Quantification of Mixing in RIM Using a Non-Diffusive

I NTERNATIONAL J OURNAL OF C HEMICAL R EACTOR E NGINEERING Volume 9

2011

Article A114

Quantification of Mixing in RIM Using a Non-Diffusive Two-Phase Flow Numerical Model

∗

Cláudio P. Fonte∗

Ricardo J. Santos†

Madalena M. Dias‡

José Carlos B. Lopes∗∗

Universidade do Porto, [email protected] Universidade do Porto, [email protected] ‡ Universidade do Porto, [email protected] ∗∗ Universidade do Porto, [email protected] ISSN 1542-6580 c Copyright 2011 De Gruyter. All rights reserved. †

Quantification of Mixing in RIM Using a Non-Diffusive Two-Phase Flow Numerical Model∗ Cláudio P. Fonte, Ricardo J. Santos, Madalena M. Dias, and José Carlos B. Lopes

Abstract Mixing in RIM is made mainly by advective mechanisms, rather than diffusion. In this paper, the advective mechanisms that enable reducing the mixing scales down to the values required for the complete chemical reaction of the two monomers inside the RIM mixing chamber are identified and studied. From Computational Fluid Dynamics (CFD) simulations of non-diffusive two-phase flow using the Volume-of-Fluid (VOF) model, a linear scale of segregation is determined as a measure of the degree of mixing and the effect of the Reynolds number is studied. KEYWORDS: mixing, reaction injection moulding, VOF, CFD

∗

Cláudio P. Fonte is in the LSRE—Laboratory of Separation and Reaction Engineering— Associate Laboratory LSRE/LCM, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. Ricardo J. Santos is in the LSRE—Laboratory of Separation and Reaction Engineering— Associate Laboratory LSRE/LCM, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. Madalena M. Dias is in the LSRE—Laboratory of Separation and Reaction Engineering— Associate Laboratory LSRE/LCM, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. José Carlos B. Lopes is in the LSRE—Laboratory of Separation and Reaction Engineering— Associate Laboratory LSRE/LCM, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. Financial support for this work was in part provided by national research grant PTDC/CTM/72595/2006. This work is partially supported by project PEst-C/EQB/LA0020/2011, financed by FEDER through COMPETE—Programa Operacional Factores de Competitividade and by FCT—Fundaça˜ o para a Ciência e a Tecnologia. Cláudio Fonte acknowledges his Ph.D. scholarship by FCT (SFRH/BD/39038/2007).

Fonte et al.: Quantification of Mixing in RIM

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Introduction Reaction Injection Moulding (RIM) is an industrial process for plastic parts production based on the mixing of two or more reacting monomers that are introduced in a cylindrical mixing chamber by opposed jets. The first RIM studies were made on cylindrical chambers, with a diameter around 1 cm, and two opposed injectors with a diameter in the range 1 to 3 mm (Malguarnera and Suh, 1977; Nam and Suh, 1980; Lee et al., 1980). Other configurations have been tried, particularly different angles between the opposed injectors (Trautmann and Piesche, 2001; Schütz et al., 2005). Figure 1 shows a typical configuration of a RIM machine. The passage times in the cylindrical chamber are in the range of 10100 ms and the viscosities of the monomers range from 20 to 1000 mPa.s. The mixture of the two monomers that leave the mixing chamber is discharged into a mould, where polymerisation occurs. The mixing of the two monomers in the chamber is the critical step of the RIM process since the mechanical properties of the obtained product depend on the degree of mixing achieved before the reactive mixture is discharged into the mould (Kolodziej et al., 1986). The study of mixing in RIM was made from: the adiabatic temperature rise in the mixing chamber (Lee et al., 1980; Kolodziej et al., 1982/1986); the final properties of the polymer (Kolodziej et al., 1982/1986; Nguyen et al., 1985/1986); a test chemical reaction (Kush et al., 1989); from mixing of fluids with a tracer or seeded with particles (Malguarnera and Suh, 1977; Tucker and Suh, 1980; Lee et al., 1980; Sandell et al., 1985; Wood and Johnson, 1991; Johnson et al., 1996; Trautman and Piesche, 2001; Schütz et al., 2005), and using laser velocimetry methods (Wood et al., 1991; Johnson et al., 1996; Johnson, 2000; Teixeira et al., 2005; Santos et al., 2008; Santos et al., 2009). Many of these studies observed that the main mixing mechanism in RIM is the formation of vortices downstream the opposed jets impingement point, these vortices engulf the fluids from the opposite jets (Lee et al., 1980; Wood et al., 1991; Santos et al., 2008). Nevertheless a thorough characterization of the mixing mechanisms in this reactor and quantification of mixing scales is not yet available. Main reasons for this are probably related to the fact that the vortices evolve very rapidly and are formed at high rates, and the fact that the tracer diffusion rapidly homogenizes the two fluids (Sandell et al., 1986). A through characterization of mixing mechanisms in RIM is here made using Computational Fluid Dynamics (CFD) simulations of two-phase flow using the Volume-of-Fluid (VOF) model. Both phases have the same physical properties and the surface tension between the two phases was set to zero. Simulations using the VOF model track the interface between both phases. This model was chosen over the approach previously used by Santos et al. (2005) due to the fact that with VOF, a value of zero for the diffusion coefficient between the

Published by De Gruyter, 2011

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International Journal of Chemical Reactor Engineering

Vol. 9 [2011], Article A114

two phases can be set. Usually, computer memory limitations and high computational times make impracticable the use of the actual molecular diffusion values, even in reactors as small as the mixing chamber of RIM machines. Santos et al. (2005) considered an unrealistically high value for the diffusion, 1000 times larger than the actual values, and even so the 2D CFD simulation took around six months to cover a period of five passage times in the mixing chamber (circa 50 ms). Furthermore, the main mixing mechanism in polymeric systems is not diffusion, neither turbulent nor molecular; the main mixing mechanism is the striation thinning of the initial laminas, i.e., the monomer laminas that are introduced by the opposed jet streams. This work measures the average scale of segregation (Danckwerts, 1952) in the RIM process, using the spatial distribution of the two phases from the CFD simulations at the mixing chamber outlet. The scale of segregation is a measure of the average width of the laminas of two fluids. This work tackles the challenge posed by Ranz (1986) regarding mixing in RIM; according to Ranz (1986) from the best available theories at that time the scale of mixing between both monomers was around 100 μm. However for the polymerisation reaction to proceed up to an extension that would allow the production of a final part in the mould a striation thickness of 15 μm would be required (Lee et al., 1980). Since RIM is a technology for the production of polymers, the theory in 1986 failed to provide answer on how the contact of both monomers occurs. The effect of the striation thickness distribution on the course of unpremixed polymerisations was studied by Fields and Ottino (1987a, b, c), particularly for the RIM process. In these studies Fields and Ottino (1987a, b, c) used a lamellar model to evaluate the extent of the polymerisation reaction in onedimensional distributions of alternating monomers, from both monodispersed and distributed material striations, and also the effect of the stretching history on monomers layers. The studies showed that the reaction time scale could be several orders of magnitude underestimated if a monodispersed striation thickness distribution is considered. The striation thickness distribution may have a severe impact on the polymer properties such as the maximum and average molecular weight achieved. From 1986 up to now the increasing availability of computational resources has allowed solving many scientific and engineering problems, although the issues of striation thickness distribution in RIM along with Ranz’s challenge has remained without an answer. An earlier attempt to tackle this challenge was made by May (1996) using turbulence models. In the study of mixing in RIM made by Santos et al. (2005) the diffusion values were too high and, as a consequence, the fluids at the outlet of the mixing chamber were too homogeneous and so it was not possible to identify any lamellar structures; thus the available data from Santos et al. (2005) could not be used to compute the scale

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of segregation. Here the scale of segregation in RIM will be computed and analysed from the results of the CFD simulations. The results of this paper identify the advective mechanisms of stretching, folding and rupture that enable the decrease on mixing scales in RIM down to the values required for chemical reaction.

Governing Equations The set of equations chosen for the flow description in the 2D model of the RIM machine mixing chamber considers incompressible fluids and neglects the body forces. The equations are the continuity equation ∇⋅v = 0

(1)

⎛ ∂v ⎞ ρ ⎜ + v ⋅ ∇v ⎟ = −∇p + µ∇ 2 v ⎝ ∂t ⎠

(2)

and the Navier-Stokes equation

where v is the velocity vector, ρ and µ are the density and the viscosity of the fluid, respectively, p is the pressure and t is the flow time. The interface between the two immiscible phases was tracked using the Volume-Of-Fluid (VOF) model. The VOF model, for two immiscible phases with the same physical properties and without surface tension and diffusional mass transfer between them, is given by ∂α A + ∇ ⋅ αAv = 0 ∂t

(

)

(3)

where α A is the volume fraction of phase A. The volume fraction of phase B, α B , is determined by using the following constraint

∑α i

i

{

= 1, i = A, B

}

(4)

that forces the sum of the volume fractions of each phase to be equal to 1.



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CFD Model The simulated 2D geometry is an axial cut of a typical industrial RIM machine mixing chamber with the same dimensions of the chamber used in the experimental work of Teixeira (2000), Santos (2003), Teixeira et al. (2005) and Santos et al. (2008). The diameter of the injectors and of the chamber are d = 1.5 mm and D = 10 mm , respectively. The chamber height is H = 50 mm and the axis of the injectors is located at a distance h = 5 mm from the top of the chamber. A drawing of the simulated geometry, respective dimensions and applied boundary conditions, is shown in Figure 1. The 2D model for the simulation of the mixing chamber has been validated over the last years for the simulation of 3D RIM machines mixing chambers (Santos, 2003; Santos et al., 2009; Santos et al., 2010). The main differences between the 2D model and the actual 3D flow field are: •

•

In the 3D flow field the jets follow a straight line up to the impingement point and after the fluids are spread radially forming a pancake like structure that promotes a large interface for the contact of both fluids (Wood and Johnson, 1991). The stretching due to the fluids spreading can reduce the initial scale of the jets to a tenth of its original value. In the 2D geometry the jets are bent towards the outlet prior to impinging each other, and immediately after the impingement point the jets keep the same velocity (Santos et al., 2005). On both 2D and 3D geometries, the self-sustainable chaotic flow regime is characterized by the formation of vortices after the jets’ impingement point. These vortices engulf fluid from both jets and evolve towards the outlet. The main 2D / 3D difference is the fact that in 3D the vortices dissipate faster, typically within a distance equal to a mixing chamber diameter from the jets’ impingement point (Teixeira et al., 2005), while in 2D the vortices evolve through longer distances.

There are some common features between the 2D and 3D flows, and this has been the basis for using the 2D model on the representation of the 3D flow. The 3D flow features that are simulated by the 2D model are: • •

A steady flow regime at lower Reynolds numbers where the two fluid streams flow segregatedly throughout the mixing chamber. A chaotic flow regime where the two fluid streams are engulfed by successive vortices that shed from the jets impingement point towards the mixing chamber outlet.



•

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The jets’ impingement point oscillates with a frequency equal to the vortices formation rate (Santos et al., 2009).

The fact that the vortices in 2D geometries evolve through longer distances of the mixing chamber was the basis for a new RIM mixing head (European Patent EP 1 732 747 and US Patent 7708918) where the mixing chamber is prismatic. The prismatic mixing chamber for RIM is an extrusion of the 2D geometry in the normal direction. Experimental results obtained on prismatic geometries were published by Sultan et al. (2010), and it was observed that the 2D model simulates the flow patterns observed in the prismatic chambers. The 2D model was chosen for its ability to capture some of the most important flow features in RIM machines, and because a 3D model would not be computationally affordable for a VOF simulation. Furthermore, the results from 2D model have an increasing interest if new RIM machine mixing chamber designs are considered. Prior to this work predictions on the mixing scales in RIM were only available from May (1996) that used a turbulent model. This work replaces turbulence assumption by the visualization of laminar mechanisms, which are more physically sounded as far as mixing in RIM is regarded. Furthermore, the VOF approach to the simulation of mixing is an important addition of this work to mixing science. Previous works on the simulation of mixing without diffusion were made using particle tracking (Aubin et al., 2003; Howes et al., 1999). Roberts (2010) using particle tracking shows the need for methods that are not affected by diffusion for the identification of the local microstructure in laminar flows. Another method for the quantification of mixing without diffusion is the conformal mapping method (Kruijt et al., 2001; Galaktniov et al., 2002; Singh et al., 2007; Meijer et al., 2009). Galaktniov et al. (2002) used the mapping method in a 3D geometry with industrial application, the Kenics static mixer. In this paper an alternative method, VOF, available in commercial CFD codes for multiphase flows is used for the simulation of diffusion-less single-phase mixing. This is a new method that adds to the two others already mentioned, and furthermore it is used in a dynamic flow field in a geometry with industrial application. In the 2D CFD simulations of Santos et al. (2002/2005), as well as experiments of (2003) and Santos et al. (2008), it was observed that advective mixing mainly occurs above a critical Reynolds number, Re = 250 , that marks the transition of the flow regime from laminar steady to a transient laminar chaotic flow. Both flow regimes will be simulated using the VOF model. The Reynolds number is defined as



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Re =

ρυ inj d µ


(5)

where d is the injectors’ width and υ inj is the average velocity at the injectors. The fluid of both phases has density ρ = 1000 kg/m 3 and viscosity µ = 0.02 Pa ⋅s .

Figure 1

2D model used in the CFD simulations: schematic drawing with geometry and boundary conditions.

The boundary conditions imposed to the 2D simulation model are summarized in Figure 1 and are described below. At the chamber walls no slip conditions are assumed, i.e. both velocity components are set to zero at the walls, υ x = υ z = 0 . At the injectors a parabolic flow profile is set. For the right injector the profile is defined as

⎛ ⎛ ⎛ d ⎞⎞ ⎛ d ⎞⎞ ⎜ z − ⎜⎝ h − 2 ⎟⎠ ⎟ ⎜ z − ⎜⎝ h + 2 ⎟⎠ ⎟ ⎝ ⎠⎝ ⎠ υ x (z) = 6 υ inj 2 d and for the left injector is defined as:


(6)


7

υ x (z) = −6 υ inj

⎛ ⎛ ⎜ z − ⎜⎝ h − ⎝

⎛ d ⎞⎞ ⎛ z −⎜h+ ⎟ ⎜ ⎟ 2⎠⎠ ⎝ ⎝

d ⎞⎞ 2 ⎟⎠ ⎟⎠

d2

(7)

The inlet stream of the right injector was set to contain only phase A and the inlet stream of the left injector only phase B. At the chamber outlet a constant and uniform pressure value was set. The CFD simulations are initially performed at steady state, i.e., the time derivatives of Equations 2 and 3 are set to zero. All the variables were set to zero in the entire domain as the initial guess of the steady state solution. For the steady state solution a wall was set at the chamber axis with shear stress equal to zero. On the left side of the shear-free wall the volume fraction of phase A was set to zero and on the right side it was set to one. The dynamic simulations were started from the steady state solution after removing the interior wall boundary condition and imposing to the system a nonsymmetric perturbation, i.e., the velocity of the right injector was varied with time as υ x (z,t * ) = υ x (z) f (t * ) , where f (t * ) is the perturbation time function affecting the steady velocity profile and t * is defined as t * = υ inj t d . Care was taken not to induce, in this way, any discontinuity in the flow field and so a smooth time function with a continuous first order derivative was chosen,

⎛ ⎛ 2π t * ⎞ ⎞ a * f (t ) = 1 + 1 − u t − b ⎜ 1 − cos ⎜ ⎟⎟ 2 ⎝ b ⎠⎠ ⎝ *

(

( (

))

(8)

)

where u t * − b is the Heaviside function. The parameter a is the maximum amplitude of the perturbation, which in this case was set to a = 0.1 . The value of b dictates the time over which the perturbation is applied to the velocity profile of the right injector and was set to a tenth of the fluid passage time in the chamber,

(

τ = DH 2υ inj d

)

(9)

This perturbation has a limited impact in time and small amplitude when compared to υ inj with a maximum value of 0.1 υ inj . After 0.1 τ no further perturbation was imposed to any of the system variables, and the boundary conditions remained constant in time. The influence of the perturbation amplitude and period on the flow dynamics was assessed by Santos et al. (2010). The


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perturbation has no impact on the flow dynamics after the transient period where the system evolves from the initial steady-state.

Discretization and Numerical Solution The simulations were performed in an Intel® Xeon® 3GHz 8-core machine running the finite-volume commercial CFD software ANSYS Fluent™ 12.1 for 64-bits Windows®. The computational grid consisted of 200 × 1000 quadrilateral cells, i.e. each cell has 50 μm of length and width. Previous works from Teixeira (2000) and Santos (2003) showed this grid to be sufficiently refined for the simulation of all the scales of the flow in the range of the studied Reynolds numbers. The time step size, Δt , was chosen according to

Δt =

Δx

υ inj,max

(10)

where Δx is the dimension of each discrete fluid element and υ inj,max = 3 2 υ inj . All simulations were performed for a flow time equal to 10 τ . The Coupled scheme was used for the pressure-velocity coupling and the PRESTO! and QUICK schemes were used for pressure and momentum discretization, respectively. An explicit first order scheme for time discretization was used and the interface between the two phases was tracked using a piecewiselinear geometric reconstruction scheme (Geo-Reconstruct) that assumes the interface to have a linear slope within each discrete fluid element. The steady state initial simulations were considered converged when a stable and constant value of the normalized residues of the continuity and momentum conservation equations was reached. In all simulations these values were lower than 10−8 . For the dynamic simulations, a normalized value of 10−4 for the residues of all variables was used as a convergence criterion. In addition to these criteria, the inlet and outlet mass flow rates were continuously compared to guarantee the mass conservation principle was being satisfied.

Scale of Segregation In order to calculate quantitatively the purely advective mixing of the two immiscible phases, a linear scale of segregation, sL , as defined by



9

Danckwerts (1952) was used. This quantity relates to the size of the clumps of each phase and was calculated in the line that defines the outlet boundary. The scale of segregation is defined as the integral of the correlation function of the volume fractions, R r , from zero, where R 0 = 1 , to a distance ξ at which

()

()

()

there is no correlation, i.e. R ξ = 0 (Danckwerts, 1952; Tadmor and Gogos, 2006):

sL =

∫ R ( r ) dr ξ

(11)

0

()

The function R r was defined by Danckwerts (1952) as

()

R r =

(α ( x ) − α ) (α ( x + r ) − α ) (α ( x ) − α ) 2

(12)

where α is the average value of the volume fraction of one of the phases at the outlet boundary, x is the position vector, and r is the distance that separates two points with volume fraction α x and α x + r .

()

(

)

In this work, the correlation function was calculated from the power spectrum of the volume fraction in the outlet using the Fast Fourier Transform technique as proposed by Tucker (1991), rather than Equation 12. The power spectrum P n is defined as

()

( ) ∫ (α ( x ) − α ) e

P n =

∞

−2π inx

−∞

2

dx

(13)

where n is the wave number with units of length-1. The coefficient of correlation function is calculated from the inverse Fourier transform of P n

()

()

∞

()

R r = ∫ P n e2π inr dn −∞

(14)



10


Results Mixing in RIM is assessed from the contour plots of the two phases introduced in the mixing chamber separately, one phase from each jet. Figures 2 and 3 show dynamic sequences of the volume fraction maps of each phase inside the mixing chamber and the maps of the component of the dimensionless vorticity normal to the 2D plane as a function of the dimensionless time t τ for Reynolds numbers Re = 100 and 300. The dimensionless vorticity, ω * , is defined as

ω* =

(

d ∇×v υ inj

)

(15)

Should be noticed that in Figure 2 and 3 the right injector is at the top-left corner and the left injector is at the bottom-left corner of each map. Both cases, Re = 100 and 300, started from a steady state, where the two opposed jets impinge in the mixing chamber axis and are bent towards the outlet. Upon impinging each other, the jets flow towards the outlet parallel to the mixing chamber axis. Santos et al. (2005) showed that mass transfer between the two liquid streams separately introduced into the mixing chamber, at the steady state, only occurs by diffusion. In the present study, since diffusion was not considered, the initial condition was set to only one of the phases in each side of the reactor. The initial steady state of the flow was perturbed at t = 0 for a period of 0.1 τ using a time function for the jets velocity at one of the inlets, see Equation 8. After being perturbed, the flow evolved to different flow regimes for Re = 100 and Re=300. At Re = 100 , see Figure 2, after the initial perturbation, the initial interface between the two phases is disturbed by the formation of vortices. As time advances, the system tends to a non-symmetrical steady state (Santos et al., 2010), which differs from the initial symmetric state, although the fluids from the two opposed jets are still kept segregated while flowing through the mixing chamber.



Figure 2

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Maps of volume fraction of phase A and dimensionless vorticity in the mixing chamber as a function of the dimensionless time for Re=100.

Increasing the Reynolds number, Re = 300 , see Figure 3, the system tends to a self-sustained chaotic flow after being perturbed. This chaotic state is characterized by the existence of two counter-rotating vortices immediately downstream the jets, each occupying half of the chamber diameter. These vortices evolve, along the chamber axis, to fully developed vortices that extend throughout the whole chamber width. At this chaotic state, strong jet oscillations occur, as observed experimentally by Teixeira et al. (2005) and Santos et al. (2009). From



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the evolution of the volume fraction maps in Figure 3, the role of these fully developed vortices is clear: when evolving throughout the chamber the fast counter-rotating vortices promote the formation of smaller portions of each phase either by the stretching and folding of the material or by the rupture of the formed laminas.

Figure 3

Maps of volume fraction of phase A and dimensionless vorticity in the mixing chamber as a function of the dimensionless time for Re=300.



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Comparing the vorticity and volume fraction maps, it is seen that each vortex is mainly formed with fluid from one of the injectors because volume fraction values in the center of the vortices are always associated to the same sign of the vorticity values. Positive values of vorticity are associated with counterclockwise rotation of the vortex formed with fluid from the right jet and so its center presents mainly phase A material. For negative values of vorticity, the opposite situation occurs. Although the analysis of the volume fraction maps from the dynamic simulation provides very good insight into the mixing mechanics, it lacks the quantification of the mixing degree. For the calculation of the scale of segregation, the history of the distribution of the two phases at the outlet for Re = 100 and 300 was recorded and it is shown in Figures 4a and 5a, respectively. From the observation of Figure 4a it is clear that for Re = 100 , after a transient state, the flow tends to a steady state with the complete segregation of both phases. For Re = 300 , Figure 5a shows the passage through the outlet of consecutive counter-rotating vortices: counterclockwise rotating vortices centres contain mainly phase A in the centre, while clockwise vortices centres contain mainly phase B. Figures 4b and 5b show the scale of segregation as a function of the dimensionless time for Re = 100 and Re = 300 , respectively. It is worth pointing that the values of the scale of segregation obtained in the first instants are in agreement with what should be obtained for a uniform stripped pattern, where the scale of segregation is equal to one quarter of the thickness of one layer sL = D 8 = 1.25 mm (Tadmor and Gogos, 2006).

(

)

The calculated scale of segregation shows the role of the advective patterns present in the self-sustained chaotic regime on the decrease of the mixing scales. For Re = 100 , see Figure 4b, the scale of segregation tends to a value very close to the one of the initial symmetric steady state, where the two phases injected by each jet are completely segregated. For Re = 300 , as shown in Figure 5b, the scale of segregation reaches values ten times smaller than the initial segregated state. The scale of segregation significantly varies with time. These variations are associated to the passage of consecutive counter-rotating vortices leaving the mixing chamber: their centres present mainly one of the phases while the borders present portions of a smaller scale of each fluid.



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Figure 4

History of the volume fraction of phase A (black) and phase B (white) at the outlet of the mixing chamber (a) and the scale of segregation at the outlet of the chamber as a function of the nondimensional time (b) for Re=100.

Figure 5

History of the volume fraction of phase A (black) and phase B (white) at the outlet of the mixing chamber (a) and the scale of segregation at the outlet of the chamber as a function of the nondimensional time (b) for Re=300.



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The VOF model with the tracking of the interface between the two phases enabled the computation of the scale to which the fluid streams are reduced while being mixed with each other. However, the map of the volume fraction of phase A overlapped with the simulation grid, see Figure 6, clearly shows that the formed lamellar structures have a thickness of the same size of the grid spacing (50 µm). Because of this, the results that are shown in this paper are not yet the smallest scales to which the fluid is striated. Figure 6 proves that further grid refinement will enable the simulation of smaller mixing scales. The scale for mass transfer is given by Batchelor scale,

λB =

λK

(16)

Sc

where λ K is the Kolmogorov scale, and Sc is the Schmidt number defined as

Sc =

µ ρ Dm

(17)

where Dm is the molecular diffusivity of a chemical species in a fluid with viscosity µ and density ρ . When there is no diffusion between the phases that are being mixed, the Schmidt number tends to infinity and the Batchelor scale tends to zero when the path over which the laminas are being striated tends to infinity. Even in limited length geometries, such as the one in this paper, the scale of mixing can rapidly be reduced to values out of reach of the available computational resources. On one hand the zero scale for mixing has no physical meaning, on the other hand a scale of practical interest can be defined and the resolution of the numerical grid can be set accordingly to it. In the present case, the simulation of the values of striation proposed by Lee et al. (1980) requires a numerical grid with a spacing of 15 μm. The image in Figure 6 is obtained at a distance D from the impingement point, and at this early stage of mixing the grid resolution was already insufficient to simulate the actual scales of mixing. The simulation at Re=300 with the current grid (200×1000 elements) needs at least 1 GB of RAM. For the simulation of laminas with 15 μm using three nodes per lamina the grid spacing would be 5 μm (2000×10000 elements), which would require around 100 GB of RAM for the CFD simulation with the VOF model. The tracer map in Figure 6 shows the formation of droplets that are generally due to the surface tension between two phases. In this work the surface tension was always set to zero and this droplets in the actual flow have the shape



16


of thin laminas. The occurrence of droplets is due to the fact that the concentration matrix stores an average value when the phases A and B occupy simultaneously a fraction of the same cell.

Figure 6

Detail of the map of the volume fraction of phase A with the computational grid overlapped.

Comparison with Experimental Data The patterns from mixing of two fluids were imaged with Planar Laser Induced Fluorescence (PLIF) at the outlet of a RIM machine mixing chamber. The pilot RIM machine is shown in Figure 7. The machine has two tanks that were filled with an aqueous solution of glycerine having a viscosity of 80 cP, in one tank the fluid was dyed with rhodamine 6G. The fluids were delivered from the tanks to the mixing chamber by two gear pumps, and were mixed in a cylindrical mixing chamber. The mixing chamber is cylindrical having diameter of 10.0 mm and height of 50 mm. The fluids were introduced in the mixing chamber from two opposite injectors having 1.5 mm diameter and located at 5.0 mm from the top of the mixing chamber. At the outlet of the mixing chamber was installed a transparent cylindrical runner with 10 mm diameter.



Figure 7

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Partial view of RIM machine and PLIF setup.

Figure 8 shows a PLIF image of the mixing patterns formed in the runner at a Reynolds number of 300. The Reynolds number was defined in Equation 5; the typical dimension, d, for this chamber was the diameter of the injectors. The flow direction in Figure 8 is from left to right. The flow at the runner is already fully developed laminar, having a parabolic velocity profile that was measured with Particle Image Velocimetry. Figure 8 clearly shows that the laminas of the two fluids are oriented in the perpendicular direction to the main flow. The vortices that engulf both fluids cause this orientation of the laminas. When the flow develops into a parabolic velocity profile this laminas are further thinned and show a parabola that is also clear in Figure 8. The CFD results show similar patterns to the PLIF image in Figure 8, i.e., the laminas are oriented in a perpendicular direction to the main flow profile. One important difference between 2D and 3D geometry is that in the 2D geometry a Poiseuille flow would be attained at positions further downstream in the mixing chamber, and so the pattern of laminas forming parabolas is not yet occurring within the 50 mm of the 2D chamber. This result validates the formation of vortices that engulf both fluid streams as the main mixing mechanism in RIM,



18


otherwise the orientation of the laminas would be parallel to the main flow direction.

Figure 8

PLIF image in the axial plan of the runner at the outlet of the mixing chamber.

Discussion and Conclusions This paper presents results of the mixing of two immiscible phases in a RIM mixing chamber from a 2D CFD numerical model. It was observed that advective mixing mainly occurs for a Reynolds number higher than a critical value placed between 100 and 300: •

For Re = 100 a non-symmetrical steady laminar flow, different from the symmetrical initial one, is obtained with complete segregation of the two phases.

•

For Re = 300 a self-sustained chaotic flow is obtained with formation of counter-rotating vortices that promote the creation of portions of each phase of smaller dimensions either by stretching and folding of the material and the rupture of the laminas.



19

These results agree with the previous CFD simulations results of Santos et al. (2005), where advective mixing mainly occurs above a critical Reynolds number of 250. From the history of the volume fractions distribution at the outlet, an average scale of segregation was calculated. It is notorious from the results the decrease of the scale of segregation from the steady laminar flow at Re = 100 to the chaotic flow obtained for Re = 300 , but further work must be done using a denser grid in order to have a complete description of the dependence of the scales of the flow on the Reynolds number. Although the smallest scales of mixing were not simulated, from these results the advective mechanisms identified in RIM are enough to explain the reduction of the lamina dimensions down to the value set by Lee et al. (1980) for the actual polymerisation to take place. The utilization of a VOF model for the analysis of mixing in single-phase reactors without diffusion was made for the first time in this study. This approach adds to the two alternative methods already reported in the literature, the use of particle tracking (Roberts, 2010) and the extended map method (Kruijt et al., 2002).

References Aubin, J., Fletcher, D.F., Bertrand, J., Xuereb, C., Characterization of Mixing Quality in Micromixers, Chemical Engineering and Technology, 2003, 26, 1262-1270. Danckwerts, P.V., The definition and measurement of some characteristics of mixtures, Applied Science Research, 1952, 3-A, 279-269. Galaktionov, O.S., Anderson, P.D., Peters, G.W.M., Meijer, H.E.H., Morphology Development in Kenics Static Mixers (Application of the Extended Mapping Method), The Canadian Journal of Chemical Engineering, 2002, 80, 604-613. Howes, T., Mackley, M.R., Roberts, E.P.L., The Simulation of Chaotic Mixing and Dispersion for Periodic Flows in Baffled Channels, Chemical Engineering Science, 1991, 46, 1669-1677.


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Johnson, D.A., Wood, P., Hrymak, A.N., The Effect of Geometrical Parameters on the Flow Field of an Opposed Jet RIM Mix Head: Equal Flow and Matched Fluids, The Canadian Journal of Chemical Engineering, 1996, 74, 40-48. Johnson, D.A., Experimental and Numerical Examination of Confined Laminar Opposed Jets. Part I Momentum Imbalance, International Communications in Heat and Mass Transfer, 2000, 27, 443-454. Kolodziej, P., Macosko, C.W., Ranz, W.E., The Influence of Impingement Mixing on Striation Thickness Distribution and Properties in Fast Polyurethane Polymerisation, Polymer Engineering and Science, 1982, 22, 388-392. Kolodziej, P., Yang, W.P., Macosko, C.W., Wellinghoff, S.T., Impingement Mixing and its effect on the Microstructure of RIM Polyurethanes, Journal of Polymer Science, 1986, 25, 2359-2337. Kruijt, P.G.M., Galaktionov, O.S., Anderson, P.D., Peters, G.W.M., Meijer, H.E.H., Analysing Mixing in Periodic Flows by Distribution Matrices: Mapping Method, AIChE Journal, 2001, 47, 1005-1015. Kusch, H.A., Ottino, J.M., Shannon, D.M., Analysis of Impingement MixingReaction Data: Use of a Lamellar Model to Generate Fluid Mixing Information, Industrial & Engineering Chemistry Research, 1989, 28, 302315. Lee, L.J., Ottino, J.M., Ranz, W.E., Macosko, C.W., Impingement mixing in reaction injection molding, Polymer Engineering Science, 1980, 20, 868874. Malguarnera, S.C., Suh, N.P., Liquid Injection Molding I. An Investigation of Impingement Mixing, Polymer Engineering and Science, 1977, 17, 111115. May, H.O., A Field Equation for the Distribution of Striation Thickness from Impingement Mixers, Polymer Engineering and Science, 1996, 36, 583585. Meijer, H.E.H., Singh, M.K., Kang, T.G., den Toonder, J.M.J., Anderson, P.D., Passive and Active Mixing in Microfluidic Devices, Macromolecular Symposia, 2009, 279, 201-209.



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Nguyen, L.T., Suh, N.P., Effect of High Reynolds Number on the Degree of Mixing in RIM Processing, Polymer Process Engineering, 1985, 3, 37-56. Nguyen, L.T., Suh, N.P., Processing of Polyurethane/Polyester Interpenetrating Polymer Network by Reaction Injection Molding (RIM) Part I: Design of a High Pressure RIM System, Polymer Engineering and Science, 1986, 26, 781-798. Ranz,

W.E., Analysis of Reaction Processes in Which Microscopic Heterogeneities Appear: Scale-Up and Scale-Down of Polymerization Reactions, Industrial & Engineering Chemistry Fundamentals, 1986, 25, 561-565.

Roberts, E.P.L., Determination of the local micromixing structure in laminar flows, Chemical Engineering Journal, 2010, 160, 267-276. Sandell, D.J., Macosko, C.W., Ranz, W.E., Visualization Technique for Studying Impingement Mixing at Representative Reynolds Numbers, Polymer Process Engineering, 1985, 3, 57-70. Santos, R.J., Mixing mechanisms in reaction injection moulding-RIM. An LDA/PIV experimental study and CFD simulation, 2003, Ph.D. Thesis, Universidade do Porto. Santos, R.J., Erkoç, E., Dias, M.M., Teixeira, A.M., Lopes, J.C.B., Hydrodynamics of the mixing chamber in RIM: PIV flow-field characterization, AIChE Journal, 2008, 54, 1153-1163. Santos, R.J., Teixeira, A.M., Costa, M.R.P.F.N., Lopes, J.C.B., Operational and design study of RIM machines, International Polymer Processing, 2002, 17, 387–394. Santos, R.J., Erkoç, E., Dias, M.M., Lopes, J.C.B., Dynamic Behaviour of the Flow Field in a RIM Machine Mixing Chamber, AIChE Journal, 2009, 55, 1338-1351. Santos, R.J., Teixeira A.M., Lopes, J.C.B., Study of mixing and chemical reaction in RIM, Chemical Engineering Science, 2005, 60, 2381-2398.


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Schütz, S., Piesche, M., Bierdel, M., Charakterisierung des Mischverhaltens von Gegenstrom-Injektions-Mischern, Chemie Ingenieur Technik, 2005, 77, 398-405. Singh, M.K., Kang, T.G., Meijer, H.E.H., Anderson, P.D., The mapping method as a toolbox to analyze, design, and optimize micromixers, Microfluidics and Nanofluidics, 2007, 5, 313-325. Sultan, M.A., Fonte, C.P., Dias, M.M., Lopes, J.C.B., Santos, R.J., Effect of geometry on mixing in T-jets mixers, ECCE 7 - 7th European Congress of Chemical Engineering, Prague, Czech Republic, 2010. Tadmor, Z., Gogos, C.G., Principles of Polymer Processing, 2006, 2nd Ed., John Wiley & Sons, New Jersey. Teixeira, A.M., Escoamento na Cabeça de Mistura de uma Máquina RIM (Flow in a RIM Machine Mixing Head), 2000, Departamento de Engenharia Química da Faculdade de Engenharia, Universidade do Porto. Teixeira, A.M., Santos, R.J., Costa, M.R.P.F.N., Lopes, J.C.B., Hydrodynamics of the Mixing Head in RIM: LDA Flow-Field Characterization, AIChE Journal, 2005, 51, 1608-1619. Trautmann, P., Piesche, M., Experimental investigation on the mixing behaviour of impingement mixers for polyurethane production, Chemical Engineering and Technology, 2001, 24, 1193-1197. Tucker III, C.L., Principles of Mixing Measurements, C. Rauwendaal, ed., “Mixing in Polymer Processing”, 1991, Plastics Engineering, Marcel Dekker, New York. Tucker, C.L., Suh, N.P., Mixing for Reaction Injection Molding. I. Impingement Mixing of Liquids, Polymer Engineering and Science, 1980, 20, 875-886. Wood, P., Hrymak, A.N., Yeo, R., Johnson, D.A., Tyagi, A., Experimental and Computational Studies of the Fluid Mechanics in an Opposed Jet Mixing Head, Physics of Fluids A, 1991, 3, 1362-1368.


Quantification of Mixing in RIM Using a Non-Diffusive

Quantification of Mixing in RIM Using a Non-Diffusive

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