crystalline structure are investigated in the study as respectively shown from Figs. 612. For each simulation, convergenceugh prescribed tolerances that are detailed in Section 3.1. The CPU time spent in these computations mainly depends on therameters, the boundary conditions and the mesh density within the calculation domain. In the study, about thirty time iterationsout per cam revolution. Average execution time of convergence requires about 4 h on the computer with a processor of Intel

    M450@2.40 GHz.Fig. 5. Model of the ow 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 ow 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.

    Fig. 6 shonear the oucrystallizatiand the sheprocess. Fighalf crystalmaterial ex

    Figs. 8 arespectivelylarger than center of thof polymer cross-sectiocrystallite ithe ow chmolecular cFig. 12 can Fig. 11. Distribution of the ow induced average crystallite radius: (a) x = 0.0 and (b) z = 0.04.ws the variation of relative crystallinity in three points respectively near the ow channel center (x = 0.0, y = 0.015, and z = 0.04),ter 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 theon rate near the die wall is relatively larger than that in the ow channel center. This is because both the supercooling degreearing action near the die wall is greater than those in the ow channel center and hence to accelerate the crystallization kinetics. 7 shows the distribution of relative crystallinity on the axial cross-section (x = 0.0) and the radial cross-section (z = 0.04) atlization time. It can be found that the relative crystallinity is larger in the downstream of the ow channel where polymericperience longer shearing and cooling effects compared with that in the upstream of the ow channel.nd 9 show the distribution of nucleation density on the axial cross-section (x = 0.0) and the radial cross-section (z = 0.04),, induced by the thermal state and the ow state. It can be found that the nucleation density near the die wall is relativelythat in the ow channel center and the ow induced nuclei is dominated compared with the thermally induced nuclei. In thee ow channel, the effect of temperature on the nucleation density is relatively slight because of the small thermal conductivitymelts. Figs. 10 and 11 show the distribution of the average crystallite radius on the axial cross-section (x = 0.0) and the radialn (z = 0.04), respectively, induced by the thermal state and the ow state. It can be found that the radius of ow induceds smaller than that induced by the thermal state. The crystallite radius near the die wall is smaller than that in the center ofannel for the reason of stronger shear effect and greater supercooling degree. With the effect of shear stress, the polymerichain would be straightened along the direction of force and the inner skin-score structure of plastic products as shown inhence 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

    1.0

    Fig. 13. Variat

    The effefurther invethe productFig. 13 showrate (Q1 = 9experience increased aextrusion pof point (x =T4 = 110 C)because of t

    6. Conclus

    The threner constitusolving metthe thermacooperatingresults of hperformed.matical modIt was founcan signicand supercodie wall is fpractical pr3025201510500.0

    0.2

    0.4

    0.6

    0.8

    Re

    lati

    ve

    cry

    sta

    llin

    ity

    time (s)

    Q1

    Q2

    Q3

    Q4

    ion of the relative crystallinity with different volume ow rate (Q1 = 9.42 106 m3/s, Q2 = 9.42 107 m3/s, Q3 = 9.42 108 m3/s, and Q4 = 9.42 109 m3/s).

    0.2

    0.4

    0.6

    0.8

    1.0

    Re

    lati

    ve

    cry

    sta

    llin

    ity

    T1

    T2

    T3

    T4201510500.0

    time (s)

    Fig. 14. Variation of the relative crystallinity with different temperature boundary (T1 = 80 C, T2 = 90 C, T3 = 100 C, and T4 = 110 C).

    cts of the processing conditions on the crystallization behavior including the volume ow rate and the temperature state arestigated as respectively shown in Figs. 13 and 14. The volume ow rate is an important processing parameter to controlion efciency in extrusion process, which can be adjusted with the extruder rotational speed and the product pulled speed.s the variation of relative crystallinity of point (x = 0.0, y = 0.015, and z = 0.04) in the ow channel with different volume ow

    .42 106 m3/s, Q2 = 9.42 107 m3/s, Q3 = 9.42 108 m3/s, and Q4 = 9.42 109 m3/s). It can be found that polymer meltseven stronger shear effect in the ow channel with the increase of volume ow rate. The ow induced nuclei number is hencend the crystallization kinetics process is accelerated. The temperature state is another important processing parameter in therocess which can be adjusted by the cooling system in the shaped mold. Fig. 14 shows the variation of relative crystallinity

    0.0, y = 0.015, and z = 0.04) in the ow channel with different temperature boundary (T1 = 80 C, T2 = 90 C, T3 = 100 C, and. It can be found that the crystallization kinetics process is accelerated with the decrease of the die wall temperature. This ishe increase of the supercooling degree of polymer melts.

    ions

    e-dimensional thermally and ow induced crystallization behavior of semi-crystalline polymers obeying Phan-Thien and Tan-tive model has been modeled and simulated by using a penalty nite elementnite difference method with a decoupledhod. The evolution of crystallization kinetics process was determined by using two sets of modied Schneider equations withl state and the ow state assumed to be two distinct driving forces. The discrete elastic viscous stress splitting algorithm in

    with streamline upwinding approach can serve as a relatively robust numerical scheme. A comparison between the numericalalf crystallization time and its corresponding experimental results in the thermally and ow induced crystallization process is

    The simulated results show satisfactory agreement with those of Koschers experimental observations. The proposed mathe-el and numerical solving method have been successfully applied in the crystallization process of polypropylene pipe extrusion.

    d that when polymer melts are extruded from the extrusion die to the shaped mold, the ow and thermo-mechanical historyantly inuence the crystallization kinetics process. Compared with that in the ow channel center, the stronger shearing effectoling degree near the die wall can increase the nucleation density and decrease the crystallite size. The crystallite near theound to be smaller than that in the ow channel center and the skin-core structure of plastic product is hence obtained inocessing conditions.

  • 204 Y. Mu et al. / Computers and Chemical Engineering 46 (2012) 190 204

    Acknowledgements

    This work is nancially supported by China Postdoctoral Science Foundation Special Funded Project (no. 201104621), the Programfor Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (no. IRT0931), the Natural ScienceFoundation of Shandong Province (no. ZR2012EEQ001) and the Specialized Research Fund for the Doctoral Program of Higher Education(no. 20090131120028).

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    Numerical investigation of the thermally and flow induced crystallization behavior of semi-crystalline polymers by using f...1 Introduction2 Mathematical modeling2.1 Balance equations2.2 Crystallization kinetics model

    3 Numerical algorithm3.1 Solution method3.2 Finite element formulations3.3 Finite elementfinite difference formulations

    4 Experimental verification5 Application to the crystallization in plastic pipe extrusion5.1 Modeling of the pipe crystallization5.2 Crystallization characteristics in pipe extrusion

    6 ConclusionsAcknowledgementsReferences

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