INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 11 (2003) 771–790 PII: S0965-0393(03)64521-3

Fluid flow aspects of twin-screw extruder process:numerical simulations of TSE rheomixing

H Tang1,2,3, L C Wrobel1 and Z Fan1,2

1 Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex,UB8 3PH, UK2 Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge,Middlesex, UB8 3PH, UK

E-mail: Hao.Tang@brunel.ac.uk

Received 10 March 2003, in final form 5 June 2003Published 16 July 2003Online at stacks.iop.org/MSMSE/11/771

AbstractThis paper presents a brief review of the twin-screw extruder (TSE) process,analyses the fluid flow aspects of a new application of TSE in rheomixingcasting, and simulates fundamental micro-mechanisms of the rheologicalbehaviour of metallurgical alloys. The TSE is widely used as a mixing toolin the polymer industry. The numerical simulation of the TSE process iscomplex and there are only few simulation packages to provide an insightinto TSE processing for viscoelastic flows. This study involves the systematicanalysis of flow features of immiscible alloys in the TSE rheomixing process, theestablishment of a simplified flow field, and simulation of the microstructure ofimmiscible alloy drops by the piecewise linear interface construction/volume-of-fluid method. It is noted that the microstructure of immiscible alloysis strongly influenced by the viscosity of the system, which dominates theinteraction between droplets. The numerical results show good qualitativeagreement with experimental results, and are useful for further optimizationdesign of prototypical rheomixing processes.

1. Introduction

The twin-screw mixing process for semi-solid immiscible alloy casting has been developedprototypically [1]. The twin-screw extruder (TSE) process has been used mainly in the polymerindustry as a dispersive mixing process before mould or extrusion die, and also for foodprocessing. After flowing around the rotating twin-screw, melt materials enter the final mouldor die whose function is to transform the circular flow section at the end of the screw intoa complex shape. Immiscible alloys are advanced materials used for bearing applicationsin the automotive industry. The application of these alloys, however, has been limited,

3 Author to whom correspondence should be addressed.

0965-0393/03/050771+20$30.00 © 2003 IOP Publishing Ltd Printed in the UK 771

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due to metallurgical problems related to liquid phase separation during casting. Because ofmixing functions of TSE [2], the rheomixing casting by TSE provides excellent metallurgicalmicrostructure characteristics in semi-solid and immiscible alloy casting.

Evidences show that the solidified microstructure of cast immiscible alloys stronglydepends on the hydrodynamic behaviour of the liquids during cooling [3]. Therefore, there isan increasing need to be able to control these complex metallurgical processes and hence, animproved capability to numerically simulate and study these processes. Numerical simulationsare, in principle, ideally suited to study these complex immiscible interfacial flows and toprovide an insight into the process that is difficult to obtain by experimentation. However,because of the limitations of numerical approaches, these flows present a challenge forsimulation due to the presence of material interfaces of arbitrarily complex topology.

2. Review of TSE process

The TSE is widely used in industry, especially in polymer processing. There are a varietyof TSE designs employed throughout the polymer industry, with each type having distinctoperating principles and applications in processing. Designs can be generally categorized asco-rotating and counter-rotating. Here, discussions will be focused on fully intermeshing,lengthwise open, screw crosswise closed, and disc crosswise open, self-wiping, two flights,co-rotating TSE as this is one of the most successful types of TSE used for polymer processing,and is also used in the rheoroute casting process.

2.1. Construction features of TSE

A description of TSE classifications and configurations is given in [4]. The basic elements of theTSE are the screws and mixing blocks, such as the kneading discs. The elements of the screw,with suitable numbers of mixing blocks, are assembled together, in so-called modular form,inside the barrel in order to reach the best mixing efficiency within the duration of the process.

The self-wiping, co-rotating screws form v-shaped wedge areas that have four to five timesmore volume. This enables material to be transferred from one screw to the other, resulting ina renewal of material layers and surfaces. The net result is a higher degree of mixing.

Self-cleaning action is important for metallic alloys mixing to prevent material fromadhering to the screw root. This is achieved by self-wiping motions. In co-rotating screws,one crest edge wipes the flanks of the other screw with a tangentially oriented, constant relativevelocity. There is a higher relative velocity in this arrangement, and hence there is a sufficientlyhigh shear velocity available to wipe the boundary layers, thus a more efficient and uniformself-cleaning action is achieved. Meanwhile, since no calender effect exists between the crestand flank of the screws as in counter-rotating screws, there is less wear. Therefore, co-rotatingscrews can be operated at much higher speeds with greater throughputs. The detailed structureand size of the design for the crest and flank of the screw could be different. Mixing blockelements can be placed in different intervals and numbers with the screws, these being entirelydependent on the material and process in order to achieve the best mixing efficiency [5]. Theconstruction of TSE for rheocasting process is depicted in [6–9].

2.2. Applications of TSE

Historically, TSE was first employed as a mixer in the early 1920s for rubber polymerizationand was described in a patent in the 1930s [4]. The excellent dispersive mixing of highlyviscous materials, continuous processing, plug flow behaviour, and multistaging capability are

Fluid flow aspects of TSE process 773

the main advantages of TSE as a blender and also as a chemical reactor in polymer, food andpharmaceutical industries. The developments and applications of intermeshing co-rotatingTSE (ICoTSE) can be traced through hundreds of patent records. The patent technology onICoTSE dates back to the beginning of the 20th century. Therefore, industrial applications ofTSE were fully dependent on patent protections and developed by several groups independently[4]. The main application fields were polymer, food and pharmaceutical industries until 1978when ICoTSE was applied in crystallization processes where crystals are broken up throughshearing caused by the relative motion of the screw and barrel [10, 11]. With the invention ofthe semi-solid metal (SSM) process, where a metal liquid is stirred during solidification ratherthan using traditional static solidification process in order to obtain spherical microstructuralcrystals, ICoTSE was then used in SSM process to produce semi-solid slurry. The first appli-cation of single screw extruder (SSE) for metal thixomoulding was described in 1992 [12] andfor rheomoulding in 1994 [13]. In the early 1980s, modular ICoTSE was developed at BrunelUniversity (BU) [14, 15]. Recently, novel applications of intermeshing co-rotating self-wipingTSE (ICoSwTSE) have been developed successfully for SSM and immiscible alloys castingprocess, the so-called TSE rheorouting casting process by Fan and his co-workers [1,16], aswell as patent applications of specially designed TSE for rheorouting casting process [6–9].

2.3. Experimental approach

The characteristics of an ICoTSE are extremely complex, and experimental studies arerelatively limited. The investigations are mainly focused on flow visualization, pumpingcharacteristics, residence time distributions (RTDs), mixing, heat transfer, devolatilization,and reactive extrusion.

The earliest reported studies of mechanisms of fluid motions in an ICoTSE are by [17],who indicated that the fluid moved in figure ‘8’ motions around the periphery of the screw[4], and in periodic forward motions in the kneading disc section. A Newtonian fluid in atransparent barrel was studied by [18] in 1978 on an ICoTSE, and figure ‘8’ motions found by[17] were reported. Later studies of the flow pattern of an ICoTSE reported on the pressureand drag flows in the kneading disc section [19]. The first investigations of fluid motions formodular screws were published by [20] in 1973 with a transparent barrel, in which alternateregions of filled and starved channels are observed. The pumping characteristics of a modularscrew were studied experimentally due to their importance for continuous extrusion processes[20–24]. Recently, velocity distributions and volume flow of an ICoSwTSE were studied byusing a laser Doppler anemometry (LDA) technique [25, 26]. The first studies of RTD inICoTSE were reported in 1966 [27]. The analysis of the RTD leads to a general and usefuldescription of the hydrodynamic pattern of a TSE, and is also useful for explaining the effectsof process parameters. Tracers used were blue dyes [28–31], radioactive tracer techniques[14, 29, 32], chemical elements Na and K [33], KCl [34] or NaCl [35], a luminescent tracer[36], an optical tracer [37], and iron powder [38]. The most recent contributions of RTD werepresented by [28, 34, 35, 38]. It is known that [34] uses potassium chloride as a tracer which ismeasured by a conductivity meter, while [38] uses iron powder as tracer measured by a magnetictechnique. Mixing effects have been investigated through studies of morphology developmentand modular screw configurations [5, 39–41]. Parallel discs were used by [41] as a simplifiedmodel to study micro-mechanisms of TSE in shear-induced flow field, and reveal that theshear rate is possibly the most important factor for the final morphology. The mechanicalenergy distribution and temperature profiles were reported by [42, 43], who compared resultsof ICoTSE with the stirred tank reactor (STR). Table 1 summarizes pioneering works in theopen literature.

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2.4. Simulation approach

TSE is a three-dimensional unsteady process, therefore, simulations of ICoTSE will be achallenge for many years to come. A few commercially available packages are listed intable 2.

Flow in screws is generally simulated by flattening out screws and performing bothanalysis and calculations in Cartesian coordinates [44]. The procedure has been adopted fromthe earliest efforts on SSE [70]. One-dimensional TSE simulators are based on simplifiedtheoretical modelling for forward and backward pumping flows, flow in kneading disc section,composite modular, and mixing. The results are given along the flow direction, althoughthese do not permit the observation of detailed flow patterns which are of great importance

Table 1. Historical perspective of ICoTSE/ICoSwTSE [4].

First events Date Objects Inventors/developers

Patent application 1901 ICoTSE A WunscheBritish patent application 1916 ICoSwTSE R W EastonPatent application 1932 Paddle disc W K NelsonDetailed description in patent 1935 ICoSwTSE F F PeaseCommercial machine 1939 ICoTSE R Colombo and LMPCommercial modular TSE machine 1950s ICoTSE Werner and PfleidererPatent for TSE moulding machine 1957 TSE injection moulding K H Baigent,Flow visualization 1964 Figure ‘8’ motion R ErdmengerMaths basis for geometry 1965 ICoSwTSE H G ZimmermannModelling flow basis for screw 1973 ICoTSE O Armstroff and H D ZettlerModelling flow basis for disc 1976 Kneading disc H WernerCrystallization process 1978 ICoTSE C Rathjen and M UlrichMaths formulation for self-wiping 1980 Self-wiping profiles M L BooyModular TSE developed at BU 1983 ICoTSE P R Hornsby et alPatents & TSE alloys casting process 1999 SSM, immiscible alloys Z Fan et al [6, 7]

Table 2. Simulation software for TSE process.

Package Developer/vendor Features References

General simulation packageBEMflow Madison Group, USA BEM Three-dimensional TSE, polymer [57]POLYFLOW Fluent Inc., Polyflow SA, FEM Three-dimensional TSE, polymer [58]

Belgium

TSE simulatorAkro-Co-Twin University of Akron, Theory One-dimensional, polymer [59]

Screw Akron, OH, USALudovic Cemef, INRA, SCC, France Theory One-dimensional, polymer [60]Screwflow-multi Plamedia, Japan Three-dimensional, polymer [61]TXS PPI, New Jersey, USA Theory Co-rotating TSE [62]Sigma University of Paderborn, Theory One-dimensional, [63]

Germany co-rotating TSETSEAP PPI, New Jersey, USA Theory Co-rotating TSE [62]

Cemef: Centre For Material Forming.INRA: French National Agronomic Research Agency.SCC: Science & Computers Consultants.PPI: Polymer Process Institute.

Fluid flow aspects of TSE process 775

Figure 1. Schematic illustrations of flow pattern within co-rotating TSE [67].

to the mixing analysis [45, 46]. Two-dimensional models [47, 48] provide less restrictivesimplifications which can offer a somewhat more precise local information of the flowbehaviour in the case of TSE analysis. A more sophisticated numerical approach to the complexgeometry of TSE is taken by using the finite element method (FEM) [49]. To account for thethree-dimensional time-dependent motion of the screws rotating about their axes, [50–52]have simulated a sequence of several instantaneous positions. Recently, computational fluiddynamics (CFD) techniques have been employed for numerical simulations of TSE polymerprocessing by using the FEM [53], flow analysis network (FAN) [54] and the boundary elementmethod (BEM) [55] in three-dimensional laminar viscoelastic flow. With CFD techniques, itis possible to analyse with deeper insight a wide range of TSE designs for optimizing theprocesses [56]. One of main simplifying assumptions is that the TSE is fully filled although alarge fraction of the TSE operates in starvation mode.

2.5. Flow features of TSE process

The flow features of TSE have been investigated since the 1960s. As mentioned in section 2.3,flow macro-mechanisms are given and classified in a series of papers [4]. Generally, the right-hand side screw transports materials, the left-hand side screw controls the pressure field, whilethe kneading disc melts and mixes materials. The flow regime is envisaged as being composedof three distinct types of flow: drag flow, pressure flow, and net flow [4, 64–66].

Since [17] described that the fluid moves in figure ‘8’ motions around the periphery of thescrew and in periodic forward motions in the kneading disc section, flow mechanisms have beeninvestigated further by a number of researchers, and material transport patterns are illustrated infigure 1 [4, 67–69]. The flow characteristics of modular TSE were first described by Armstroffand Zettle in 1970, and subsequently by Werner in 1976, White and co-workers in 1987, 1989,1990 and others [4]. They observed that modular co-rotating TSE under starve-fed conditionsprocess sequential fully filled and starved regions. This characteristic was associated with theinteraction of the pumping capabilities of the different elements along the screw axis consistentwith the die pressure [70]. Current experimental and numerical studies have focused on themacro-scale motion, but only few authors described the micro-mechanisms of shearing flow byTSE process in experimental [41] and numerical approaches [71]. These authors reported thatthe fundamental mechanisms of shearing-induced flow in the TSE process are influenced byshearing forces and viscosity ratio. However, flow mechanism studies by those authors are forlaminar viscoelastic flows of highly viscous materials in polymer processing. No studies have

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Figure 2. Schematic illustrations of a TSE machine for rheomixing.

been reported for immiscible liquid alloys and SSM alloys [72] until TSEs were employed forcasting processes [73].

TSE flow macro-mechanisms may be different for various modular types and materials, afew comparisons have been made [4] for intermeshing counter-rotating TSE. Generally, somecommon characteristics are achieved, a much broader distribution of residence times, a muchgreater distributive mixing ability, kneading disc mixing elements tend to broaden distributionsin co-rotating TSE. However, it is difficult to go much further since much depends on themodular twin-screw design.

3. Rheomixing approach by TSE process in metallurgical material fields

TSE has been successfully applied to alloy casting processes by Fan and his co-workers [73],such as rheomoulding [16], rheomixing [1], and rheoextrusion [73]. However, applicationsof TSE in immiscible alloys casting process have different flow mechanisms, which resultin the desired solidified microstructure of cast immiscible alloys. The study of immiscibleZn–Pb binary alloys in shear-induced turbulent flows is significantly important to providemore detailed information into the rheomixing process, to increase our control of the rheologyof an emulsion and its microstructure. An understanding of the fluid dynamics inside andaround a suspended drop is necessary for delineating the mechanisms of microscopic transportand microstructure of immiscible alloys casting.

3.1. Rheomixing process for immiscible alloys casting

Figure 2 shows a TSE machine for rheomixing process, which is equipped with a speciallydesigned modular TSE for mixing immiscible alloys. Here, we present a numerical analysisof the hydrodynamic behaviour of an immiscible metallic drop in a shear-induced turbulentflow, which is the main flow feature in the TSE rheomixing process. The emphasis of thepresent work is to investigate hydrodynamic behaviours and to draw qualitative differences inthe features produced by different shearing approaches and properties.

Rheomixing is an integrated process for production of near-net shape components fromimmiscible alloys, such as Zn–Pb, Al–Pb, Ga–Pb systems. Rheomixing starts from meltingmetal ingots and pouring the melts into the rheomixer above the miscibility gap. Molten metalis then sheared through the miscibility gap to form a uniformly mixed paste, and transferred

Fluid flow aspects of TSE process 777

into a shot sleeve. This is subsequently injected into a die. Advantages of the rheomixingprocess include near-net shape production, one-step production, microstructural management,improved production efficiency, improved property combinations, lower overall cost.

The twin-screw used in the rheomixing process has a 16 mm diameter at tip and 3 mmgroove with a special profile [6] to achieve high shear rate and enhance the positive displacementpumping action. The maximum rotation speed of the screw is designed at 1000 rpm, whichcorresponds to a maximum shear rate of 4082 s−1 in the gap between the tip of the screw flightand the barrel. The rotation speed for the experimental work in the BCAST laboratory wasset at 800 rpm. The status of the immiscible alloy liquid during the twin-screw processingis controlled by a temperature control system which ensures a proper viscosity of the matrixphase from start to finish [1]. The thermophysical properties of immiscible metallic Zn–Pbbinary alloys are taken from [74], while phase equilibrium data of Zn–Pb binary alloys aretaken from [75].

3.2. Rheological behaviour of flow in TSE rheomixing process

Understanding and simulating the flow around the twin-screws has always been a difficult andchallenging task due to the complex geometry in the vicinity of the interscrew regions and thelarge deformations induced by the two rotating screws. Understanding the fundamental flowmechanisms on a microstructural scale will help to optimize the mixing processes.

The flow fields of a self-wiping TSE can be divided into two hypothetical regions, one inwhich the two screws intermesh (nip or intermeshing region) and the other far away from thenip, referred to as the convey region. The flow in the convey region is considered to be similarto the flow in the channel of a SSE. The advantages of a TSE over a SSE are often attributedto the nip region, which is considered to have a positive displacement character [76].

Basically, the main feature of a TSE is a strong shear flow field produced by co-rotatingintermeshing screws. Droplets are created on a microscopic scale, and the turbulent flowenhanced by mixing, swirling and pumping actions in a macroscopic scale. Two immiscibleliquid alloys are stirred before pouring into a TSE, and the minority phase Pb liquid is in the formof larger droplets. The main concern for the rheomixing process is droplet interaction, includingrupturing, coalescence, and suspension, which affect the microstructure of immiscible alloycastings.

The flow configurations along the axis of TSE in rheomixing process are shown in figure 3.The procedures for rheomixing can be described in four stages, according to rheologicalbehaviour:

1. Formation of Pb drop in Zn matrix liquid phase: Pb drops formed from two liquid phasesby shear stresses, induced by the action of mechanical devices; the minority phase is inthe form of large drops.

2. Rupture and coalescence of Pb drops: Pb drops breakup into small droplets by a twin-screwat high speeds.

3. Pb liquid droplet and Zn solid particle collision: collisions occur when the temperature isbelow Tm, the Pb drop is broken up into fine droplets by collision with Zn solid particlesin turbulence flow activated mainly by mixing block elements, to prevent the coarseningof Zn particles.

4. Stability of dispersion and suspension: Pb droplets are dispersed by turbulence andsuspended in the Zn matrix phase due to the increased viscosity of the Zn matrixphase in semi-solid state. The viscosity of the matrix phase in semi-solid slurry state

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Figure 3. Flow regions across the phase diagram along the TSE axis.

counterbalances the Stokes and Marangoni motion (caused by gravity and thermocapillaryforces, respectively) of Pb droplets.

3.3. Simplified flow fields

Model experiments of parallel discs were performed in order to study the fundamentalmechanisms of immiscible polymeric materials in a TSE [41]. A numerical simulation ofsimplified shear-induced flow for TSE process was presented (same as shown in figure 4a)by [71]. Numerical simulations provide advantages since various shear rate profiles canbe established for setting up initial and boundary conditions, and more complex forcescan be easily imposed to reflect the special operating conditions and screw configuration.Here, the essential micro-mechanism of immiscible Pb–Zn liquid alloys in rheomixingprocess is presented. The rupturing, interaction and dispersion of droplets, the essentialmicroscopic mechanisms of the TSE, are investigated to improve further our understanding ofthe rheomixing process. The initial and boundary conditions are simplified and categorized asshown in figure 4. These include one-sided and two-sided shear; rapid profile, linear profileand nil profile of initial shear rate.

Fluid flow aspects of TSE process 779

v scr ew su r fa ce

v scr ew su r fa ce

v screw surface

v screw surface

v screw surface

barrel surface

v screw su r fa ce

ba r r el su r fa ce

Figure 4. Schematics of simplified flow configuration for the initial and boundary conditions inthe different areas of TSE. Schematics a and b show linear profile of shear rate. Schematics c andd show rapid profile of shear rate. Middle figure shows the velocity field [84].

4. Numerical simulation

4.1. Numerical methods

The motion of the interface between two immiscible liquids of different density and viscosityin the volume-of-fluid (VOF) method [77] is defined by a volume fraction function C. Thefollowing three conditions are possible:

C = 0 fluid 1, the cell is empty of fluid 2 (1)

C = 1 fluid 2, the cell is empty of fluid 1 (2)

0 < C < 1 the cell contains the interface between the two fluids (3)

Based on the local value of C, the appropriate properties and variables are assigned to eachcontrol volume within the domain.

The volume fraction function C is governed by the volume fraction equation∂C

∂t+ u · ∇C = 0 (4)

where u is the velocity of the flow.The flow is governed by a single momentum equation, as shown below, with the resulting

velocity field shared among the phases

ρ

(∂u∂t

+ u · ∇u)

= −∇p + µ∇2u + ρg + F (5)

where F stand for body forces, and g for gravity acceleration.The properties appearing in the momentum equation are determined by the presence of

the component phase in each control volume, that is, the average value of density and viscosityare interpolated by the following formulae

ρ = Cmρm + (1 − Cm)ρd (6)

µ = Cmµm + (1 − Cm)µd (7)

where subscript m represents the matrix phase, and d the drop phase.

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The formulation of the VOF model requires that the convection and diffusion fluxesthrough the control volume faces be computed and balanced with source terms within thecell itself. The interface is approximately reconstructed in each cell by a proper interpolatingformulation, since interface information is lost when the interface is represented by a volumefraction field.

The geometric reconstruction piecewise linear interface construction (PLIC) scheme[78] is employed because of its accuracy and applicability for general unstructured meshes,compared to other methods such as the donor–acceptor, Euler explicit, and implicit schemes.

The pressure drop across the surface depends upon the surface tension coefficient σ ,

�p = σ

(1

R1+

1

R2

)(8)

where R1, R2 are the two radii, in orthogonal directions, to measure the surface curvature. Inthe continuum surface force (CSF) formulation [79], the surface curvature is computed fromlocal gradients in the surface normal at the interface. The normal n is defined by

n = ∇Cd (9)

where Cd is the secondary phase volume fraction.The curvature k is defined in terms of the divergence of the unit normal n:

k = ∇ · n = 1

|n|[(

n

|n| · ∇)

|n| − (∇ · n)

](10)

where

n = n

|n| (11)

The surface tension can be expressed in terms of the pressure jump across the interface,that is expressed as a volume force added to the momentum equation

F = 2σkCd∇Cd (12)

The solution algorithm involves the use of a control-volume-based technique to convertthe governing equations to algebraic equations that can be solved numerically. Non-lineargoverning equations are linearized by an implicit scheme to produce a system of equations forthe dependent variable in every computational cell. A point implicit Gauss–Seidel linearequation solver is then used, in conjunction with an algebraic multigrid (AMG) method,to solve the resultant scalar system of equations for the dependent variable in each cell[80]. The pressure–velocity coupling is achieved by using the pressure-implicit with splittingof operators (PISO) scheme [81]. The standard k–ε turbulence model is employed forturbulence-imposed flow.

4.2. Simulation of two-dimensional liquid metallic drop in shear-induced mixing process

4.2.1. Deformation evaluation of a Pb metallic drop. The initial Pb phase drop is suspendedin the matrix phase of liquid or semi-solid slurry of Zn. The metallic drop is assumed tohave an undeformed radius rd and viscosity µd, while the matrix phase has viscosity µm. Theliquid immiscible metallic Pb drops breakup into small droplets in shear-induced flow, withsmall daughter drops forming in areas of high local shear. The initial breakup factor Kr isdefined as the ratio of the capillary number of daughter drop to parent drop, and the elongativelength factor KL is defined as the ratio of the elongative length of daughter drop to parent

Fluid flow aspects of TSE process 781

Figure 5. A sequence of deformation leading to the breakup of a Pb metallic drop. Grid 128 × 32,domain 16 × 4. Case 1 and case 2 are for λ = 1, Ca = 3.2 for one side shear-induced flow anddouble side shear-induced flow, respectively, and with enhanced initial shear rate near wall. Case 3is for Ca = 1.17, Re = 0.0 initial flow field.

drop diameter. These factors will be used for measuring the characteristics of initial breakupevolution:

Kr = Cad

Cap= γ rddµm/σ

γ rdµm/σ(13)

KL = elongative length of daughter drop

parent drop diameter(14)

where rdd denotes the daughter drop radius, γ is the imposed shear rate, given by the equationγ = 2nπ(rs/δ − 1) in rheomixing processes [65], rs is the screw radius and n is the screwrotation speed, δ corresponds to the channel depth scale of the TSE between barrel and screwsurface. The shear rate in the twin-screw rheomixing process is distributed in the flow fieldwith an area of high shear rate located between the flank top of the screw and barrel wall, aswell as an area located near the tip of two flights. These are defined as one-sided shear andtwo-sided shear imposed flows, with a low shear rate area in the middle of the screw channel.

The deformation of a Pb metallic drop in one-sided shear flow is shown in figure 5-case 1.For two-sided shear-induced flows, results of the simulations are similar to case 1. Daughterdrops were born on both sides as shown in figure 5-cases 2 and 3. The deformation of the Pbmetallic drops in the three cases is quite different from previous studies [82]. The deformationsare strongly influenced by the ratio of viscosity λ, as shown in figure 6. The effect of viscosity isillustrated in figure 6 through cases 4 and 5. The initial breakup factor Kr = 0.167, KL = 3.833for case 4, and 0.125, 2.667 for case 5. Case 4 reaches the full breakup much faster than case 5,and the shape of daughter drops in case 5 is longer than in case 4. The viscosity of the Znmetal matrix phase is increased during the rheomixing process. When the temperature is closeto the monotectic temperature, the Zn matrix phase will be in semi-solid state, with muchhigher viscosity than in the liquid metal state. According to the observations in cases 4 and5, the shearing action should be completed before the temperature drops to the monotectictemperature. The rheomixing process will be more efficient if the operating temperature isnot close to the monotectic temperature. The shearing time will be shorter and the daughterdroplets will be finer.

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Figure 6. A sequence of deformation leading to the breakup of a Pb metallic drops in pure shearflow, grid 128 × 32, domain 16 × 4, Ca = 2.5. Case 4 is for λ = 1, case 5 is for λ = 0.5.

Figure 7. Comparison of rupturing behaviour of a fully broken up Pb drop in turbulent flow. Case 6is for λ = 1, case 7 is for λ = 0.5.

4.2.2. Rupturing behaviour of a Pb metallic drop. The deformation of a Pb metallic dropfinally leads to its breakup into droplets. The full rupturing process of a Pb drop in shear-induced flow is shown in figure 6. The rupturing is faster in two-sided shear-induced flow thanin one-sided shear-induced flow. However, the shape of the daughter drops is a long rice shapewhich becomes longer with increasing shear rate, as shown in figure 5 case 2 and case 3. Thedroplets have smaller size and spherical shape in case 1, but it takes double the time of case 2and case 3 to complete the full breakup. The effects of the viscosity ratio on the full breakupevolution are shown in figure 7, where the maximum size of the daughter drop is increased withdecreasing viscosity ratio λ. The ratio of the maximum size of daughter drop to the diameterof parent drop at full breakup is Kr max = 1.292, 0.792, 0.725, 1.042, and 1.083 for cases 1, 2,3, 4 and 5, respectively.

4.3. Summary of rheological behaviour of metal drop in two-dimensional shear-induced flow

Table 3 summarizes the initial breakup factors Kr, KL, Kr max, and Taylor number D for thetime of the initial breakup of the first daughter drop and full breakup. Case 1 has the minimum

Fluid flow aspects of TSE process 783

Table 3. Initial breakup factors Kr, KL and Taylor deformation number D.

Case 1 2 3 4 5 6 7

Kr 0.083 0.125 0.333 0.167 0.125 0.333 0.4KL 2.167 3.147 5.542 3.833 2.667 2.125 2D for KL 0.529 0.783 0.936 0.859 0.711 0.522 0.5Kr max 1.292 0.792 0.725 1.042 1.083 0.625 0.625D for Kr max 0.771 0.810 0.706 0.852 0.857 0.648 0.630

The Taylor deformation parameter D = (L − B)/(L + B), where B = rd(1 − D) is the minordeformation axis of the drop and L = rd(1 + D) is the major one (figure 9).

Figure 8. Comparison of the initial breakup scale factors (IBSF) Kr max from the time the firstdaughter drop is formed until full breakup for cases 1–7.

L

B

rd )(

)(

BL

BLD

+−=

Figure 9. Scalar measure of deformation and orientation.

size of the first daughter drop (Kr = 0.083) and the largest size of daughter drop at fullbreakup (Kr max = 1.292). The minimum size of the daughter drop has occurred in turbulentflow (cases 6 and 7). Taylor numbers for KL show the deformation of the parent drop at thetime of formation of the first daughter droplet in each case. Taylor numbers for Kr max showthe shape of the largest daughter droplet at full breakup. The minimum Taylor numbers appearin turbulent flow (cases 6 and 7), meaning that the shape of the droplets is close to a sphericalshape than in other cases.

Figure 8 shows Kr max, the ratio of the largest size of daughter drop to the diameter of parentdrop, from the time of formation of the first daughter drop to the full breakup for cases 1–7.

784 H Tang et al

Figure 10. Shearing time from the time the first daughter drop is formed until full breakup forcases 1–7.

Figure 11. A sequence of deformation of a viscous drop in shear-induced flow in three-dimensionaldomain.

The shearing time span from first daughter drop forming to the full breakup for cases 1–7are quite different, as shown in figure 10. The duration in the one-sided shearing flow (case 1)is longer than for two-sided shearing flow (cases 2 and 3). The low λ value (high viscosity ofthe matrix phase, case 5) needs much more time than equal viscosity flow (case 4) to reachfull breakup. Turbulent flow (case 6) leads to the formation of the first daughter drop earlierthan in laminar flow. The low λ value turbulent flow (case 7) also needs more time than theequal viscosity turbulent flow (case 6) to reach the full breakup.

4.4. Comparisons

Several numerical experiments were performed in order to demonstrate the versatility of theVOF method used in the present study.

The deformation and rupturing behaviour of a Pb metallic drop in Zn matrix phase aredifferent from a viscous drop in shear-induced flow in polymer processing. This can be seenin figures 11 and 12, which show the drop deformation for each case.

Fluid flow aspects of TSE process 785

Figure 12. A sequence of deformation of a metallic drop in shear-induced flow.

Figure 13. Comparison of the smoothness of the volume function, choosing c = ρ. Right graphis original work [79], left is similar model used by present authors.

The algorithms of the VOF method are fully dependent on mesh size to implement propersurface reconstruction. However, they are also influenced by the computation of the interfacenormal ni,j . Figure 13 depicts the static interface reconstruction of a two-dimensional dropcomparing with original work by [79] for the same mesh size.

The non-linear development of Rayleigh–Taylor instability is used to investigate thetopologies of interface and numerical convergence since it is used widely in the validationof the VOF method. In this problem, an air/helium system in a rectangular 0.01 × 0.04 mdomain is computed to time 0.025 s. Grid sizes are 32 × 128, 64 × 256, adapt 64 × 256(adaptive mesh based on 32 × 128 grid), adapt 128 × 512 (adaptive mesh based on 64 × 256grid). The interface is initially a sine wave with amplitude 0.5 mm. As shown in figure 14,the interface development is tracked properly. With a fine grid, detailed interface featuresare clearly produced which are blurred in a coarse grid. The results are compared with thoseof a high order solver [83] on the same mesh size (figure 15). Similar interface propagationis achieved although there is a slight delay in the development of the mushroom cap. Thereconstructed interface retains all the small-scale features of the flow. For this type of unstableinterface problem, it is noted that beside mesh grid sizes the procedure is sensitive to the timestep. Probably, the CFL number (umax�t/�x) needs to decrease below the usual value 0.25.

5. Conclusions

This paper has presented studies of fundamental hydrodynamic mechanisms of immisciblemetallic droplets in shear-induced turbulent flow. The simulations are conducted as simplified

786 H Tang et al

Figure 14. The topologies of the Rayleigh–Taylor unstable interface with different grid solutionsat different time steps.

Figure 15. Comparison of the development of the interface at different time steps with [83] andvelocity vector field at time t = 0.112 s

problems through the VOF method with the PLIC scheme, AMG approach and k–ε turbulencemodel. The metallic drop deformation and rupturing, the essential microscopic mechanisms ofthe rheomixing process, are investigated to increase our understanding of the basic behaviourof immiscible metallic drops in a prototypical rheomixing process.

It was found that the first daughter drop is the smallest one. This may indicate that theTSE rheomixing process is most suitable for mixing immiscible alloys, because materialsare reoriented continuously during shearing. TSE rheomixing process provides good micro-mixing ability. In order to increase the renewal of materials during rheomixing, optimizationof the screw configuration is important, and a suitable number of mixing elements such askneading discs will help to achieve an efficient reorientation action and more surface renewal.

Fluid flow aspects of TSE process 787

It is also noted that Pb metallic drops can easily breakup in thin viscous flow, and their shapebecome more spherical in thick viscous turbulent flow, as well as achieving a more homogenousdispersion. The deformation and breakup can be controlled by the capillary number and theviscosity ratio. The results reveal that in an immiscible Pb–Zn binary alloy system, Pb dropswill easily breakup under equal viscosity or high shear rate conditions, turbulence will speedup the breakup process and will lead to the formation of spherical, smaller and uniform sizedroplets, and homogenous dispersion. Increasing the viscosity of the matrix phase will leadto the formation of spherical droplets, delay the formation of the first daughter droplet anddelay full breakup. Possible suggestions for optimizing the rheomixing process may be givenas follows: start shearing immiscible metallic binary alloys under enhanced turbulence andtemperature above Tm. Both phases have the same viscosity value, which might cause fastbreakup and fine droplets, shortening the time until full breakup, saving power consumption. Ifthe shearing remains at temperature Tm, this will result in spherical droplets as the viscosity ofthe matrix phase is increased. Droplets will also be dispersed stably in thick matrix phase andwill be suspended homogenously in the matrix phase. Control of the viscosity of the matrixphase by temperature is important because it will balance the Stokes and Marangoni motionof droplets (that cause sedimentation and migration, resulting in stratification and segregationin the microstructure, respectively), which are the main obstacles to the industrialization ofimmiscible alloy casting.

The studies reveal a wealth of interesting rheological and microstructural features thatprovide qualitative insights into rheomixing, which are consistent with previous experimentalwork [85].

Acknowledgments

We acknowledge financial support from EPSRC grant GN/N14033, Ford Motor Co., PRISM(Lichfield, UK) and the Mechanical Engineering Department at Brunel University. We arealso grateful to researchers in the CFD group and in BCAST (Brunel Centre for AdvancedSolidification Technology) for the helpful discussions on the numerical approaches and theTSE rheomixing casting process.

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