Numerical simulations of fluid mixing andoptimization of mixers

O.S. Galaktionov, P.G.M. Kruijt, P.D. Anderson, G.W.M. Peters, H.E.H. Meijer

Dutch Polymer Institute, Materials Technology, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

email: [alexei,peterk,patricka,gerrit,han]@wfw.wtb.tue.nl

phone: +31 (0)40 2474823, fax: +31 (0)40 2447355

Abstract

A computational tool for fast simulation of laminar fluid mixing, ca-pable of dealing with industrial mixer geometries is presented. Thismethod uses extensive computations to create a sparse mapping ma-trix (set of matrices) that describes fluid transport between the cellscovering the flow domain. These extensive computations necessitateparallel processing, which is easily realizable due to the huge number ofindependent steps involved. Then, mixture evolution on a (large) timestep is computed using matrix-vector multiplications. We evaluate thecapability of the technique to cope with industrial mixer geometries.This forms the basis for optimization of mixer design.

1 Introduction

Fluid mixing processes receive significant attention because of their widespreadoccurrence in nature and especially due to their importance for industrial ap-plications. Although in many cases mixing is associated with turbulent fluidmotion, mixing of viscous fluids in laminar (low Reynolds number) flows consti-tutes an important class of mixing phenomena. This type of mixing is typicalfor polymer blending, food processing etc. While significant advances in under-standing of underlying mechanisms of laminar mixing have been achieved (seee.g. [6]), numerical simulations remain computationally expensive, especially in3D, therefore, require special methods.

The basis for studying mixing is an accurate determination of the velocityfield. But unlike many other hydromechanical problems, this is only a first,although very important, step. A variety of techniques based on the trackingof deforming individual fluid volumes (see, for example, reviews in [7] and [10])can be used to study mixing. The authors recently developed an adaptive three-dimensional front tracking technique that takes stretching and curvature of theinterface into account. The method, originally reported at PVM-MPI’97 [2]strongly benefits from parallelization of the code. Its details were later elabo-rated in [1, 4]. Such direct techniques, however, remain expensive and are limitedonly to initial stages of mixing.

Most practically interesting mixing flows exhibit temporal or spatial period-icity. A recently developed mapping technique [1] exploits this repetitive nature

of the flow. It is based on the accurate tracking of fluid volumes, but uses itonly once to create a large mapping matrix that describes fluid exchange betweensmall finite size sub-domains during a large time step. This stage is computa-tionally expensive and necessitates parallel computing as well as efficient andflexible way of sparse matrix storage. After that the evolution of concentration(described by a concentration vector) is modeled with a sequence of matrix-vector multiplications. Authors demonstrated the applicability of this techniquefor simulation and optimization of mixing in 2D and 3D prototype flows. Nowwe evaluate the capability of this technique to treat complex industrial flows.

2 Basics of mapping technique

The details of the technique are described in [3]. Here we briefly outline itsprinciples. The mapping method is based on two simple ideas. First, insteadof a continuous field of concentration (or an other volume-averageable materialquantity) the mixture is described by locally averaged concentrations in the cellsof a properly chosen spatial grid. Second, the flow is presented as a sequence ofbasic discrete steps, for which a mapping from an initial grid to a final grid iscomputed. A flow domain Ω is subdivided into n non-overlapping sub-domainsΩi with boundaries Γi (not related to the mesh on which the velocity field iscomputed). The boundaries of the sub-domains are tracked from t = t0 to t =t0+∆t using an adaptive front tracking technique [4] that refines the boundariesdepending on their local stretching and curvature. The coefficient Φij of themapping matrix Φ is defined as the fraction of the deformed sub-domain Ωj attime t = t0 + ∆t that is found in the reference (as at t = t0) sub-domain Ωi:

Φij =∫

Ωj |t=t0+∆t

⋂Ωi|t=t0

dV

/ ∫Ωj |t=t0

dV. (1)

Figure 1 shows an example of the intersection of a non-deformed (initial) sub-domain at t = t0 with a deformed subdomain ∆t later in the three-dimensionalcase. Computation of the intersection volume involves an appropriate refinementof the surface grid, describing the deformed cell and the subsequent collapse ofall parts of the surface that lie outside the sub-domain with which intersectionis being computed.

The most important property of the mapping matrix (that enables mappingcomputations with high spatial resolution) is that the matrix Φ is sparse dueto the fact that, given the small time step ∆t, the fluid from one sub-domain istransported to a limited number of resulting sub-domains.

Once the mapping matrix is computed, quantities, related to sub-domainscan be mapped. Such quantities should be additive and must not change theflow field. Since the mapping method assumes an uniform distribution of themapped quantity within each sub-domain, contributions from different donorsub-domains are averaged. This averaging on every step leads to a systematicerror, that can be called a “numerical diffusion” (since it blurs the sharp bound-aries). It sets the limits on the acceptable maximum size of sub-domains.

⇒

Fig. 1: Computation of the elements of the mapping matrix Φ: determining thevolume of the intersection of the deformed cell with initial cells.

The quantity being mapped in this work (unless stated differently) is averagedconcentration of the marker fluid. The concentration distribution is describedby a vector C, its components Ci are the locally averaged concentrations inthe sub-domains Ωi. If the initial distribution at t0 = 0 is described by theconcentration vector C0, the concentration after time ∆t can be computed asC1 = ΦC0. If the same flow is continued, the concentration after n steps shouldbe defined as Cn = ΦnC0. The idea of the mapping method was formulatedby Spencer and Wiley [9], who proposed to analyze the mixing behaviour bystudying Φn. Such an approach is not suitable for high spatial resolution, sincethe matrix Φn will not be sparse and, thus, becomes prohibitively large. Insteadof this, the concentration vector Cn is computed in a sequence

Ci+1 = ΦCi, (2)

without direct evaluation of Φn.To validate the mapping approach the results of mapping computations were

compared to the explicit front tracking results in a prototype flows in a cubiccavity, driven by the motion of two opposite walls [3] (see figure 2). The initialvolume of marker fluid (c = 1 for marker fluid and c = 0 everywhere else)was mapped for a few flow steps using the 100 × 100 × 100 discretization ofthe flow domain. Then, the boundary of the marker blob was recovered as anisoconcentration surface c = 0.5 (figure 2c) . It closely matches the shape of theexplicitly tracked blob surface (figure 2b).

3 Mapping matrix computations

Although the mappings themselves are performed rather fast, computation ofthe mapping matrix remains a computationally extensive task, since numeroussub-domain boundaries must be tracked. In earlier 2D versions of the techniqueall boundaries were tracked simultaneously (using a parallel algorithm, describedin [2]), and boundaries of individual deformed sub-domains were extracted af-terwards. For complex 3D flows such an approach turns out to be restricted

(a) (b) (c)

Fig. 2: Comparison of front tracking and mapping results in a 3D cavity flow.The flow is generated by four consecutive displacements of front and back wall inX and Z directions. a) initial test volume; b) results of adaptive front tracking;c) interface shape recovered from the mapping results.

by memory and storage capabilities. Each sub-domain is being tracked indi-vidually and the related mapping coefficients are computed. Since tracking ofeach sub-domain can be performed independently, such computations are easilyparallelized.

The mapping technique requires the storage and manipulation of very largesparse matrices (typically 106 × 106 as in [3]). We are using a special typeof sparse storage that not only enables the storage of large matrices but alsofacilitates the parallel computations. The storage is column-oriented, thus, toretrieve a particular element the search is performed along the column of thematrix. Unlike a standard approach, however, the elements of the column are notstored consecutively in the array. Instead, along with every non-zero element theindex pointing to the location of the next non-zero element of the same columnis stored. Thus, for a user the storage appears ordered. New elements are addedto the end of the list. As a result, the order in which the elements of the sparsematrix are computed and stored can be arbitrary and does not cost extra whenhandling the large sparse matrix. All the information about the sparse matrix(the values of non-zero elements and indexing) is stored in the same integer array(real values occupying the place of two integers), which facilitates the usage ofmany different sparse matrices within the same program unit.

Parallel computations of the mapping matrix are performed in a master-slave model. The master program initializes the storage for the sparse matrix,i.e. generates indexing for initially empty matrix. After that the list of the initialsub-domains, for which the mapping coefficients still have to be computed, iscreated, forming a pool of elementary tasks for slave processes. If the poolof tasks is not empty, slave processes are spawned. General parameters, likespatial resolution of the sub-domain grid, initial and final time for the sub-domain tracking and precision of the numerical integration of the equations ofmotion, are broadcasted to the slaves.

Started slave processes receive a buffer with general parameters and each of

them loads the velocity field data. Then the slave process is ready to handlerequests from the master process. These requests normally contain the index ofthe sub-domain to be tracked. Knowing this number, the mesh describing theboundary of the coresponding sub-domain is generated by the slave process. Itis tracked in the flow using an adaptive front tracking technique. Then, intersec-tions of the deformed sub-domain with undeformed sub-domains are computed.For all initial sub-domains whith which an intersection is found, the index andintersection volume are packed in a buffer, which is sent to the master. Afterthat the slave is ready to handle new requests.

Note that in this layout the master program only deals with the storageof the big sparse matrix and handles workload distribution among slaves. Slaveprocesses, in contrast, acquire the velocity field but do not store the matrix itself,only computing its coefficients on request and returning them to the master.Communications between program units is minimal, resulting in low overhead.Separating velocity field and mapping matrix into different programs helps toavoid memory problems when the memory available per process is too restricted.The possibility to store the coefficients of the sparse matrix in an arbitrary orderis essential, since the sequence in which data is returned from the slaves cannot be predicted beforehand (complexity of individual tracking tasks can varysignificantly).

The master program initially sends one task to each slave. Upon receivingdata back from any slave, new coefficients are stored in a sparse matrix and anew task is immediately sent to the same slave (rather simple kind of dynamicalload balancing). Thus, all of the slave processes are kept busy until the pool oftasks is emptied. After every N (specified by user) sub-domain data received, themaster program saves data on disk. It is also possible for a user at any momentto force the master program to save data and quit, stopping all slaves (specialmessage is broadcasted to them). Computations can be restarted afterwards.

4 Transport section of twin screw extruder

The mapping technique was recently tested in a prototype flow with simplegeometry [3]. In the current work we apply the same technique to the Stokesflow of a viscous fluid in a co-rotating twin screw extruder (TSE) [5]. Devicesof this type are widely used for melting, pumping and blending of polymers.Within this paper we consider only the transport section of an extruder (othertypes of screw sections are used to enhance mixing performance). Four conveyingelements of the TSE are shown in figure 3. Screws are tightly intermeshing andhave a self-wiping shape, scraping each others surface during rotation. Bothscrews (see figure 3) rotate counter-clockwise viewed from the bottom in thedirection of the flow.

A fictitious domain technique, implemented in the SEPRAN finite elementpackage [8] was used to compute the velocity field inside this mixing device. Anunstructured mesh, consisting of tetrahedra was used to compute the velocityfield. The helical shape of the screws allows to use only a single computed

S4

S3

S2

S1

exit ⇑

⇑ entry

(a)

Sub-domain grid and mapping matrix

property value

nx × ny × nz 184× 100× 48# sub-domains:

- total 883,200- used 291,284- % used 33 %

Full matrix size 1, 766, 400× 883, 200(≈ 1.56× 1012)

non-zero elements 13, 011, 100matrix density 8.34× 10−6

(b)

Fig. 3: a) Transport section of the twin screw extruder. Four elements areshown. b) Grid resolution and properties of the mapping matrix.

velocity field, accounting for the angle of screw rotation in time by means of anappropriate axial shift of the reference coordinate system.

Since the extruder is an open system, application of the mapping techniquehas some special features. The grid of initial sub-domains covers one mixingelement (i.e. one period of the screws). Since the net transport of fluid takesplace along the mixer, the resulting mesh, however, must cover at least 2 periodsof the screw: the same element as the initial mesh plus one downstream. We useda structured rectangular grid, tightly enclosing the barrel of extruder (some cellsinevitably are falling beyond the flow domain) as the initial sub-domain grid.The grid covering one screw element has resolution 184 × 100 across the barreland 48 cells along it. Due to the fact that general rectangular mesh is used,only 1/3 of totally 883, 200 sub-domains are actually inside the fluid and haveto be tracked. “Out of fluid” sub-domains leave the corresponding columns inthe matrix empty.

Since two-lobe screws are used, the time period of the flow corresponds to a180 rotation of the screws. However, using the symmetry of the system, every-thing can be expressed in terms of deformation produced by just 90 rotation:after that the shape of the flow domain becomes like initial one, but rotated by180. The mapping matrix actually describes the transport of fluid, containedin one element of the mixer, caused by 90 turn of the screws.

The details concerning the mapping matrix are summarized in the table infigure 3b. Computations of the mapping matrix were performed on an OriginTM-2000, using 32 processors. The total amount of CPU time required to computethe mapping matrix (not measured exactly) is estimated as nearly one year. Onlyparallelization made it affordable. Note that most of the CPU time is spent ontracking the surfaces of deforming boxes in the flow. The performance can besignificantly improved if a better approximation (higher order) for the velocity

field is used. Some other ways to speed-up mapping matrix computations willbe discussed later.

Note that, although determining the mapping matrix is computationally ex-pensive and necessitates the use of a parallel algorithm, the mapping itself israther fast: a single mapping is performed within 3 seconds on a single CPU.

5 Results: concentration and residence time distributions

The mapping matrix describes the transport of fluid from one mixing element(one spatial period of the screws) to the same and the next element. When themapping is performed for the long transport section of extruder, the spatialperiodicity is used: the same mapping matrix operates on every mixer element.Although it is necessary to store fluid concentration in the whole extruder beingsimulated, the huge matrix that would describe complete transformation is neverbuilt. Since the mapping involves only the source element and the next one, thealgorithm starts applying mapping from the last (exit) element and proceeds tothe first (entry) one. This allows to use the same storage for distributions beforeand after a mapping step. In this paper we show only first four elements of TSE.

Treatment of the entry element is somewhat different: it not only gets thematerial from itself but also receives a prescribed input (what would be otherwisetransported from the previous element). To prescribe the input the difference ismade between the material entering through the left and right half of the barrelin the entry cross-section. There are three types of fluid in the mixer in oursimulation: fluid 1 entering on the left, fluid 2 entering on the right and matrixfluid 3 filling the whole extruder at t = 0. The mapping algorithm can handleany number of components simultaneously. Presenting black-and-white imagesin the text, we show fluid 1, entering on the left, black (see figure 3a), the restremaining white.

Figure 4 shows how the material propagates along the extruder. The con-centration of marker fluid is shown in the cross-section made through the axesof both screws. It takes approximately 3.5 rotations before the first markerfluid reaches the exit, but the concentration pattern will be close to stable afterapproximately 6 rotations.

The next figure shows the (nearly stable) concentration patterns achievedin the extruder after 6 turns. The slices are made perpendicular to the axis at3/4 height of each element (their location is shown in figure 3a). The imagesin figure 5 show how a layered structure is developing along the mixer, andwhere the material folds and thicker layers are likely to be found. The exactdistribution, certainly, depends on the inflow conditions.

An even more interesting possibility offered by mapping method is its abilityto deal with residence time as well. The time that the material spends inside thedevice is important for evaluation of the mixer design. Stagnation zones shouldbe avoided. Moreover, residence time should be minimal not only to increasethroughput but also to prevent thermal degradation of some molten polymers.From another point of view the material should reside in a mixer long enough to

0.5 turns 1 turn 1.5 turns 2 turns

2.5 turns 3 turns 3.5 turns 4 turns

Fig. 4: Axial concentration slices in TSE.

section S1 section S2

section S3 section S4

Fig. 5: Concentration slices in TSE after 6 screw rotations. See figure 3a for thelocation of the sections S1 – S4.

achieve a desired mixture homogeneity. The redidence time distribution shouldbe narrow, but not too narrow in order to smooth out fluctuations in the feed.

Computation of residence time involves a slight modification of mapping.The residence time is not just carried with material as concentration but is in-cremented on every time step. The material entering the mixer by definition has

0

1

2

3

4

5

6

7

axial cross-section

section S1

section S4

section S3

section S2

Fig. 6: Residence time (measured in screw rotations) in the transport part ofa twin screw extruder. The drawn white and black lines correspond with adimensionless residence time of 2 and 4 respectively.

zero residence time. Figure 6 shows the residence time distribution in differentcross-sections of the extruder. To ensure a stable (self-repeating after each half-turn) distribution, 50 turns of the screws were performed. This figure reveals thezones where the oldest material is likely to be found. Such zones occur not onlynear the walls, which is expected, but also in the bulk of the flow. Interesting isthat material with a high residence time is more likely to be found on the screwsrather then on the barrel. This can be explained by the fact that the barrel isbeing wiped by screws more efficiently: screws have contact lines with the barrelbut only localized contacts with each other.

6 Conclusions and future work.

The mapping technique for numerical simulations of laminar mixing wasdeveloped. Computation of the so-called mapping matrix in realistic mixer ge-ometries requires large amount of CPU time and necessitate the use of parallelalgorithms. An efficient way of storage of large sparse matrices is required. Thetechnique being used here allows an arbitrary sequence of filling non-zero matrixcoefficients and facilitates parallel computations with dynamical workload distri-

bution (done using message passing). Physically interesting results are obtainedafterwards at low computational cost using a single CPU.

The next step of this work will be an optimization of the mixer design. Forthis, different type of the screw elements (transport, kneading, pressure-buildingetc.) should be considered and a set of corresponding mapping matrices needs tobe computed. Combining them will make it possible to find better sequence ofscrew elements to achieve optimal mixing performance. Numerical optimizationis likely to be more cost effective then extensive experiments with different screwdesigns. To reduce the computational cost, better (high-order) approximationfor the velocity field should be used. Rectangular sub-domain grid used in thiswork should be replaced by the grid specially tailored for twin screw extrudergeometries, having less “dead” sub-domains. After mapping matrices are finallyformed, they can be converted to more compact, physically ordered storage.

From a computational point of view, giving the master program also theslave functions of tracking sub-domains can be beneficial for efficient use ofprocessors (master program has idle states). This, however, results in largermemory requirements. Authors were using PVM for message passing but areplanning to switch to MPI due to the latter’s wide acceptance and support.

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