- Email: [email protected]
Numerical evaluation of immiscible metallic Zn–Pb binary alloys in shear-induced turbulent flow
Materials Science and Engineering A365 (2004) 325–329
Numerical evaluation of immiscible metallic Zn–Pb binary alloys in shear-induced turbulent flow H. Tang a,b,∗ , L.C. Wrobel a , Z. Fan b b
a Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex UB8 3PH, UK Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge, Middlesex UB8 3PH, UK
Abstract This paper presents a numerical investigation of the breakup of Pb metallic drops into a Zn matrix phase in shear-induced turbulent flow. This is the main flow feature in the rheomixing process, which has been successfully developed for casting immiscible metallic engineering alloys. A numerical evaluation by a volume-of-fluid (VOF) method with a turbulence model is presented for the studies of fundamental hydrodynamic mechanisms of immiscible metallic droplets. It is noted that turbulence speeds up the drop breakup process and leads to the formation of spherical droplets. It was found that the deformation and rupture of immiscible metallic drops could be predicted essentially via the capillary number. It was observed that the Pb metallic drops could be more easily broken up in thin viscous fluids and get a more spherical shape in thick viscous turbulent flow. The initial breakup scale factors (IBSF) Kr , KL and Krmax are introduced for measuring the breakup evolution for the parametric study. © 2003 Elsevier B.V. All rights reserved. Keywords: Immiscible alloy; Drop breakup; VOF; Shear-induced turbulent flow; Rheomixing
1. Introduction Drop deformation and rupture in simple shear flow is the most common feature in emulsions, and occurs widely in various industrial processes, such as food processing, pharmaceuticals, biomedicine, as well as in materials processing. The microstructure of two immiscible metallic liquids in shear-induced turbulent flow is interesting as the rheological behaviour of immiscible drops, broken up into droplets and dispersed in a matrix phase flow, is of fundamental importance to the casting of immiscible alloys. Based on experimental research in the rheomixing process that has been successfully developed for mixing immiscible alloys [1], shear-induced turbulent flows, which is the main flow feature of rheomixing process, are investigated numerically. The study of immiscible Zn–Pb binary alloys in shear-induced turbulent flows is important to provide more detailed information into the rheomixing process, to increase our control of the rheology of an emulsion and its microstructure. An understanding of the fluid dynamics inside and around a suspended drop is necessary for delineating the mechanisms of ∗ Corresponding author. Tel.: +44-1895-274000; fax: +44-1895-256392. E-mail address: [email protected] (H. Tang).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.09.059
microscopic transport and microstructure of immiscible alloys casting. The hydrodynamic behaviour of a single viscous drop in shear flow has been well reported [2,3]. The drop formation and rupture by shear can be generated by numerical simulations [4]. However, deformation and breakup for immiscible metallic alloy drops have not been reported. The solidified microstructure of cast immiscible alloys strongly depends on the hydrodynamic behaviour within the melt state during cooling [5]. Recently, the effects of the dynamics of droplets of the Pb metal phase on the microstructure of immiscible Al–Pb binary alloys casting has been reported by Zhao [6]. The numerical simulations presented here are simplified problems conducted by using the VOF method with piecewise linear interface construction (PLIC) scheme [7]. The metallic drop deformation and rupturing, the essential microscopic mechanism of the TSE (twin-screw extruder) rheomixing process, are investigated with a view to improving our understanding of the basic behaviour of immiscible metallic drops in a prototypical rheomixing process. The investigations have revealed a wealth of interesting rheological and microstructural features, providing qualitative and quantitative insights into the rheomixing flow which are consistent with experimental work.
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H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
The motion of the interface between two immiscible liquids of different density and viscosity in the VOF method is defined by a volume fraction function C. The following 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
2. Physical parameters A computational model was developed for studying the essential microstructure mechanism of immiscible metallic Zn–Pb binary alloys, which is shear-induced turbulent flow. The computational domain is depicted in Fig. 1. The initial Pb phase drop is suspended in the matrix phase of liquid or semisolid slurry of Zn. The metallic drop is assumed to have an undeformed radius rd and viscosity µd , while the matrix phase has viscosity µm . The distance between upper and lower walls is δ, which corresponds to the channel depth of the twin-screw extruder between barrel and screw surface. The velocity of the moving wall is U = γy, ˙ where γ˙ is the imposed shear rate, given by the equation γ˙ = 2nπ(rs /δ−1) in rheomixing processes [8], where rs is the screw radius and n the screw rotation speed. There are four dimensionless parameters for the numerical simulations: (1) the capillary number Ca = γµ ˙ m rd /σ, where σ is the interfacial tension and an average shear rate is defined by γ˙ = U/δ; (2) the viscosity ratio λ = µd /µm ; (3) the drop Reynolds number Red = ρm γr ˙ d2 /µm ; where ρm is the matrix density; (4) the Taylor deformation parameter D = (L − B)/(L + B), where B is the minor deformation axis of the drop and L the major one. The thermophysical properties of immiscible metallic Zn–Pb binary alloys are taken from [9], while phase equilibrium data of Zn–Pb binary alloys are taken from [10]. 3. Numerical methods
the cell contains the interface between the two fluids
(3)
The volume fraction function C is governed by the volume fraction equation ∂C + u · ∇C = 0 (4) ∂t 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 ∂u ρ + u · ∇u = −∇p + µ ∇ 2 u + ρg + F (5) ∂t where F stands for body forces, and g for gravity acceleration. 3.2. Geometric reconstruction scheme for the VOF method The geometric reconstruction PLIC scheme 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. A VOF geometric reconstruction scheme is divided into two parts: a reconstruction step and a propagation step. The key part of the reconstruction step is the determination of the orientation of the segment. This is equivalent to the determination of the unit normal vector n to the segment. Then, the normal vector n and the volume fraction C uniquely determine a straight line. Once the interface has been reconstructed, its motion by the underlying flow field must be modelled by a suitable algorithm [7].
3.1. The equation of motion
3.3. Implementation of surface tension
Several free surface modelling methods [4] can be used in the study of flow composed of two immiscible liquids of different densities and viscosities. There are classified into tracking methods (Lagrangian methods), including moving-mesh, front tracking, boundary integral and particle scheme; and capturing methods (Eulerian methods), including volume tracking, volume-of-fluid (VOF), continuum convection, level set, and phase field. Each of these methods has its own advantages and disadvantages. The VOF method provides a simple way of treating topological changes of the interface, such as merging and folding, which may be difficult to handle accurately and directly by others.
Surface tension acts to balance the radially inward inter-molecular attractive force with the radially outward pressure gradient across the surface. Implementation of surface tension was reviewed by Scardovelli and Zaleski [4]. Here, the continuum surface force (CSF) model proposed by Brackbill et al. [11] is employed. The addition of surface tension to the VOF method is modelled by a source term in the momentum equation. The CSF model allows for a more accurate discrete representation of surface tension without topological restrictions, and leads to surface tension forces that induce a minimum in the free surface energy configuration.
H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
3.4. Numerical approach The solution algorithm involves the use of a controlvolume-based technique to convert the governing equations to algebraic equations that can be solved numerically. Non-linear governing equations are linearized by an implicit scheme to produce a system of equations for the dependent variable in every computational cell. A point implicit Gauss–Seidel linear equation 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. The pressure–velocity coupling is achieved by using the pressure-implicit with splitting of operators (PISO) scheme. The standard k–ε turbulence model is employed for turbulence-imposed flow.
4. Results and discussion 4.1. Deformation evolution of a Pb metallic drop The deformations of Pb metallic drops in the rheomixing process are evaluated in simplified computational domains as depicted in Fig. 1. The liquid immiscible metallic Pb drops breakup into small droplets in shear-induced flow, with small daughter drops forming in areas of high local shear. The initial breakup factor Kr is defined as the ratio of the capillary number of daughter drop to parent drop for measuring characteristics of initial breakup evolution: Kr =
Cad γr ˙ dd µm /σ = Cap γr ˙ d µm /σ
(6)
where rdd denotes the daughter drop radius. The shear rate in the twin-screw rheomixing process is distributed in the flow field with an area of high shear rate located between the flank top of the screw and barrel wall, as well as an area located near the tip of two flights. These are defined as one-sided shear and two-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 these areas is quite different. The computational domain is twodimensional, size 16 × 4, grid 128 × 32, Ca = 3.2, λ = 1, Re = 13.9. The Pb metallic drop is given an initial radius
327
rd = 1, and the initial shear rate was imposed on the flow field according to the operation condition of the rheomixing process. The deformation of a Pb metallic drop in one-sided shear flow is shown in Fig. 2—case 1. The sharp elongative end pinching was formed on the shear-induced side; consequently, the first daughter drop formed during breakup has quite a small radius. The ratio of the first daughter radius to the parent drop radius is Kr = 0.083, while the ratio of the elongation length to the parent drop radius is KL = 2.167. Drop breakup continues with further elongation process; however, later daughter drop radii are much bigger than the first few ones, and have a long rice shape, not the near spherical shape that the first daughter drop has. For two-sided shear-induced flows, results of the simulations are similar to case 1. Daughter drops were born on both sides as shown in Fig. 2—case 2 and case 3. The ratio of the first daughter radius to the parent drop radius Kr is equal to 0.0125, while the ratio of the elongation length to the parent drop diameter KL is equal to 3.417 for case 2. The drop breakup is 16.67% quicker than in one-sided shear flow. The drop full breakup into droplets is also easier than in one-sided shear flow, as the drop was much more elongated. A rapid shear-induced flow can lead to the quicker breakup of Pb metallic drops into finer droplets. For case 3, the ratio of the first daughter radius to the parent drop radius Kr is 0.333, the ratio of the elongation length to the parent drop diameter KL is 5.542. Case 1 and case 3 need longer for breakup than case 2. The deformation of the Pb metallic drops in the three cases is quite different from previous studies of a viscous drop [12]. The deformations are strongly influenced by the ratio of viscosity λ. It appears that the breakup is easier in thin viscous fluids than in thick viscous fluids. This means that the breakup should be completed in the fluid state with low viscosity of the matrix phase in order to obtain fine droplets and short processing time. According to the observations in cases of unequal viscosity, the shearing action should be completed before the temperature drops to the monotectic temperature. The rheomixing process will be more efficient if the operating temperature is not close to the monotectic temperature. The shearing time will be shorter and the daughter droplets will be finer.
Fig. 2. A sequence of deformation leading to 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 and double-side shear-induced flow, respectively, and with enhanced initial shear rate near wall. Case 3 is for Ca = 1.17, Re = 0.0 initial flow field.
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Fig. 3. A Pb metallic drop breakup in turbulent flow at the time when the first daughter droplet is formed, and at the time of full rupturing, grid 128 × 32, domain 16 × 4, Ca = 3.2, one-side shear-induced flow with initial velocity flow field, near wall initial shear rate. Case 6 is for λ = 1, case 7 is for λ = 0.5.
4.2. Effects of the dynamic flow and turbulence During materials processing, drops are in constant movement within the matrix phase fluid. The material transfer direction is opposite to the direction of screw rotation in the rheomixing process, which means that the shear rate direction is reverse to the direction of fluid velocity. The next simulation was performed by imposing an initial flow field with the material transfer velocity. Results show that drop breakup is much faster in a dynamic flow than in pure shear flow, as illustrated in Fig. 3. The initial breakup factor Kr = 0.333, KL = 2.125 for case 6, and Kr = 0.4, KL = 2 for case 7. The first daughter drop was born at t = 1.6 ms, and full rupturing was reached at t = 6.0 ms in case 6. However, case 7 needs more than twice the time required for case 6.
Comparing the shape of the daughter drop with cases 1–5, droplets are much closer to spherical than elliptical shape. The turbulence leads to forming a round shape as well as speeding up rupturing. The ratio of the smallest size of the daughter drop to the diameter of parent drop, from the time of formation of the first daughter drop to the full breakup time Kmin for cases 1–7, is shown in Fig. 4. The minimum size of daughter droplets is decreased before the parent drop reaches full breakup. However, the maximum size of daughter droplets is increased during this period. The sizes of droplets are various, large differences between the size of the smallest droplet and the largest droplet are observed, the distribution of droplets is inhomogeneous in the channel. Further refinement and dispersion of droplets occur at later stages.
Fig. 4. Comparison of the initial breakup scale factors (IBSF) Kmin from the time the first daughter drop is formed until full breakup for cases 1–7.
H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
329
5. Conclusion
Acknowledgements
The hydrodynamic behaviour of a Pb metallic drop deformation and breakup in various shear flow fields was examined. It is noted that Pb metallic drops can be broken up more easily in thin viscous fluids, and get a more spherical shape in thick viscous turbulent flow. The deformation and breakup can be controlled by the capillary number and the viscosity ratio for materials with similar properties. The results reveal that in immiscible Pb–Zn binary alloy systems, the Pb drop will easily breakup under equal viscosity or high shear rate conditions. Turbulence will speed up the breakup process and will lead to the formation of spherical droplets. Increasing the viscosity of the matrix phase will delay the formation of the first daughter droplet and extend the time until full breakup. Possible suggestions for optimising the rheomixing process may be given as follows: start shearing immiscible metallic binary alloys under enhanced turbulence and temperature above Tm . Both phases have the same viscosity value, which might cause fast breakup and fine droplets, shorten the duration of the full breakup, which means saving power consumption. If the shearing remains at temperature Tm , this will result in spherical droplets as the viscosity of the matrix phase is increased, droplets will also be dispersed stably in thick matrix phase. The initial breakup scale factors (IBSF) Kr , KL and Kmin are introduced for the breakup evolution. These parameters measure the drop breakup characteristics directly and conveniently for parametric studies.
This work is supported by EPSRC grant GR/N14033, Ford Motor Co., and PRISM (Lichfield, UK). The first author would like to thank Dr. S. Ji and Dr. X. Fang for helpful discussions on the experiments and the TSE rheomixing casting process. References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10]
[11] [12]
Z. Fan, S. Ji, J. Zhang, Mater. Sci. Technol. 18 (2001) 837. B.J. Bentley, L.G. Leal, J. Fluid Mech. 167 (1998) 241. T.G. Mason, J. Bibette, Langmuir 13 (1997) 4600. R. Scardovelli, S. Zaleski, Annu. Rev. Fluid Mech. 31 (1999) 567. L. Ratke, S. Diefenbach, Mater. Sci. Eng. RI5 (1995) 263. J.Z. Zhao, S. Drees, L. Ratke, Mater. Sci. Eng. A 282 (2000) 262. D.L. Youngs, in: K.W. Morton, M.J. Baines (Eds.), Numerical Methods for Fluid Dynamics, Academic Press, New York, 1982, p. 273. C. Rauwendaal, Polymer Extrusion, third ed., Hanser Publisher, New York, 1985. T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Oxford University Press, New York, 1988. R. Hultgren, R.L. Orr, P.D. Anderson, K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metals Park, OH, 1973. J.U. Brackbill, D.B. Kothe, C. Zemach, J. Comput. Phys. 100 (1992) 335. J. Li, Y.Y. Renardy, J. Non-Newtonian Fluid Mech. 95 (2000) 235– 251.
Numerical evaluation of immiscible metallic Zn–Pb binary alloys in shear-induced turbulent flow H. Tang a,b,∗ , L.C. Wrobel a , Z. Fan b b
a Department of Mechanical Engineering, Brunel University, Uxbridge, Middlesex UB8 3PH, UK Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge, Middlesex UB8 3PH, UK
Abstract This paper presents a numerical investigation of the breakup of Pb metallic drops into a Zn matrix phase in shear-induced turbulent flow. This is the main flow feature in the rheomixing process, which has been successfully developed for casting immiscible metallic engineering alloys. A numerical evaluation by a volume-of-fluid (VOF) method with a turbulence model is presented for the studies of fundamental hydrodynamic mechanisms of immiscible metallic droplets. It is noted that turbulence speeds up the drop breakup process and leads to the formation of spherical droplets. It was found that the deformation and rupture of immiscible metallic drops could be predicted essentially via the capillary number. It was observed that the Pb metallic drops could be more easily broken up in thin viscous fluids and get a more spherical shape in thick viscous turbulent flow. The initial breakup scale factors (IBSF) Kr , KL and Krmax are introduced for measuring the breakup evolution for the parametric study. © 2003 Elsevier B.V. All rights reserved. Keywords: Immiscible alloy; Drop breakup; VOF; Shear-induced turbulent flow; Rheomixing
1. Introduction Drop deformation and rupture in simple shear flow is the most common feature in emulsions, and occurs widely in various industrial processes, such as food processing, pharmaceuticals, biomedicine, as well as in materials processing. The microstructure of two immiscible metallic liquids in shear-induced turbulent flow is interesting as the rheological behaviour of immiscible drops, broken up into droplets and dispersed in a matrix phase flow, is of fundamental importance to the casting of immiscible alloys. Based on experimental research in the rheomixing process that has been successfully developed for mixing immiscible alloys [1], shear-induced turbulent flows, which is the main flow feature of rheomixing process, are investigated numerically. The study of immiscible Zn–Pb binary alloys in shear-induced turbulent flows is important to provide more detailed information into the rheomixing process, to increase our control of the rheology of an emulsion and its microstructure. An understanding of the fluid dynamics inside and around a suspended drop is necessary for delineating the mechanisms of ∗ Corresponding author. Tel.: +44-1895-274000; fax: +44-1895-256392. E-mail address: [email protected] (H. Tang).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.09.059
microscopic transport and microstructure of immiscible alloys casting. The hydrodynamic behaviour of a single viscous drop in shear flow has been well reported [2,3]. The drop formation and rupture by shear can be generated by numerical simulations [4]. However, deformation and breakup for immiscible metallic alloy drops have not been reported. The solidified microstructure of cast immiscible alloys strongly depends on the hydrodynamic behaviour within the melt state during cooling [5]. Recently, the effects of the dynamics of droplets of the Pb metal phase on the microstructure of immiscible Al–Pb binary alloys casting has been reported by Zhao [6]. The numerical simulations presented here are simplified problems conducted by using the VOF method with piecewise linear interface construction (PLIC) scheme [7]. The metallic drop deformation and rupturing, the essential microscopic mechanism of the TSE (twin-screw extruder) rheomixing process, are investigated with a view to improving our understanding of the basic behaviour of immiscible metallic drops in a prototypical rheomixing process. The investigations have revealed a wealth of interesting rheological and microstructural features, providing qualitative and quantitative insights into the rheomixing flow which are consistent with experimental work.
326
H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
The motion of the interface between two immiscible liquids of different density and viscosity in the VOF method is defined by a volume fraction function C. The following 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
2. Physical parameters A computational model was developed for studying the essential microstructure mechanism of immiscible metallic Zn–Pb binary alloys, which is shear-induced turbulent flow. The computational domain is depicted in Fig. 1. The initial Pb phase drop is suspended in the matrix phase of liquid or semisolid slurry of Zn. The metallic drop is assumed to have an undeformed radius rd and viscosity µd , while the matrix phase has viscosity µm . The distance between upper and lower walls is δ, which corresponds to the channel depth of the twin-screw extruder between barrel and screw surface. The velocity of the moving wall is U = γy, ˙ where γ˙ is the imposed shear rate, given by the equation γ˙ = 2nπ(rs /δ−1) in rheomixing processes [8], where rs is the screw radius and n the screw rotation speed. There are four dimensionless parameters for the numerical simulations: (1) the capillary number Ca = γµ ˙ m rd /σ, where σ is the interfacial tension and an average shear rate is defined by γ˙ = U/δ; (2) the viscosity ratio λ = µd /µm ; (3) the drop Reynolds number Red = ρm γr ˙ d2 /µm ; where ρm is the matrix density; (4) the Taylor deformation parameter D = (L − B)/(L + B), where B is the minor deformation axis of the drop and L the major one. The thermophysical properties of immiscible metallic Zn–Pb binary alloys are taken from [9], while phase equilibrium data of Zn–Pb binary alloys are taken from [10]. 3. Numerical methods
the cell contains the interface between the two fluids
(3)
The volume fraction function C is governed by the volume fraction equation ∂C + u · ∇C = 0 (4) ∂t 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 ∂u ρ + u · ∇u = −∇p + µ ∇ 2 u + ρg + F (5) ∂t where F stands for body forces, and g for gravity acceleration. 3.2. Geometric reconstruction scheme for the VOF method The geometric reconstruction PLIC scheme 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. A VOF geometric reconstruction scheme is divided into two parts: a reconstruction step and a propagation step. The key part of the reconstruction step is the determination of the orientation of the segment. This is equivalent to the determination of the unit normal vector n to the segment. Then, the normal vector n and the volume fraction C uniquely determine a straight line. Once the interface has been reconstructed, its motion by the underlying flow field must be modelled by a suitable algorithm [7].
3.1. The equation of motion
3.3. Implementation of surface tension
Several free surface modelling methods [4] can be used in the study of flow composed of two immiscible liquids of different densities and viscosities. There are classified into tracking methods (Lagrangian methods), including moving-mesh, front tracking, boundary integral and particle scheme; and capturing methods (Eulerian methods), including volume tracking, volume-of-fluid (VOF), continuum convection, level set, and phase field. Each of these methods has its own advantages and disadvantages. The VOF method provides a simple way of treating topological changes of the interface, such as merging and folding, which may be difficult to handle accurately and directly by others.
Surface tension acts to balance the radially inward inter-molecular attractive force with the radially outward pressure gradient across the surface. Implementation of surface tension was reviewed by Scardovelli and Zaleski [4]. Here, the continuum surface force (CSF) model proposed by Brackbill et al. [11] is employed. The addition of surface tension to the VOF method is modelled by a source term in the momentum equation. The CSF model allows for a more accurate discrete representation of surface tension without topological restrictions, and leads to surface tension forces that induce a minimum in the free surface energy configuration.
H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
3.4. Numerical approach The solution algorithm involves the use of a controlvolume-based technique to convert the governing equations to algebraic equations that can be solved numerically. Non-linear governing equations are linearized by an implicit scheme to produce a system of equations for the dependent variable in every computational cell. A point implicit Gauss–Seidel linear equation 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. The pressure–velocity coupling is achieved by using the pressure-implicit with splitting of operators (PISO) scheme. The standard k–ε turbulence model is employed for turbulence-imposed flow.
4. Results and discussion 4.1. Deformation evolution of a Pb metallic drop The deformations of Pb metallic drops in the rheomixing process are evaluated in simplified computational domains as depicted in Fig. 1. The liquid immiscible metallic Pb drops breakup into small droplets in shear-induced flow, with small daughter drops forming in areas of high local shear. The initial breakup factor Kr is defined as the ratio of the capillary number of daughter drop to parent drop for measuring characteristics of initial breakup evolution: Kr =
Cad γr ˙ dd µm /σ = Cap γr ˙ d µm /σ
(6)
where rdd denotes the daughter drop radius. The shear rate in the twin-screw rheomixing process is distributed in the flow field with an area of high shear rate located between the flank top of the screw and barrel wall, as well as an area located near the tip of two flights. These are defined as one-sided shear and two-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 these areas is quite different. The computational domain is twodimensional, size 16 × 4, grid 128 × 32, Ca = 3.2, λ = 1, Re = 13.9. The Pb metallic drop is given an initial radius
327
rd = 1, and the initial shear rate was imposed on the flow field according to the operation condition of the rheomixing process. The deformation of a Pb metallic drop in one-sided shear flow is shown in Fig. 2—case 1. The sharp elongative end pinching was formed on the shear-induced side; consequently, the first daughter drop formed during breakup has quite a small radius. The ratio of the first daughter radius to the parent drop radius is Kr = 0.083, while the ratio of the elongation length to the parent drop radius is KL = 2.167. Drop breakup continues with further elongation process; however, later daughter drop radii are much bigger than the first few ones, and have a long rice shape, not the near spherical shape that the first daughter drop has. For two-sided shear-induced flows, results of the simulations are similar to case 1. Daughter drops were born on both sides as shown in Fig. 2—case 2 and case 3. The ratio of the first daughter radius to the parent drop radius Kr is equal to 0.0125, while the ratio of the elongation length to the parent drop diameter KL is equal to 3.417 for case 2. The drop breakup is 16.67% quicker than in one-sided shear flow. The drop full breakup into droplets is also easier than in one-sided shear flow, as the drop was much more elongated. A rapid shear-induced flow can lead to the quicker breakup of Pb metallic drops into finer droplets. For case 3, the ratio of the first daughter radius to the parent drop radius Kr is 0.333, the ratio of the elongation length to the parent drop diameter KL is 5.542. Case 1 and case 3 need longer for breakup than case 2. The deformation of the Pb metallic drops in the three cases is quite different from previous studies of a viscous drop [12]. The deformations are strongly influenced by the ratio of viscosity λ. It appears that the breakup is easier in thin viscous fluids than in thick viscous fluids. This means that the breakup should be completed in the fluid state with low viscosity of the matrix phase in order to obtain fine droplets and short processing time. According to the observations in cases of unequal viscosity, the shearing action should be completed before the temperature drops to the monotectic temperature. The rheomixing process will be more efficient if the operating temperature is not close to the monotectic temperature. The shearing time will be shorter and the daughter droplets will be finer.
Fig. 2. A sequence of deformation leading to 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 and double-side shear-induced flow, respectively, and with enhanced initial shear rate near wall. Case 3 is for Ca = 1.17, Re = 0.0 initial flow field.
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H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
Fig. 3. A Pb metallic drop breakup in turbulent flow at the time when the first daughter droplet is formed, and at the time of full rupturing, grid 128 × 32, domain 16 × 4, Ca = 3.2, one-side shear-induced flow with initial velocity flow field, near wall initial shear rate. Case 6 is for λ = 1, case 7 is for λ = 0.5.
4.2. Effects of the dynamic flow and turbulence During materials processing, drops are in constant movement within the matrix phase fluid. The material transfer direction is opposite to the direction of screw rotation in the rheomixing process, which means that the shear rate direction is reverse to the direction of fluid velocity. The next simulation was performed by imposing an initial flow field with the material transfer velocity. Results show that drop breakup is much faster in a dynamic flow than in pure shear flow, as illustrated in Fig. 3. The initial breakup factor Kr = 0.333, KL = 2.125 for case 6, and Kr = 0.4, KL = 2 for case 7. The first daughter drop was born at t = 1.6 ms, and full rupturing was reached at t = 6.0 ms in case 6. However, case 7 needs more than twice the time required for case 6.
Comparing the shape of the daughter drop with cases 1–5, droplets are much closer to spherical than elliptical shape. The turbulence leads to forming a round shape as well as speeding up rupturing. The ratio of the smallest size of the daughter drop to the diameter of parent drop, from the time of formation of the first daughter drop to the full breakup time Kmin for cases 1–7, is shown in Fig. 4. The minimum size of daughter droplets is decreased before the parent drop reaches full breakup. However, the maximum size of daughter droplets is increased during this period. The sizes of droplets are various, large differences between the size of the smallest droplet and the largest droplet are observed, the distribution of droplets is inhomogeneous in the channel. Further refinement and dispersion of droplets occur at later stages.
Fig. 4. Comparison of the initial breakup scale factors (IBSF) Kmin from the time the first daughter drop is formed until full breakup for cases 1–7.
H. Tang et al. / Materials Science and Engineering A365 (2004) 325–329
329
5. Conclusion
Acknowledgements
The hydrodynamic behaviour of a Pb metallic drop deformation and breakup in various shear flow fields was examined. It is noted that Pb metallic drops can be broken up more easily in thin viscous fluids, and get a more spherical shape in thick viscous turbulent flow. The deformation and breakup can be controlled by the capillary number and the viscosity ratio for materials with similar properties. The results reveal that in immiscible Pb–Zn binary alloy systems, the Pb drop will easily breakup under equal viscosity or high shear rate conditions. Turbulence will speed up the breakup process and will lead to the formation of spherical droplets. Increasing the viscosity of the matrix phase will delay the formation of the first daughter droplet and extend the time until full breakup. Possible suggestions for optimising the rheomixing process may be given as follows: start shearing immiscible metallic binary alloys under enhanced turbulence and temperature above Tm . Both phases have the same viscosity value, which might cause fast breakup and fine droplets, shorten the duration of the full breakup, which means saving power consumption. If the shearing remains at temperature Tm , this will result in spherical droplets as the viscosity of the matrix phase is increased, droplets will also be dispersed stably in thick matrix phase. The initial breakup scale factors (IBSF) Kr , KL and Kmin are introduced for the breakup evolution. These parameters measure the drop breakup characteristics directly and conveniently for parametric studies.
This work is supported by EPSRC grant GR/N14033, Ford Motor Co., and PRISM (Lichfield, UK). The first author would like to thank Dr. S. Ji and Dr. X. Fang for helpful discussions on the experiments and the TSE rheomixing casting process. References [1] [2] [3] [4] [5] [6] [7]
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