La modelisation et la silumation de l'etape de melangeage pour le procede de moulage par injection de poudres a ete entreprise. Il consiste a melange une poudre principale avec differents liants facilitant le demoulage, la diminution de la viscosite et la mouiabilite du melange. La simulation a ete realise par resolution dans le logiciel consol. La détermination du taux de cisaillement et de la viscosité à partir de la vitesse de rotation des vis de mélangeage et du couple a ete realisee. Les resultats sont prometteurs pour la prediction des differentes valeurs

BARRIERE, Thierry

DJOUDI, Hamza

GELIN, Jean-Claude

I-Revues

 
 
20
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 Congrès Français de Mécanique                                       Besançon, 29 août au 2 septembre 2011 
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Numerical simulation of polymers-carbon nanotubes 
mixing process 
H. Djoudi, T. Barrière, J. Gelin 
FEMTO-ST,applied mechanics lab., 24 Epitaphe Avenue, 25000 BESANCON (France) 
Abstract 
This paper focuses on the numerical simulation of polymer mixing process. The 
polypropylene has been mixed with different MWCNTs contents. A two dimensional 
numerical simulation of the polypropylene/ multi-walled carbon nanotubes (PP/MWCNT) has 
been investigated showing the flow into a twin-screw mixer. The velocity, the shear-rate and 
the temperature has been carried out with taking into account the thermal coupling. 
 
Keywords: Mixing process, thermal coupling, velocity filed, shear-rate filed, 
polypropylene/multi-walled carbon nanotubes. 
 
Résumé 
Cette communication concerne le procédé de mélangeage de polymères avec des nanotubes 
de carbone. Nous proposons une simulation numérique bidimensionnelle de l’écoulement des 
polymères chargés en  nanotubes de carbone dans le mélangeur indiquant le champ de  
vitesse, le champ de température et le taux de cisaillement dans la chambre de mélangeage, 
en prenant en compte le couplage thermique. Ensuite on aborde la, et nous avons calculé le 
champ de vitesse et de température durant la phase de mélangeage. 
Mots clés: Procédé de mélangeage, champ de vitesse, taux de cisaillement, champ de 
température, couplage thermique, polypropylène chargé par des nanotubes de carbones 
 
1. Introduction 
Since their official discovery in 1991 by Iijima [1], carbon nanotubes characteristics are 
growing in the field of science and engineering: their mechanical and physical properties 
make CNT as attractive candidates for composite materials (polymer/CNT) and applications. 
These mechanical performances strongly depend on the composite melting. Many studies 
have been carried out by several authors and referenced by Coleman et al. [2] and Breuer and 
Sundaraj [3]. 
One of the most important parameters during the elaboration of polymers/CNT composites is 
the nanotubes dispersion in the polymer matrix. Indeed it is critical for reinforcing the 
efficiency because the strong tendency of nanotubes to nanotubes aggregation when they are 
mixed significantly affects the mechanical properties of the resulting composite material. 
Many investigations on carbon nanotubes dispersion in the polymer matrix have been carried 
out and a part of them are related in this paper.  
Comsol® software has been chosen for modeling the CNT-polymer mixing process because 
of its capabilities for solving multiphysic coupled problems. A proper description of the 
 
 
20
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 Congrès Français de Mécanique                                       Besançon, 29 août au 2 septembre 2011 
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PP/MWCNT physics flow is obtained. This FEM analysis appears as a first step for the 
simulation of three dimensional mixing of PP/MWCNT composites. In our research, the 
mixing experiments were carried out using a Brabender® mixer (Plastograph EC W50EHT, 
Fig. 1), that allows to process with batch mixtures up to 40 cm
3
. The temperature in the 
mixing cavities is regulated. The blades are made of special steel. Counter-rotation towards 
each other at different speeds provides excellent compounding and mixing characteristics, the 
speed ration driven at idle blade is about 2:3. As it is shown in Fig. 3, the fluid is contained in 
the cavity enclosed by the frame and the two screws. The left screw rotates in the clockwise 
while the right one rotates in the counter clockwise. Both mixing torque and the temperature 
are measured using sensors and software 
 
 
 
 
 
 
FIG. 1 - Twin screw mixer (velocity max. 150 rpm, max. temp 500°c) 
2 Modeling and numerical simulation 
2.1 Governing equations 
An Arbitrary Lagrangian–Eulerian (ALE) approach with moving mesh on both    and 
  sub-domains is used to discritize the problem as shown by Thiébaud F, Gelin J [4]. By 
considering the above geometrical data, the solution of the coupled fluid flow–thermal 
problem allows us to calculate: 
The velocity field:                                       
 The pressure field :         . 
 The temperature field :         . 
 
The velocity field and the pressure are governed by the following Navier–Stokes equations: 
The temperature variation is governed by the heat equation, given by the Eq. (3): 
                                                         into                                                            (1) 
                                               
    
  
                                                                                 (2) 
                     
  
  
                                                 into                          (3) 
where    is the specific heat coefficient,         is the shear viscosity,    is the equivalent 
shear rate and    is the shear rate tensor. The Fourier’s law is: 
                                                                                                                                            (4) 
where    is the heat flow,   is the thermal conduction coefficient. The heat equation becomes: 
                    
  
  
                                             into                              (5) 
 
 
20
ème
 Congrès Français de Mécanique                                       Besançon, 29 août au 2 septembre 2011 
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and   is given by the mixture rule: 
                                                                
 
  
 
   
  
 
  
                                                        (6) 
 
where   is the mass fraction of the MWCNT,    is the Multi-walled Carbon Nanotube 
(MWCNT) density and    is the density of polypropylene. 
 
The boundary conditions to be considered for this problem are the following: 
 
Frame and screws walls adhesion:                          into the local frame. Open 
boundary condition at the mixer inlet:            . Imposed temperature on the screws and 
the frame:        on            . External heat transfer at the inlet of the mixer is: 
                                  on the    , where     is the stress tensor defined vs viscosity 
and pressure.  
                                                                                                                                (7) 
where      is the external temperature fixed at 293 K, Furthermore, continuity equations must 
be considered at the moved/fixed frames boundaries      and     . These equations can be 
written as following for the mechanical problem 
                                                                                                                                 (8) 
                                                                                                                                 (9) 
where           is the stress tensor of         
and in the following form for the thermal problem 
                                                                                                                           (10) 
                                                                                                                           (11) 
where           is the heat flow of          
The initial conditions are: 
                                                                                                                             (12) 
                                                                                                                            (13) 
                                                                                                                                 (14) 
 
where   ,      and    are respectively the initial velocity, the initial temperature    fixed to 
    , and the initial pressure. 
The equations of the moving frame are: 
                                                         
                                              
                                                         
                                                                                         (15) 
     
where         are the distances between   and the extremity        of the domain       , 
       are the Cartesian coordinates, t is the time, and    angular speed of each blade. 
2.2 FEM simulations 
COMSOL Multiphysics (formerly FEMLAB) is a finite element analysis, solver and 
Simulation software / FEA Software package for various physics and engineering 
 
 
20
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 Congrès Français de Mécanique                                       Besançon, 29 août au 2 septembre 2011 
4 
 
applications, especially coupled phenomena, or multiphysics. COMSOL Multiphysics also 
allows for entering coupled systems of partial differential equations. Comsol Multiphysics 
software has been used to implement the proposed model and to perform the 3D simulation. 
2.3 Two dimensional mixing simulation 
The geometry that is treated is associated to the mixer described in figure 1. The model is 
based on the Navier-Stokes equations formulated with a rotating frame in the inner sub-
domain and in fixed coordinates in the outer one. At the fixed boundaries no-slip boundary 
condition is applied. At the inlet and outlet, the normal stress is set to zero. The 
implementation, using the Rotating Navier-Stokes predefined multiphysics coupling, is quite 
straightforward. The characteristics of the polymer are shown in the table below. The 
computational time corresponds to 1.58min. The computer’s characteristics are: CPU : Intel® 
Pentium® Dual 2.00GHz and RAM: 2.00Go.  
In that section one relates the neat PP and PP/MWCNT composites flow into the Brabender® 
mixer during the mixing stage. A two dimensional (2D) problem is considered as an 
approximation of the section screw, one considers a laminar and incompressible flow without 
external (polymer + fiber) loading as shown by Haas A, Scholle M, Aksel N, Thompson H M, 
Hewson R W, Gaskell P H [5]. The material domain   is divided into three sub-domains 
described in figure 3. 
 
 
 
 
 
 
 
 
 
 
 
 
 
FIG. 3 - Moving and fixed frame, sub-domains and boundaries definition. 
Physical characteristics Value Nomenclatures 
        
    
              
       
   
        
   
      
     
      
     
    
  
Specific heat coefficient 
Thermal conduction coefficient 
Density of MWCNT 
Density of polypropylene 
Mass fraction of MWCNT 
TAB. 2 - Thermal parameters for the neat PP for PP/MWCNT composite. 
                                                                                                                                                 
The fluid–thermal model has been implemented and can be assimilated to a multiphysic 
coupled problem between the ‘‘fluid flow module” (for the fluid problem), ‘‘heat transfer 
module” (for the thermal problem) and the ‘‘ALE moving mesh module” (for the moving 
meshes). The number of quadratic elements in   is 241, in    is 2046 and in    is 2090, the 
total number of elements is 4377 with 2523 nodes (Fig. 4). An explicit time stepping scheme 
is used for the time dependant solver algorithm. The solution is given on the actual frames at 
each time step until tf =3.5s. 
 
 
20
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 Congrès Français de Mécanique                                       Besançon, 29 août au 2 septembre 2011 
5 
 
 
FIG. 4 - Mesh definition related to the twin screws mixer FE model (30795 DOF). 
2.4 Results and discussions 
The following results are related to the PP/MWCNT composite with 8 wt.% MWCNT 
content. One can notice that the maximum value for the velocity field is located at the middle 
of the frame and at the blades extremity. In addition, wall adherence conditions on the frame 
        are respected. 
 
FIG. 5 - Velocity fields (in mm/s) at several time steps for the 8 wt.% PP/MWCNT 
The shear rate is represented in Fig. 6, at t = 5 s. These figures reveal that the shear rate is 
approximately constant in the cavity except near the screws extremity where the shear rate is 
very higher than in the cavity. One can affirm that the mixing phase is effective near the 
screw extremity (at high shear rate). Furthermore, the shear rate field is practically constant in 
time. 
 
FIG. 6 – Shear-rate (in 1/s) at t= 5s for the PP/MWCNT 
The temperature fields are related in Fig.7. At the beginning of the simulation, the 
PP/MWCNT composite temperature is globally equal to 423°K except on the frame and 
screws walls, where the temperature is imposed (T = 473°K). During the simulation, the 
temperature in the cavity increases up to limit value equal to 473°K, corresponding to the 
imposed mixing temperature on the frame and screws walls. This simulation clearly reveals 
that the mixing temperature is quickly reached, which guarantees optimal mixing conditions. 
The heat transfer influence at the mixer inlet is related in figure that relates that a significant 
 
 
20
ème
 Congrès Français de Mécanique                                       Besançon, 29 août au 2 septembre 2011 
6 
 
variation of the temperature in the middle of the cavity, where the temperature is practically 
equal to the PP melting temperature. 
FIG. 7 - Temperature field (in K) at several time steps for the 8 wt.%  PP/MWCNT 
3. Conclusions and prospects 
The physical modeling and numerical simulations of PP/MWCNT composite flow in a twin-
screw mixer at high shear rates has been investigated in this paper, based on an appropriate 
finite element modeling of the composite flow during the mixing process. The reported finite 
element simulations allow to account the velocity effects, as well as temperature contours and 
provide a well founded basis for the simulations of the PP/MWCNT composite in the mixer 
according a multiphase model developed at the microscale. 
The complete 3D modeling and simulation problem now is ongoing and it will permit to 
establish an accurate modeling tool for the development of a computational methodology of 
carbon nanotubes loaded polymers, as well as for establishing the resulting physical and 
mechanical properties with mixing and injection modeling. 
References: 
[1] Iijima S., Helical microtubes of graphitic carbon, Nature, 354, 56-58, 1991. 
[2] Coleman N, Kahn U, Blau WJ, Gun’ko YK., Small but strong: a review of the mechanical 
properties of carbon nanotube–polymer composites. Carbon, 44, 1624-1652, 2006. 
[3] Breuer O, Sundaraj U., Big returns from small fibers: a review of polymer/ carbon 
nanotube composites. Polym Compos, 630, 25-45, 2004. 
[4] Thiébaud F, Gelin J., Characterization of rheological behaviors of polypropylene/carbon 
nanotubes composites and modeling their flow in a twin-screw mixer. Composites science 
and technology, 70, 647-656, 2010. 
[5] Haas A, Scholle M, Aksel N, Thompson H M, Hewson R W, Gaskell P H., Laminar 
Thermal Mixing in Coating Flows. 


Numerical simulation of polymers-carbon nanotubes mixing process

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Numerical simulation of polymers-carbon nanotubes mixing process

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