A New Adaptation of Mapping Method to Study Mixing of Multiphase
Flows in Mixers with Complex Geometries
Arash Sarhangi Fard a,b, Navid M. Familia,*, Patrick D. Andersonb
b
a
Polymer Engineering Group, Tarbiat Modares University, P.O.Box 14115-111, Tehran, Iran
Materials Technology, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
_______________________________________________________________
Abstract
The main objective of the present work is to modify the traditional mapping method for the
simulation of distributive mixing of multiphase flows in geometries involving moving parts such as,
internal mixers or twin-screw extruders without a limitation on their geometrical periodicity. The
periodicity condition, limits the results of traditional mapping method to tracking mapping
mesh between specific discrete time intervals or distances for that geometry is repeated, hence, result
is only for fixed orientation of rotors. Imaginary Domain Method is introduced to track mapping
mesh from one state to the next free of geometrical periodicity limitations. In this work the method is
introduced and its applicability and accuracy is discussed in details. A two-dimensional (2D)
simulation of mixing of two Newtonian fluids with different viscosities in an intermeshing internal
mixer is used as a test case study. In this example the key issues of ability to predict mixing state in
details for all orientations of rotors is presented. To reduce diffusion errors of mapping method in the
boundaries of the rotors, mapping mesh refinement technique that relies upon one single reference
mesh is also presented.
Keywords: Distributive Laminar Mixing; Mapping Method; Imaginary Domain Method; Two Phase Flow
_______________________________________________________________
1. Introduction
Mixing as an important part of polymer processing could be found in almost all of the
operations in this industry. Various examples could be found in polymerization or polymer
processing industries. Primarily, mixing in polymer processing refers to two operations:
mixing of polymer with various additives such as fillers, color pigments, reinforcement
agents, blowing agents… and blending of different polymers in melt state for the production
of polymer alloys and/or blends with desired chemical and/or mechanical properties.
The term mixing refers to operations that have a tendency to reduce non-uniformities or
gradients in concentration, temperature, size of a dispersed phase, or other properties of
materials (Baird & Collias, 1995). Since the ultimate physical and mechanical properties of
the blends depend on the final morphology of the phases, the structure control of the two or
more phases is the basic goal in the production of these mixtures. The progress of phase
morphology development clearly depends on the material volume ratio, interfacial tension,
and viscosity ratio as well as the nature of the flow during processing. Mixing modeling
shows the effect of these factors on the morphology development of the final blend.
To achieve the fine-scaled mixture, mixing process first begins from macromixing stage, as
it progresses from large to small scale, micromixing approach dominates. In the
macromixing stage of mixing, large masses of the phases are embedded in each other and
under flow field they are stretched and folded over. At this stage the interfacial forces do not
*Corresponding author.
E-mail address: nfamili@modares.ac.ir (Navid M.Famili).
play a significant role because viscous forces are dominant. With the progression of mixing;
extended blobs break into many smaller drops and interfacial forces effects become
important. The microstructural phenomena such as breakup and coalescences compete
against each other and finally this competition determines the final droplet size distribution
or mixture morphology (DeRoussel et al., 2001).
We can characterize mixing in the macro scale by defining mixing parameters and
calculation of these parameters using velocity field. Therefore, the analysis of the macromixing stage is based on evaluation of the velocity fields (Tadmor & Gogos, 1979;
Middleman, 1977; Wetzel & Tucker, 1999).
Characterization of initial stages of mixing could be achieved by either implicit or explicit
models: In the implicit modeling, mixing parameters such as residence time distribution,
strain distribution, and local measures of extensional or rotational flow is directly calculated
from the velocity field. However, these parameters are not useful for mixing analysis of
multiphase flows and in general are not flexible enough to allow for optimization of mixing
processes. The applications of these mixing parameters in engineering and design of mixers
are limited and these models mostly applicable for one phase flows (Kruijt et al., 2001a). We
refer the reader to Bravo, Hrymak and Wright (2004) and Yao & Manas-Zloczower (1996).
Explicit models are more useful for modeling multiphase flows because they contain
variables that represent the instantaneous state of mixing as well as their evolution in the
flow field. These parameters such as concentration distribution of phases could be used for
the evaluation of morphology development in the last stages of mixing (Kruijt et al., 2001a).
The most common method in the modeling of the early stages of mixing in multiphase flows
is to solve the momentum and the mass conservation equations numerically. A number of
methods have been used for this purpose, such as; front capturing methods (Noh &
Woodward, 1976): see e.g. VOF method (Hirt & Nichols, 1981), the method of Young
(Young et al., 1980), front tracking methods (Tryggvason & Unverdi, 1990; Tryggvason et
al., 2001; Unverdi & Tryggvason, 1992): see e.g. marker and cell method (Aulisa et al.,
2004), adaptive front tracking method (Galaktionov et al., 2000) and mapping method
(Kruijt et al., 2001a).
Of these the mapping method shows the most flexibility in application while keeping the
calculation efficiency and required accuracy. The method is based on the repetitive nature of
mixing flows. Mapping method has been used for mixing analysis in complex geometries
and flows where mixing parameters such as concentration, residence time distribution and
other statistical parameters where evaluated (Kruijt et al., 2001b; Kruijt et al., 2001c;
Anderson et al., 2000; Galaktionov et al., 2002; Le Guer & Schall, 2004).
Kruijt et al. (2001c) applied the mapping method in the analysis of mixing in a closely
intermeshing self wiping, co-rotating twin screw extruder. They developed a three
dimensional model of mixing of two Newtonian fluids with same properties (only differ in
color) in a twin screw extruder. Velocity field was calculated using a fictitious domain mesh
generation finite element based method. Since the geometry of two-lobes screw elements are
symmetric at 90º,180º,270º and 360º rotations, they were able develop the mapping matrix
for 90º rotations of screws. Therefore mixing parameters such as concentration distribution
could be evaluated only for each 90º rotation of the screw. Hence, the development of the
velocity field in between is ignored. The method could only be used for symmetric corotating screw elements.
The mapping method could be used only on repeating geometries where tracking of mapping
mesh is possible between fixed geometrical orientations. This will limit the application of the
method on un-symmetric mixers like most of twin screw extruders. The major goal of this
work is to develop a method of mixing modeling of multiphase flows based on mapping
method in the mixers which have some moving parts such as twin screw extruders or internal
mixers. The geometry of these mixers at best could be repeated only after some time period
(time periodic) or after some distance (space periodic) where this periodicity condition
restricts the normal application of mapping method only for some fixed orientations of
geometry. Imaginary Domain Method (IDM) is introduced for mapping mesh tracking free
of periodicity of geometry limitation. With using IDM, time intervals that is needed for
calculation mapping matrix is not restricted to periodicity of geometry and is limited only to
application of quasi state condition of flow between time intervals that mapping matrix is
calculated, and controlling the diffusion error of mapping method. Using IDM, simulation of
fluids with different rheological properties is also possible. In order to reduce diffusion error
of mapping method refine of mapping mesh was introduced.
In this study we used an internal mixer as a test case study but this method can be extend to
other continuous mixers such as twin-screw extruders with using some numerical and
geometrical techniques such as slice method (Sastrohartono et al., 1990; Sastrohartono et al.,
1995; Kajiwara et al., 1996; Bravo et al., 2000; Rauwendaal, 2001).
2. Imaginary domain method
Mapping method is based on repeatability nature of mixing flows. Time or space periodicity
of mixer geometry is necessary condition for applying mapping method. In mapping method
the flow domain, Ω is subdivided into n sub-domains, Ωi with boundaries Γi with no gap or
overlapping between sub-domains. Using velocity field, boundaries of elements are tracked
from t=t0 to t=t0+∆t. Two meshes are obtained, initial mesh and deformed mesh (Fig. 1).
From superposition of two meshes, distribution or mapping matrix, Φ is computed where Φij
is defined as the area of overlapping between the deformed element j and initial element i,
divided by the area of deformed element j (Kruijt et al., 2001a).
∫ dΩ
Φ ij =
Ω j | t = t o + ∆t ∩ Ω i | t = t o
∫ dΩ
(1)
Ω j | t = to
Figure1. (a) Initial mapping mesh at, t=to, (b) deformed mapping mesh at, t=to+∆t,(c) Superposition
of two meshes for evaluation of Φij for flow in a cross-section of co-cylindrical geometry with inner
rotating cylinder
The quantity which is mapped in interval ∆ti should not influence flow field or its influence
should be negligible, in other word, flow field in the interval ∆ti must be steady state or quasi
steady state. In other hand, mapping method assumes a uniform distribution of mapping
quantity over every initial sub-domain. This causes small error in the second step and
propagation of the error in the subsequent mapping steps. In order to reduce this numerical
diffusion error the number of mapping steps should be reduced or in the other word the ∆ti
should be chosen as large as possible. Therefore an optimal time interval, ∆ti for evaluation
of mapping matrix must be chosen (Kruijt et al., 2001a).
Coarse-grain concentration (C) is regarded as the mapping quantity. If column vectors Cn
and Cn+1 represent concentration distribution of sub-domains in steps n and n-1, respectively,
and Φij represents the mapping matrix in interval ∆t, then:
(2)
C n = Φ ∆t ⋅ C n − 1
Figure 2 shows a cross section perpendicular to the rotor axes of an internal mixer (with two
lobes fully intermeshed rotors).
Figure 2. (a) Geometry of internal mixer rotors, (b) cross section of rotors
If we consider internal mixer as a time periodic mixing device, the rotors will return to their
initial positions after every half rotation of 180°. They also show a mirror symmetry at every
quarter rotation of 90° (figure 3).
Figure3. Time periodicity of internal mixer rotors with two lobes forθ rotation angle of rotors
Therefore, mapping matrix should be calculated for each 90º rotation and tracking of the
mapping mesh is possible at 90° rotations. Consequently the state of mixing is available only
at 90° orientation intervals. In this case application of mapping method is restricted to
mixture of fluids with similar rheological properties, since between two 90º rotations,
concentration distribution of two fluids enormously varies and for two fluids with different
rheological properties velocity field will change during calculation of mapping matrix.
For an accurate analysis of mixing and predicting the changes of morphology of a
multiphase flow, ∆ti should be chosen such that the quasi steady state condition of the flow
field is not relaxed. Hence, rotation angle of rotors for calculation of mapping matrix should
be rather small, far less than 90º rotation. As such mapping method could not be used for
complicated geometries such as internal mixers or twin screw extruders for conditions where
no geometrical symmetry could be established. To solve this problem the Imaginary Domain
Method (IDM) is introduced, where the problems related to the un-symmetric nature of
internal mixers or twin screw extruders are solved.
In IDM the cross section of a mixer is discretized into two parts (figure 4). The first part is
the fixed geometry region where the cross section does not change in all orientations of
rotors (rotor shaft and casing). The second part is the imaginary domain where their
geometries for each orientation of rotors are changed (lobes of screws or rotors). Imaginary
domains are considered as a inelastic fluid with high viscosity (about 10 6 − 107 order of real
fluid viscosity). In effect the overall mapping domain contains both the imaginary and the
real fluids and concentration distribution of the imaginary fluid in each step of cross section
represents orientation of flights.
Figure 4. (a) Profile of an internal mixer screws is discretized into two parts, (b) fixed geometry and
(c) imaginary domain
Hence, mapping mesh involves both real Ωr and imaginary fluids Ωi regions (figure 5). All
cells in both regions simultaneously are tracked at ∆ti time intervals equivalent to ∆θ
rotation angle intervals.
Figure 5. Mapping discretization of computational domains (Ωi ∪ Ωr) with triangular cells
3. Implementation techniques
To show efficiency of IDM in mapping analysis, we choose two dimensional incompressible,
isothermal, Stokes flow of a two phase Newtonian fluids with different viscosities in a corotating internal mixer where the rotors have infinitesimal thickness ∆z and a cross section
shown in figure 2-b. Computational domain Ω was subdivided into two general regions, the
first region Ωr, consists of two real fluids where their flow describes mixing state and the
other region Ωi, consists of one imaginary fluid that its flow describes shape of screws
flights in different orientation. Hence, the moving boundary is the root of screws (Γ2). The
spatially discretized computational domain consists of a flow domain (Ωi ∪ Ωr) with
stationary and moving boundaries Γ1 and Γ2 respectively. The boundary conditions on Γ1
and Γ2 are incorporated in finite element analysis as shown in figure 2.b, as:
on Γ1
. ux = 0 , u y = 0
u x = −2π Rr N sin ω p , u y = 2π Rr N cos ω p where: ω p = tan −1
x p = x p − Lc
yp
xp
on Γ2
(3)
If x p > Rr
Rr, N, ωp, (xp,yp) and Lc are respectively radius of screws(rotors) root, angular velocity of
rotors, counter-clockwise angle of an arbitrary point p on the rotor roots (boundary Γ2) with
x-axis, the coordinate of any point p on the rotor roots and centerline distance of rotors.
Galerkin finite element method (Reddy, 1993; Nassehi, 2002) is used to solve the
conservation equations. The finite element mesh with second order elements is employed to
solve both the imaginary and the real domains (Ωi ∪ Ωr) with boundaries Γ1 and Γ2 (figure
6).
Figure 6. Finite element discretization of computational domain with stationary boundary Γ1 on
barrel surface and moving boundary Γ2 on the root of screws
C1e |θ , C 2e |θ and (1 − C1e |θ −C 2e |θ ) denote coarse grain concentrations of fluid 1, imaginary
domain and fluid 2, respectively. For every arbitrary ∆θ rotation these concentrations are
evaluated using mapping method in the eth finite element mesh at the screw orientation θ.
Mixed viscosity at the same elements are evaluated using mixture rule as:
µ e |θ = C1e |θ µ1 + C 2e |θ µ 2 + (1 − C1e |θ −C 2e |θ ) µ 3
(4)
Where µ1 , µ 2 and µ 3 are viscosities of fluid 1, imaginary fluid (about 10 6 − 10 7 of real fluid
viscosity) and fluid 2 respectively.
Since in macro-mixing the effects of interfacial tensions are less than viscose forces
interfacial tensions are neglected (DeRoussel et al., 2001). For a viscous fluid, flow during a
∆θ rotation angle (figure7) could be described through employing Stokes and continuity
equations as:
− ∇p + µ ∇ 2 u = 0
in Ωr ∪ Ωi
(5)
∇.u = 0
Where u , p and µ are velocity, pressure and mixed viscosity of flow, respectively.
It is assumed that during a ∆θ rotation angle the concentration distribution of fluids are fixed;
therefore the quasi steady–state condition is assumed. However, from one ∆θ rotation to the
next, mapping matrix Φ is computed and velocity field is recalculated according to new
concentration distribution:
C1 |θ + ∆θ = Φ.C1 |θ
C 2 |θ + ∆θ = Φ.C 2 |θ
(6)
C3 |θ + ∆θ = 1 − C1 |θ + ∆θ −C 2 |θ + ∆θ
C1 |θ , C 2 |θ and (1 − C1 |θ −C 2 |θ ) denote coarse grain concentrations of fluid 1, imaginary
domain and fluid 2, respectively in the mapping mesh at the screw orientation θ.
Figure 7. Mapping matrix is computed for each ∆θ rotation of screws
It is necessary to assume a concentration distribution of real fluids at θ = 0° as an initial
condition.
Generally there are two meshes; one is finite element mesh and another is mapping mesh.
Concentration values in the elements of finite element mesh is interpolated with using the
element shape functions, as:
C1e |θ = ∑ j =1 C1i j |θ ⋅ψ ej
n
C 2e |θ = ∑ j =1 C 2i j |θ ⋅ψ ej
n
(7)
C3e |θ = 1 − C1e |θ −C 2e |θ
At any θ rotation angle position, C1i j |θ and C 2i j |θ represent fluid 1 and imaginary fluid
concentrations, of the jth node of eth element of finite element mesh within the ith cell of
mapping mesh. C1e |θ , C 2e |θ and C 3e |θ are interpolated concentration distributions of fluid 1,
imaginary fluid and fluid 2 for the eth element of finite element mesh, respectively. ψ ej is the
shape function of the element e which has n nodes.
The nodes of subdomains that are partly outside or on the boundary of the flow domain are
translated slightly into the flow. The reason to do so is that the boundaries of the flow
domain are first approximation of the actual curved boundaries, and it not guaranteed that the
points are actually located inside the discretized flow domain.
4. Accuracy
If each cell of mapping mesh contains only imaginary fluid (each component of matrixC2
equals 1 or 0), then result of a single mapping during a ∆θ rotation angle is exact
(Galaktionov et al., 2002). On the other hand if coarse-grained concentration of imaginary
fluid in the cell of mapping mesh is less than unity (0<C2<1) because of immiscible nature of
imaginary fluid with other real fluids, the ideal solution may be that imaginary fluid goes to
one recipient cell and the other real fluids go to other. However on the second and
subsequent mapping steps for progressive rotation angles θ+∆θ, θ+2∆θ… small errors
appear in the boundaries of screw lobes. Because the mapping method only knows the
average concentration in the cells, so it transfers an average concentration of imaginary fluid
mixed with other real fluids to the recipient cells. This introduces numerical diffusion of
imaginary domains to the regions of real fluids thus boundaries of imaginary domains after
further mapping steps take a shadow effect (figure8). Since in these shadow regions,
elements of finite element mesh contain high viscosity imaginary fluid (order of 106-107of
real fluid viscosity), numerical error during the calculation of velocity field using finite
element method is observed. In the regions between flights tips and barrel surface these
errors are more distinct.
Figure 8.Due to the numerical diffusion of imaginary domain into the regions of real fluid, a shadow
effect is generated on the boundaries of the imaginary domain in the subsequent mapping steps
Refining of mapping and finite element mesh for each ∆θ rotation is done for reducing these
errors. For the refinement of the elements the modification of the Zienkiewicz non-standard
refinement method (Zienkiewicz, 1977) proposed by Fortin and Tanguy (Fortin & Tanguy,
1984; Bertrand et al., 2003) is used. To employ this refinement technique, at first N control
points on the flights surfaces at θ=0° (boundary of imaginary domain at θ=0°) is generated,
and then, their position is tracked since the shape of flights for each rotation angle θ, is
known. (figure 9-a). Using these control points the initial triangular elements is decomposed
into four triangular sub-elements as shown in figure 9-b. This decomposition is continued till
an accurate shape of the flights is obtained.
Figure 9. (a) boundary of imaginary domains, (b) non-standard decomposition of elements
After mesh refining, the concentration of metal in the new elements which are inside the
imaginary domains Ωi is equal unity. The metal coarse-grained concentration of other new
elements which are inside the real domains Ωr become zero therefore the concentration of
real fluids in these new elements where are in the real domains, are equal:
Ω
C1e1 = e .C1e
Ω e1
(8)
C 2e = 1 − C1e1
Where, C1e1 and C 2e1 are coarse-grained concentrations of real fluids 1 and 2 in the new
element e1 respectively which appears from decomposition of base element e with coarsegrained concentration C1e .
With mesh refining, coarse-grained concentration of metal (imaginary domains) in all
elements with good approximation is equal zero or unity. After mapping mesh refinement
new mapping mesh is tracked and mapping matrix is recalculated. Refinement of mapping
mesh increases number of cells which consumes large memory and numerical calculation
time. Therefore after mesh refinement for one revolution of rotors the decomposed mapping
mesh returns to its initial state and coarse grained concentration of the fluid k in the initial
mesh cell e which has been decomposed is equal:
n Ω
ej
(9)
C ke = ∑
⋅ C kej
j =1 Ω e
Figure10. Solution procedure from θ = 0°to θ = θd
Where, C kej are coarse-grained concentrations of fluid k in the ejth cell with area Ωej which has
been obtained from jth decomposition of eth cell. n is total number of decomposed elements
of eth element. Initial mesh for new revolution is tracked and is refined for mapping steps
similarly. Figure 10 represents the flow chart of above mesh refinement procedure.
The mapping method is based on tracking which directly depends on the velocity field. In
order to reduce numerically errors in the elements that the metal concentration inside them
are between zero and unity (0 < C 2e < 1) and obtain an accurate velocity field, finite element
mesh is decomposed using the standard decomposition method (Bertrand et al., 2003) near
the boundaries of the imaginary domains.. This method to determine flow field is similar to
fictitious domain method (Glowinski et al., 1994; Bertrand et al., 1997).
Finally, the accuracy of the tracking of subdomains is important. We used Rung-Kutta
Merson method with dynamic step size (Gerald & Wheatley, 1999). The area conservation
error εA, is defined as:
| A − Atracked |
εA = o
(10)
Ao
Where A۪ is the area of flow domains (approximately area of initial mesh Ωr ∪ Ωi) and
Atracked is the area of tracked subdomains. In order to reduce area conservation errors,
velocity field was computed using second order elements and the sub-domains that fall
entirely outside the flow domain (barrel) or inside one of the two rotor roots are ignored for
the computations.
5. Validation and numerical results
A number of tests was performed to validate the technique and to assess its robustness and
precision. We choose an internal mixer as a test case study to validate the efficiency of
presented numerical techniques. There are different types of rotor elements for internal
mixers but in this article an intermeshing rotor, which is studied in many pervious works, is
considered. In figure 11 the cross section geometry of intermeshing rotor with infinitesimal
thickness ∆z has been showed.
Geometrical parameters of
rotors
Number of tips
Rotor diameter (D)
Centerline distance of
rotors (Lc)
Helix angle (ψ)
Intermeshing angle (αi)
Clearance distance (δ)
2
4cm
3.34cm
10º
33.31º
0.05cm
Table1. Rotor geometry parameter
Figure11. Profile of intermeshing rotor internal mixer
By using Boy’s kinematics principles the angle of intermeshing αi is related to the nip angle
by (Booy, 1978):
αi =
π
4
−
αt
2
(11)
αt is tip angle. With considering clearance between rotor and chamber, if circumferential
angle θ starts at the beginning of the flight flank the channel depth H(θ) as a function of
angle θ can be written as:
( D + 2δ )
1
⎛
⎞
H (θ ) =
(1 + cos θ ) − ⎜ L2c − ( D + 2δ ) 2 sin 2 θ ⎟
(12)
2
4
⎝
⎠
D, δ and Lc respectively are rotor diameter, clearance distance and centerline distance of
rotors. Values of rotor geometry parameters were listed in table 1.
It assumed that two different fluids which are shown with yellow ( µ1 = 8000 Pa.s) and blue
( µ 2 = 7000 pa.s) colors are fed into the chamber at orientation θ =0° (as a t=0s), one flow is
fed to right part of the chamber cross-section and the other through the left part. The
viscosity of imaginary fluid was considered about 106 order of yellow fluid viscosity. The
mapping matrix Φ, using the geometry and boundary conditions as described in the pervious
sections, is computed for ∆θ=7.2° rotation for angular rotor velocity 0.1 rps. To evaluate the
influence of mesh-size on the IDM quality, four mapping and finite element mesh sizes have
been tested (table2). Figure12 shows the concentration distribution in some orientations of
rotors (different θ) for the type 4 mesh, described in table 2.
Figure 12A. Concentration distribution at various cross-sections orientations
Table2. Different orders of mapping and finite element meshes
Mesh
Finite element mesh
Finite element mesh
Mapping mesh
Mapping mesh
ds (*)
Number of nodes
Number of triangles
Number of nodes
Number of triangles
1
1965
900
1979
3500
0.11
2
5187
2480
3979
7500
0.09
3
9523
4620
6479
12500
0.05
4
4581
7120
7729
5000
0.02
*Mean distortion of triangle cells in imaginary domain
Figure 12B. Concentration distribution at various cross-sections orientations
Mixing begins from the interface of two phases in the intermeshing region. Green contrast
shows regions that mixing has occurred. In the intermeshing region because of change in the
velocity direction of flow, elongational flow dominates (figure 13). This also causes
elongational deformation of mapping meshes (figure 14).
Figure13. Velocity field in the intermeshing
region
Figure14. Mapping mesh deformation in the
intermeshing region under elongational flow
field
With reducing channel depth the effect of shear flow increases, hence, in the conveying
sections shear flow is dominant. This shear flow reduces the thickness of the phase which
was transferred from one side to another under the elongational flow at the intermeshing
region (figure 15). With increasing shear rate and history of strain the interfacial of two
phases is increased (figure 16).
Figure15. Transport of material in the intermeshing
region
Figure16. Shear mixing in the conveying
section
Cells of mapping mesh inside the imaginary domain should not deform under flow field.
This could be verified employing surface deformation ds of triangles associated with a nodal
point (Le Guer & Schall, 2004):
d
i
1 n Sj −Sj
ds = ∑
(13)
n j =1 S ij
Where, n is the number of triangles associated with the nodal point, Sij being the initial
triangle area and Sdj being the deformed triangle area. For one revolution of screw the
average stretching over triangles which are in the imaginary domains has been calculated for
different mesh sizes (table2).
The intensity of segregation is a measure for the deviation of the local concentration Ci of
domain Ωi from the ideal concentration C (Kruijt et al., 2001a).
1 n (C i − C ) 2
I = ∑i =1
Ωi
(14)
Ω
C (1 − C )
Where Ω is total area of mixing domain and Ωi is local area of mapping elements. Intensity
of segregation decreases with rotation of rotors (figure 17).
Figure 17. Intensity of segregation at various orientation of rotors
6. Conclusion and Discussion
Objective of this article was to introduce a novel technique based on the mapping method
and mesh refinement procedures for the simulation of a macro-scale, laminar mixing of
multiphase flows in complex geometries involving moving parts. Traditional application of
mapping method could be applied to only the repeated orientations of rotors, hence, it could
not give information on the mixing state for in between rotor orientations. An Imaginary
Domain Method (IDM) is introduced as a new adaptation to the traditional mapping method
to ease this limitation. The method was applied for simulation of mixing in an intermeshing
rotor internal mixer. The state of mixing was obtained for all orientation of rotors. Hence, it
was shown that IDM is a suitable method for applying mapping method free of geometrical
periodicity restriction. The extension of this method to the 3D case for twin screw extruders
is in progress and will be the topic of a forthcoming article.
Employing IDM we were also able to simulate the mixing of a mixture of two fluids with
different viscosities. To treat this problem we updated the velocity field between mapping
steps using quasi-steady state condition for small angle of rotation ∆θ (or ∆t at which the
mapping matrix is calculated). Increasing the number of mapping steps increases numerical
diffusion errors. To solve this problem, a traditional mesh refinement procedure is adapted to
the mapping method.
7. Acknowledgements
The financial support for this work was provided by the Research & Development
department of Iran Petrochemical Co., grant number 83113098.
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