Modeling of the Complex Mixing Process in Internal Mixers - Industrial

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Ind. Eng. Chem. Res. 2007, 46, 7328-7339

Modeling of the Complex Mixing Process in Internal Mixers Laurent Adragna,† Franç oise Couenne,† Philippe Cassagnau,‡ and Christian Jallut*,† UniVersite´ de Lyon, UniVersite´ Lyon 1, Laboratoire d’Automatique et de Ge´ nie des Proce´ de´ s, UMR CNRS 5007, ESCPE, 43 bd du 11 NoVembre 1918, 69622 Villeurbanne, Cedex, France, and UniVersite´ Lyon 1, ISTIL, Laboratoire des mate´ riaux polyme` res et biomate´ riaux, UMR CNRS 5627, 43 bd du 11 NoVembre 1918, 69622 Villeurbanne, Cedex, France

The objective of this work is to describe a model allowing the representation of the mixing phenomenon between two fluids having very different viscosities in an internal mixer, typically a cold plasticizer and a hot molten polymer. We present a dynamic model based on a simplified view of the flow and on mass, energy, and momentum balances. As far as the mass transfer is concerned, we consider simultaneously the mechanical action of the rotors and the diffusive process. Heat transfer has to be considered because of the important temperature difference between the two components and the viscous dissipation. Finally, a momentum balance allows us to calculate the velocity profile and the torque time evolution by considering that the viscosity is concentration and temperature dependent. A sensitivity study has been done to check for the influence of some model parameters on the simulated results. The latter will show a good agreement with experimental results. 1. Introduction There are numerous reasons to be interested in mixing. An understanding of viscous mixing is of considerable technological importance in the context of material processing, food processing, and reactive and nonreactive polymer processing. From a polymer processing point of view, complex formulations often involve the incorporation of low viscosity liquids that may be thermodynamically miscible or not with molten high viscosity polymers. The large mismatch between additive viscosities and molten polymers greatly magnifies the difficulties associated with compounding these components in a laminar flow. Even if restricted to the case of miscible fluids, mixing is far from being completely understood. However, a host of new experimental results and theoretical developments lead to increasing understanding and predictions.1 It could be then argued2 that the treatment of laminar or viscous mixing is in good shape due to advances in fluids mechanics and new experimental results. For example, mixing investigations of shear-thinning viscoelastic fluids have been already studied although authors of these studies must admit that the mechanisms responsible for mixing of such fluids are not fully understood.3 Nevertheless, the problem is further exacerbated when one of the fluids has a significantly lower viscosity than the other and has the tendency to segregate into the high shear rate regions (lubrication effect) of the mixing equipment. In addition, polymer reactive or nonreactive processes are highly complicated to design and to control since one has to deal with a great number of highly nonlinear and coupled phenomena. For example, the flow generated by the screw rotation in an extruder is basically laminar but it is very difficult to simulate due to its non-steady-state nature. Furthermore, molten polymers may be non-Newtonian and their properties may change along the processing machine (spatial and temporal evolution) due to the mixing process and/or chemical reaction * To whom correspondence should be addressed. E-mail: jallut@ lagep.univ-lyon1.fr. † Laboratoire d’Automatique et de Geńie des Proce´de´s, UMR CNRS 5007. ‡ Laboratoire des mate´riaux polyme`res et biomate´riaux, UMR CNRS 5627.

progress. Furthermore, the diffusion process has also to be considered since it can become a significant limiting step at the molecular scale of mixing, and as far as chemical reactions are present, their rates have to be known. Finally, heat transfer has to be taken into account due to the viscous dissipation and to the chemical reactions. All these transport and chemical phenomena are coupled at least by the temperature and composition dependence of the transport properties (viscosities, diffusion coefficients, etc.). Within the framework of extrusion process studies, internal or batch mixers are commonly used for preliminary studies of the mixing problems. Studies on internal mixers are usually carried out with two main objectives. On the one hand, the conversion from torque-rotor speed steady-state data to rheological data (viscosity, shear stress, and strain rate) for pure or premixed molten polymers can be addressed. This problem was treated in 1967 by Goodrich and Porter4 and Blyer and Daane5 in the case of a Brabender torque rheometer. Laguna et al.6 and Bocayuva et al.7 extended these works to the case of a Haake rheometer. Bousmina et al.8 validated the Goodrich group’s work by generalizing their model for the estimation of the strain rate from rotor speed and torque steady-state measurements. The model generally used by these authors consists of considering the flow between the rotors and the mixer wall as a simple Couette steady-state flow and by defining an equivalent internal radius. Tadmor and Gogos9 developed a model based on a more realistic representation of the mixer geometry. Yang et al.10 worked on this model to estimate the strain rate as a function of the rotor speed. More detailed models based on computational fluid dynamics (CFD) have also been published. Cheng and Manas-Zloczower11-14 have used the fluid dynamics analysis package (FIDAP) to calculate velocity, pressure, or temperature profiles at steady or transient states. Kim and White15-18 and Nassehi and coworkers,19-22 thanks to the finite element method, are able to represent velocity, temperature, and pressure fields for Newtonian or non-Newtonian rheological laws. These models are all based on a mesh design for a specific position of the rotor in the chamber wall. So, a complete revolution of the rotors in

10.1021/ie0704795 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/03/2007

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Figure 1. Torque variations (N ) 50 rpm, Tw ) 200 °C, 7.5% of -caprolactone).

the chamber is decomposed into different geometry, each one being analyzed separately using the finite element method. On the other hand, internal mixers are used to study the mixing process between species by considering the time dependence of the torque. As far as we know, the problem of the mixing modeling between miscible or nonmiscible species in an internal mixer has been less addressed in the literature. This mixing process involves momentum and heat transfers but also diffusion processes. Some works have been devoted to the mixing operation of a polymer with carbon black.17,21-23 Only a few works have been published concerning the mixing of high viscosity molten polymers with low viscosity liquids.24-27 However, some reactive extrusion processes require the mixing of fluids having very different viscosities like a monomer and a molten polymer. A dynamic model of a reactive extrusion process in the case of bulk homopolymerization has been previously proposed within the framework of process automatic control.28-29 Our goal is to extend this work to the case of reactive extrusion processes involving the mixing of small molecules with a molten polymer. To this end, we propose in the present paper a preliminary study of the mixing process in a Haake internal mixer. A dynamic model of the mixing process is proposed and compared to the experimental result. As previously explained, this mixing process is highly complicated, and we will use a chemical engineering approach based on a simplified view of the flow in the mixer. On the basis of this view, we then propose an original dynamic model to represent the torque trajectory during the mixing process between a polymer and a plasticizer, with fluids having an important viscosity ratio. The application of mass, energy, and momentum balance equations to this flow model allows us the representation of concentration, temperature, and velocity profiles as well as torque time evolution. As far as the mass transfer is concerned, we consider at the same time the mechanical action of the rotors and the diffusive process between the small molecules of plasticizer and the molten polymer. The energy balance in the system has also to be considered because of the sudden introduction of the cold plasticizer in the internal mixer as well as the viscous dissipation

effect. Furthermore, from the momentum balance, we will be able to simulate the torque variation due to the mixing process of a low viscosity liquid with a high viscosity molten polymer. From the understanding of these highly complicated mixing situations that we have obtained from our modeling study, we intend in the future to extend our extrusion dynamic model28-29 to these situations. 2. Experimental Study 2.1. Experimental Setup. Experiments have been performed with -caprolactone as plasticizer mixed with polycarbonate (PC). These two species are considered to be miscible. The -caprolactone is a Lancaster product, and the polycarbonate Makrolon is provided by Bayer. The internal mixer, a Haake Rheomix 600p with two roller rotors R600, one directly fitted on the rotor shaft and the second one geared to the shaft, is equipped with the software Polylab V4-1. The gear ratio is 2/3. The polymer is injected first to be molten. Then, the monomer is injected quickly. At a given rotor speed, we measure the evolution of torque according to the time. We have also performed experiments by introducing a BASF 58-5850 Paliogen red dye in the plasticizer and by taking pictures at different mixing times. During all these experiments, the internal mixer itself is heated according to a constant temperature setpoint Tw. 2.2. Experimental Results. 2.2.1. Torque Variations and Dye Experiment. An example of the torque variation due to the mixing process between -caprolactone monomer and polycarbonate is represented in Figure 1. The torque time evolution shown in Figure 1 can be divided into five periods represented in Figures 2 and 3. First, pellets of PC are introduced in the mixing chamber (period 1). The rise of the torque is due to the PC pellet friction, which softens them and makes a homogeneous molten polymer under thermal and mechanical actions (period 2). The low viscosity -caprolactone is then introduced into the mixing chamber. According to the viscosity ratio between the polymer and the monomer, a lubrication phenomenon is observed leading

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Figure 2. (a) Steady-state torque value corresponding to the molten pure PC in the mixing chamber. (b) Sudden decrease of the torque after -caprolactone injection.

Figure 3. (a) Image taken during the rise of torque and after the -caprolacton injection. (b) PC/-caprolactone blend at steady state.

Figure 4. Blend at the end of the experiment.

to a sudden decrease of the torque (period 3). Then, the mixing action of the rotors allows the blend homogenization so the torque increases (period 4) to a constant final steady-state value (period 5). In order to get a more realistic view of the mixing process, we have introduced a red dye into the -caprolactone and performed mixing experiments by taking mixing chamber pictures at different steps of the mixing process after stopping and taking it apart. During this operation, a part of the matter remained stuck to the removable vertical wall of the mixer or fell down: this is the reason why the mixing chambers seem to be partly filled in the pictures given in Figures 3 and 4. In fact, the experiments are performed with fully filled mixing chambers according to the total quantity of matter that is introduced and the available volume of the mixer. Period 1, corresponding to the polymer melting process is out of interest here and is not represented. Figure 2a and b are related to the second and the third periods. Figure 2a corresponds to the end of the pure PC melting process: the polymer is distributed according to two volumes. In the gap between the rotor top and the mixing chambers wall, the PC appears as a more or less stagnant layer while it appears to be mixed in the volume situated between the rotors. Figure 2b is a picture taken just after the -caprolactone introduction when the torque suddenly decreases (period 3 in Figure 1). The -caprolactone, darker than the PC in the picture because of the red dye, is also distributed according to the free volume between the rotors and the gap between the top of the rotors and the mixing chamber wall. At the initial stage of the mixing

process, the molten PC and the -caprolactone form a double layer within this gap, the -caprolactone being situated at the rotor side. During the fourth period (Figure 3a), the torque increases as the monomer mixes with the polymer. Although the initial double layer is less and less visible due to the mixing process, a layer of matter remains present in the gap but some pieces of matter are removed from this layer toward the free volume between the rotors. This transfer process is due to the scrapping action of the rotors at the gap surface. In the steady-state period, the homogeneous polycarbonate/ -caprolactone blend occupies the major part of the free volume of the chamber (Figure 3b). The last picture (Figure 4) shows a piece of the blend taken from the external wall mixing chamber where it was stuck. We observe a red front from the top of the rotor toward the mixer wall. Although the torque seems to be constant according to its measurement sensitivity, the mixing process is not completely achieved. From this last picture, a diffusion process is assumed within the layer of matter situated between the mixing chamber wall and the rotor top. 2.2.2. Torque Evolution as a Function of the -Caprolactone Mass Fraction. Previous studies on the case of miscible blend mixing process in an internal mixer have been carried out by Burch and Scott.27 Blends with viscosity ratios from 1 to 1 × 10-4 were pigmented and visually analyzed. Observations led to the conclusion that the lower the ratio, the greater the importance of the mixing time, with an additional phaseinversion-like process for the lowest viscosity ratio blends. A similar increase in torque caused by a phase inversion was observed by Cassagnau et al.24 for reactive system. Low viscosity fluids migrate to the high shear rate regions of the internal mixer where a lubrication phenomenon takes places. The viscous dissipation, creating energy able to transform polymer pellets to a molten fluid, is maximum in these regions. The presence of lubricant decreases the energy available for the transformation and, hence, decreases the rate of mixing. In our experiments, introduction of the lubricant takes place when the polymer is already molten. The torque is related to the current viscosity of the blend so it depends on the -caprolactone quantity that is introduced and mixed with the molten polymer. We express this quantity as the global mass fraction of -caprolactone in the system. It is also the uniform mass fraction of the plasticizer in the final mixture when equilibrium is reached. Figure 5, which represents the torque evolution versus time for injections of different -caprolactone quantities, logically shows that the higher the -caprolactone mass fraction, the lower the viscosity (and torque) at the end of the experiment. 3. Qualitative Analysis of the Mixing Process In the case of a pure molten polymer, Bousmina et al.8 used an analogy between the Haake internal mixer and a flow structure called “double-Couette” to model the steady-state flow in the mixer. Within this simplified view of the flow, the rotors are represented by cylinders and the molten polymer is supposed to stay between the rotor top and the external wall of the mixer (Figure 6). From this simplified view of the flow and from the experimental results described above, we propose the qualitative representation of the mixing process represented in the Figure 7. As far as the real shape of the rotor is concerned, we propose to divide the free volume according to the following scheme.


Figure 5. Torque variation according to the -caprolactone mass fraction (N ) 50 rpm, Tw ) 200 °C).

Figure 6. Analogy between the Haake internal mixer and a “doubleCouette” flow.

the plasticizer introduction (lubrication phenomena). Then, the process leading to the time torque evolution has to be represented. This process involves mass and energy transfer since the viscosity of the mixture depends on its temperature and its composition. The effect of the temperature can be significant since the plasticizer is cold when it is introduced in the mixer. 4.1. Plasticizer Material Balances. The plasticizer material balances are written under their volumetric form since we suppose the blend density to be constant. Within the gap, the mass transfer is supposed to act only in the radial direction (see Figure 7) so that only diffusion is taken into account:

( )

∂Φp 1 ∂ ∂Φp )D r ∂t r ∂r ∂r

Figure 7. Qualitative representation of the mixing process in an internal mixer.

Vp is the volume of the gap between the top of the rotors and chamber wall. Only diffusion will be considered as the mixing process within this gap. V is an accessible volume to the blend outside the gap. This zone is represented by a continuously stirred tank reactor (CSTR) and will prove to act as a reservoir of plasticizer for the mixing process. In order to represent the scraping action of the rotor, the surface of the layer located in the gap is represented by an equivalent CSTR of volume Vc having a uniform composition Φs and a global flow q being recycled between Vc and V. This flow rate q is a distributive mixing efficiency parameter. The application of mass, energy, and momentum balance equations to this simplified flow structure allow the representation of concentration, temperature, and velocity profiles as well as the torque time evolution. 4. Model Equations The purpose of this model is to represent the torque evolution from the introduction of the plasticizer (periods 4 and 5 of Figure 1). To this end, one has to define the initial conditions in order to calculate the lowest torque that is observed immediately after

(1)

where Φp(r,t) is the volumetric fraction of the plasticizer within the gap and D is its diffusion coefficient that is assumed to be constant because of the small quantities of plasticizer introduced. The action of the rotors is described according to the following mechanism: • A convective mass transfer is assumed to occur between the gap surface and the mixing chamber according to the total volumetric flow rate q. • A scraping action of the rotor is assumed to occur on the gap surface. This mechanism is represented by using two CSTRs respectively devoted to describe the mixing chamber (volume V) and the scraped surface of the gap (volume Vc) (see Figure 7). The corresponding plasticizer material balances are then written as follows:

{

dΦ ) q(Φs - Φ) dt dΦs ∂Φp Vc ) q(Φ - Φs) + DAs dt ∂r

V

( )

(2)

r)ri

The plasticizer material balance is expressed as a function of its volumetric fractions in the mixing chamber and at the gap surface, denoted by Φ and Φs. The balance at the gap surface (r ) ri) includes the diffusion flux toward the gap from its surface. Before the introduction of the plasticizer, the molten polymer is mainly present within the gap (see Figure 2a). When the plasticizer is introduced in the mixer, a lubrication phenomenon is immediately observed. Figure 8 represents the torque value immediately after the plasticizer introduction as a function of the plasticizer global mass fraction. It is a part of Figure 5

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Figure 8. Initial torque values depending on the plasticizer concentration (N ) 50 rpm, Tw ) 200 °C).

balance can be considered to be at steady state. Neglecting the gravity and assuming a uniform pressure, the momentum balance is then as follows:

∂ 2 (r τrθ) ) 0 ∂r Figure 9. Detail of the initial conditions within the gap.

According to the interval of the shear rate existing in the mixer, a Newtonian behavior can be assumed:

beginning at the lowest value of the torque after the injection of the plasticizer: the higher the mass of plasticizer, the more important the lubrication phenomenon. Within the framework of our model, we do not try to represent the spreading process of the plasticizer leading to the lowest torque that is observed. We only fix the initial conditions in order that the calculated initial value of the torque is equal to the measured one. To this end, the mass of the plasticizer is distributed between: • the volume V representing the mixing chamber; • the volume Vc representing the action of the rotor on the gap surface; • a layer situated at the top of the gap (see Figure 9). This is motivated by the pictures showing two different layers when introducing the -caprolactone into the internal mixer (see Figure 2b). As far as the boundary conditions are concerned, we consider that there is not any flux at the chamber wall (r ) re) and that the volumetric fraction at the gap surface (r ) ri) is equal to the volumetric fraction in the CSTR representing the scraping phenomenon:

{

∂Φp (r ) re, t) ) 0 ∂r Φp(r ) ri, t) ) Φs(t)

(4)

(3)

The second condition allows imposing at the gap surface a perfect mixing condition. 4.2. Momentum Balance. In order to calculate the torque, the momentum balance has to be solved within the gap where a velocity gradient is supposed to exist (see Figure 6). The velocity is assumed to be in the θ direction only and to be a function of r only (νθ(r) ) V). By comparing the kinematic viscosity of the mixture and the plasticizer diffusion coefficient, one can conclude that the diffusion time constant is much more important than the momentum transfer one, so the momentum

( ∂r∂ (Vr))

τ ) τrθ ) τθr ) -η r

(5)

The viscosity is not constant since it depends on Φp, the plasticizer volumetric fraction, and Tp, the temperature. Then, by combining eqs 4 and 5, we obtain the following differential equation:

{(

∂ 2 ∂ V r -η(Φp, Tp)r ∂r ∂r r

( ))} ) 0

(6)

At the initial stage of the process, a large velocity gradient exists because of the lubrication phenomenon created by the plasticizer on the molten polymer layer present between the rotors and the chamber wall. According to the way the initial conditions for the mass transfer model are fixed (see above), we assume that, at t ) 0 s, the gap can be seen as two layers having uniform compositions and viscosities (see Figure 9). In this case, eq 6 can be solved analytically:

{

c1 + b1r, ri < r < rc r c2 V2 ) - + b2r, rc < r < re r

V1 ) -

(7)

According to the following boundary conditions,

{

r ) ri w V1 ) Ωri r ) r e w V2 ) 0 r ) rc w V1 ) V2, τ1 ) τ2

(8)

we can easily find the four constants c1, c2, b1, b2, and the expression of the velocities within the two layers:

{

((

)( ( )( )) ) ( ( ) ( ))


-rc2Ω

rc2Ω 1 1 + Ω + r, 2 2 2 2 r ri < r < rc η1 r c rc η1 r c rc ri2 -1 - 2-1 -1 - 2-1 2 2 η2 r η2 r ri ri e e 2 2 η1 rc Ω rc Ω η1 1 V2 ) r, + 2 2 2 2 η2 η r r ηr η r rc < r < re rc rc2 1 c 2 e 1 c 1 1 1 1 η2 r 2 η2 r 2 ri2 ri2 e e

V1 )

(

)( ) ( )( )

These two expressions are included in the model as the velocity initial conditions. 4.3. Energy Balances. The temperature of the wall of the mixing chamber is regulated, and we introduce the plasticizer at ambient temperature; so, we consider the following processes (see Figure 7): • heat transfer between the blend and the chamber wall as well as the rotorssthe rotors and the chamber wall are assumed to be at the same constant and uniform temperature Tw; • heat transfer by conduction within the gap; • viscous dissipation within the gap. According to the assumptions that have been made to establish the material balances (see section 4.1), the energy balances are written similarly. The energy balance within the gap is written as follows:

Cp

(( )

( ) ) ( ( ))

∂Tp V ∂V ∂V 2 V2 1 ∂ ∂Tp -2 +k r )η + ∂t ∂r r ∂r r r ∂r ∂r

(10)

where Cp is the volumetric heat capacity of the blend and k is its thermal conductivity. The energy balances of the two CSTRs used to describe the mixing chamber and the scraped surface of the gap (see Figure 7) are as follows:

{

dT ) qCp(Ts - T) + UAs(Tw -T) dt (11) dTs ∂Tp VsCp ) qCp(T - Ts) + UAs(Tw -Ts) + kAs |r)ri dt ∂r

VCp

where U is a global heat transfer coefficient assumed to be constant over the mixing chamber wall. In the two CSTRs and layer 1 (see Figure 9), the initial temperature of the plasticizer is assumed to be the ambient temperature. The temperature of the molten polymer within layer 2 (see Figure 9) is assumed to be uniform initially and equal to Tw. As far as the boundary conditions are concerned, the temperature at r ) ri is equal to the temperature in the CSTR representing the scraping phenomenon while the temperature at the external radius r ) re is equal to the setpoint temperature chosen for the experiment:

{

T(r ) ri, t) ) Ts(t) T(r ) re, t) ) Tw

(12)

5. Viscosity Model As a function of the composition and the temperature, the viscosity couples the three balances within the gap. It is then necessary to find a unique model that can express the viscosity of the blend from the pure plasticizer to the pure polymer as a function of the temperature. The following equation based on

(9)

the work of Marin et al.30 and extended by Gimenez et al.31 is used in this work:

η(Φp, Tp) ) [Φpηplasticizer + ηpolymer(1 - Φp)4] × Ep 1 1 exp R Tp Tw

((

))

(13)

The wall set point temperature Tw is used as a reference temperature for the calculation of the viscosity dependence with respect to the temperature. Actually, this equation predicts the variation of the zero shear viscosity only. Nevertheless, as far as the shear rate that is imposed to the blend is concerned, a Newtonian behavior can be assumed. We have compared the theoretical values of the viscosity as given by eq 13 only to those that we have measured for the blends that were obtained after our mixing experiments. The experiments were carried out using a Rheometrics RMS 800 rheometer equipped with parallel plate geometry. In addition, the absolute complex viscosity was expressed as the shear viscosity according to the Cox-Merz law. The results of the comparison are shown in Figure 10. We cannot really conclude for the validity of eq 13 since a systematic study according to the blend composition should be done: such a study was beyond the scope of our work mainly devoted to the analysis and modeling of the mixing process. As far as this objective is concerned, eq 13 appears to be satisfactory since it allows representing the huge influence of the blend composition on the viscosity and it is compatible with the values that we have measured. 6. Parameters of the Model In order to perform a simulation study, we have determined the parameters of the model (see Table 1). They are divided according to • physical parameters; • geometrical parameters. As far as physical parameters are concerned, values from the literature have mainly been used: • The thermal conductivity k is calculated by Zhang et al.32 • The heat capacity Cp comes from Legrand et al.33 • The overall heat transfer coefficient is an average value from different works.34-36,12,18 • The diffusion coefficient D has been determined using the free volume theory.37-42 • The activation energy of the blend is calculated from data cited by Gimenez et al.31 for the -caprolactone and by Legrand et al.33 for the polycarbonate. The thermodynamic and transport properties Cp, k, and D have been taken constant contrary to the viscosity η that has been represented as a function of Φp and Tp. This is motivated by the fact that the viscosity has a direct influence on the calculation of the torque that is the measured variable while the influence of the other properties is indirect through the

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Figure 10. Viscosity model, T ) 240 °C.

Figure 11. Plasticizer volumetric fraction evolution.

Figure 12. Time evolution of velocity profiles.

material and energy balances. As far as the influence of the temperature is concerned, the values that are given in Table 1 have been calculated at Tw.

In order to estimate the equivalent internal radius ri, we have used the method proposed by Bousmina et al.8 for the same internal mixer. The values of L and re, the depth of the mixer


Figure 13. Temperature time evolution.

Figure 14. Viscosity evolution.

and the external radius of each chamber, were directly measured. Knowing the total volume, V could be calculated with re, ri, and L as well as the area of the cylinder representing the rotor As. This area is supposed to represent also the surface which is in the middle of the mixing chamber represented by a CSTR. The initial layer thickness of plasticizer depends on the amount of -caprolactone introduced in the internal mixer and an assumption was made concerning the volume Vc: this volume is equal to the volume of the initial layer of plasticizer and it is the volume “scraped” in 1 s in order to have a basis for the value of the volumetric flow rate q. 7. Simulation Results and Comparison with Experimental Data The system of equations representing the model has been solved using the orthogonal collocation method.43 One hundred collocation points (fifty for the first layer and fifty for the second

one) have been taken due to the highly nonlinear character of this problem. The simulations were performed by using the parameters that are given in Table 1. As far as the torque is concerned, we have also compared the model simulations with experimental data obtained with the -caprolactone/polycarbonate mixture but some of the parameters were modified in order to obtain a good fit of the model to the experimental results. 7.1. Evolution of Plasticizer Volumetric Fractions. Figure 11 shows the concentration profile within the gap. On Figure 11, one can see the following: • The decrease of the volumetric fraction of plasticizer represents the evolution in each CSTR and in layer 1 initially made of monomer (see Figure 9). Concentration decreases to the stationary regime. A slight increase is observed for the evolution closest to the polymer layer, due to the mixing phenomenon.

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Figure 15. Comparison between experimental and modeled results.

Table 1. Parameters of the Model physical parameter volumetric flow rate q (m3‚s-1) diffusion coefficient D (m2‚s-1) thermal conductivity k (W‚K-1‚m-1) volumetric heat capacity Cp (J‚m-3‚K-1) overall heat transfer coefficient U (W‚m-2‚K-1) activation energy Ep (J‚mol-1) plasticizer viscosity (Pa‚s) polymer viscosity (Pa‚s)

ref value

ref value

radius ri (m)

17.6 × 10-3

radius re(m)

20 × 10-3

2.3 × 10-1

depth L (m)

4.6 × 10-2

2 × 106

area As (m2)

5.08 × 10-3

volume V (m3)

11.9 × 10-6

1×

10-7

geometrical parameter

2 × 10-11

250 31 000 1 × 10-3 3000

• The volumetric fraction of plasticizer in the layer initially made of pure polymer increases. This increase is all the quicker as we consider a position close to the gap surface. 7.2. Velocity Profiles. Figure 12 shows the velocity profiles indexed by time. The one corresponding to the initial conditions is calculated analytically and is made of two parts: the first one is the profile in the plasticizer layer. As far as the second one is concerned, the velocity values are too small to be represented on the same graph. The initial lubrication phenomenon is well-represented by this initial profile. The steady-state profile tends to the Couette profile, corresponding to a Newtonian homogeneous blend. 7.3. Temperature Evolution. These curves (Figure 13) represent the temperature evolution in the two CSTRs and in the gap. The boundary condition at the external radius is the set point temperature Tw ) 478 K. At t ) 0 s, the temperature in the CSTRs and the layer initially made of pure monomer is 298 K, because the plasticizer is introduced at ambient temperature. It is then increased to reach the steady-state temperature which is slightly greater than Tw due to the viscous dissipation effect. This process is relatively fast. We see also a polymer transient cooling effect due to the introduction of the cold monomer. This effect is more or less pronounced according to the position in the polymer layer: the closer the position to the initial monomer layer, the more important the decrease.

7.4. Viscosity Evolution. Figure 14 shows the viscosity evolution in the layer initially made of polymer at three different positions. All curves start at the same point, the pure polymer viscosity. The first curve, representing the evolution close to the initial monomer layer, decreases directly because of the introduction of a low viscosity liquid. We can see a slight increase before the fall for the second one, situated in the middle of the initial polymer layer. It is due to the cooling effect observed previously. If the temperature goes down, the viscosity goes up. For the last one, close to the external radius, a decrease is observed, slower than for the first one, because the monomer needs time to diffuse in the polymer matrix. This figure reveals a competition for the viscosity evolution between the plasticizer diffusion and the heat conduction. 7.5. Torque Evolution and Comparison with Experimental Results. The torque associated to one rotor can be calculated by the following equation:

Γ ) (2πrL)rτrθ

(14)

L is the depth of the internal mixer, and τrθ is the shear stress depending on the viscosity and the velocity profile. The final torque in the entire mixing chamber can be calculated by considering the total mechanical power coming from the two rotors:

ΓΩ1 ) Γ1Ω1 + Γ2Ω2

(15)

Then, it is possible to represent the evolution of the torque for different concentration of -caprolactone. Figure 15 shows the experimental and simulated torque for two different plasticizer concentrations: 5% (upper curve) and 15% (lower curve). The final value of the torque depends on the total amount of plasticizer: it is more important with less plasticizer. A good agreement is observed between the model and the experiments: the dynamical part is mainly governed by the volumetric flow rate q and the diffusion coefficient D whereas the steady-state depends preferentially on the viscosity. 7.6. Sensitivity Study to the Key Parameters q and D. Depending on their values, the dynamical part of the model will be different as one can see in Figures 16 and 17.


Figure 16. Torque evolution with different values of q.

Figure 17. Torque evolution with different values of D.

The importance of q is visible in Figure 16 and more particularly on the increasing part of the torque model. If q is too small, the plasticizer coming from the CSTR moves more slowly and a peak is observed due to the high viscosity of the blend mainly constituted of polymer in the first moments. This is not the case if the flow rate is more important, and the plasticizer is moving faster in the gap. As for the flow rate, the value of D has an impact on the dynamical part of the simulation. The lower the diffusion coefficient, the slower is the rise of the torque. In order to obtain a good fit of our model to the experimental results as shown in Figure 15, a rather high value of the diffusion coefficient (10-9 m2‚s-1) had to be taken when compared to the diffusion coefficient that is given by the free volume theory (see Table 1). The plasticizer fraction can have an influence on the diffusion coefficient, but such an influence is probably not sufficient to explain the result. Another point is the influence of the ri parameter. It is an apparent radius conditioning the time constant of the diffusion process through the gap thickness. We did not consider the apparent radius as an adjustable parameter, but for

a given time constant, another value of ri will give another value of D. 8. Conclusion Unsteady-state mixing experiments have been performed in an internal mixer between a monomer acting as a plasticizer, the -caprolactone, and a molten polymer, the polycarbonate. On the basis of pictures obtained by using a red dye mixed with the monomer, a simplified flow model of the mixing process is proposed. From this flow model and from the mass, momentum, and energy balances, we are able to simulate the mixing phenomenon between two fluids having an important viscosity ratio in an internal mixer. This model is close to experimental results since it takes into account not only the diffusion phenomenon of small molecules of plasticizer into the molten polymer but also the rotors action in the mixing process: this rotor action has been described as a distributive mixing process. The highly nonlinear system of equations has been solved using a collocation method in a Fortran code. The mass, energy, and momentum balances gave a complete

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representation of the volumetric fraction, temperature, and velocity evolutions. The simulated torque has been favorably compared to the experimental results. Acknowledgment This work has been realized in collaboration with the French research agency (CNRS) and Rhodia Company as part of a research program devoted to reactive processing of polymers (CPR “Mate´riaux - Mise en oeuvre des polyme`res”). Our acknowledgments go to the Rhodia Company and CNRS for their financial support. Notations Roman Alphabet As ) area of one rotor and of the middle mixing chamber surface (m2) Cp ) volumetric heat capacity (J‚m-3‚K-1) D ) diffusion coefficient (m2‚s-1) Ep ) activation energy (J‚mol-1) k ) Thermal conductivity (W‚deg C-1‚m-1) q ) total volumetric flow (m3‚s-1) rc ) radius of the initial plasticizer layer (m) ri ) rotor equivalent radius (m) re ) external mixing chamber radius (m) T ) temperature of the blend in the CSTR representing the mixing chamber (K) Tp ) temperature in the gap (K) Ts ) temperature of the blend in the CSTR representing the scraping phenomenon (K) Tw ) set temperature of the wall (K) U ) overall heat transfer coefficient (W‚m-2‚K-1) V ) velocity within the gap V1 ) velocity in the initial layer 1 V2 ) velocity in the intial layer 2 V ) volume of the CSTR representing the mixing chamber without the gap (m3) Vc ) volume of the CSTR representing the “scraping phenomenon” and the initial plasticizer layer in the gap (m3) Vp ) total gap volume (m3) Greek Alphabet ηplasticizer ) plasticizer viscosity (Pa‚s) ηpolymer ) polymer viscosity (Pa‚s) τ ) shear stress (Pa) Φ ) plasticizer volumetric fraction in the CSTR representing the mixing chamber without the gap Φs ) plasticizer volumetric fraction in the CSTR representing the scraping phenomenon Φp ) plasticizer volumetric fraction in the gap Ω ) rotation speed (rad‚s -1) Literature Cited (1) Jana, S. C.; Metcalfe, G.; Ottino, J. M. Experimental and computational studies of mixing in complex Stokes flows. J. Fluid Mech. 1994, 256, 199. (2) Ottino, J. M. Mixing and chemical reactions: A tutorial. Chem. Eng. Sci. 1994, 49 (24A), 4005. (3) Arratia, P. E.; Shinbrot, T.; Alvarez, M. M., Muzzio, F. J. Mixing of Non-Newtonian Fluids in Steadily Forced Systems. Phys. ReV. Lett. 2005, 94, 1. (4) Goodrich, J. E.; Porter, R. S. A rheological interpretation of torquerheometer data. Polym. Eng. Sci. 1967, 45. (5) Blyer, L. L.; Daane, H. An analysis of Brabender Torque Rheometer Data. Polym. Eng. Sci. 1967, 178.

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ReceiVed for reView April 3, 2007 ReVised manuscript receiVed July 20, 2007 Accepted July 26, 2007 IE0704795

Modeling of the Complex Mixing Process in Internal Mixers - Industrial

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