Simulation of metal filling progress during the casting process

Journal of Materials Processing Technology 100 (2000) 224±229

Simulation of metal ®lling progress during the casting process S. Sulaiman, A.M.S. Hamouda*, S. Abedin, M.R. Osman Department of Mechanical and Manufacturing Engineering, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia Received 12 April 1999

Abstract Metal ®lling in pressure die-casting is a complex process where performance is governed by a number of design variables. The ®lling analysis of non-Newtonian ¯uid is based on the Navier±Stokes equations. In this paper a complete analysis is accomplished by combining the network element method and ¯uid ¯ow analysis to describe an incremental ¯ow front movement. It has been demonstrated that the developed two-dimensional numerical scheme for simulating mould ®lling behaviour can provide reliable results; such as pressure, velocity and temperature variation within the cavity. The temperature and pressure are important to the ®nished product quality and may be used to optimise the moulding process. The effect of draft angle over metal ®lling process is also investigated. The results obtained from the analysis have been veri®ed against analytical results and published data. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Die-casting; Metal ®lling; Draft angle; Network element

Nomenclature A d D g L P Q R t u z m t r

cross-sectional area hydraulic diameter diameter gravity length inertial pressure flow rate duct radius time initial velocity axial co-ordinate direction viscosity density shear stress

even for the experienced mould/die designer. During the ®lling process the factors in¯uencing ¯ow are viscosity, pressure and the physical shape of the passages. When liquid ¯ows in an enclosed passage, its viscosity and the nature of the ¯ow will determine the extent to which drag is imposed by the passage walls with a consequent pressure build-up. Filling pressures are high in pressure die-casting. This is widely alleged to be for the purpose of increasing surface ®nish and de®nition, i.e. the ability of the metal to ®ll small radii so as to reproduce ®ne detail. The mould is usually taken as a simple shape for enhancing the pressure in the casting during freezing to reduce porosity [1]. To achieve improved product shape as well as for smooth surface ®nish of the product, ¯uid ®lling analysis and transient heat-transfer analysis are incorporated in the developed system.

1. Introduction

2. Fluid ¯ow modelling

Metal ®lling is an important phase of the casting process, which is technically complicated and dif®cult to predict

As explained in [2±5], the Navier±Stokes equations in cylindrical co-ordinates for laminar Newtonian ¯ows give   @u @u @u ‡u ‡u r @t @r @z  2  @p @ u 1 @u u @ 2 u ÿ ‡ ‡ (1) ‡ gr ˆÿ ‡m @r @r 2 r @r r 2 @z2

*

Corresponding author. Tel.: ‡603-948-6101-2083; fax: ‡603-9488939. E-mail address: [email protected] (A.M.S. Hamouda).

0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 4 0 9 - 4

S. Sulaiman et al. / Journal of Materials Processing Technology 100 (2000) 224±229

for radial motion and   @w @w @w ‡u ‡w r @t @r @z  2  @p @ w 1 @w u @ 2 w ÿ ‡ ‡ ˆÿ ‡m ‡ gz @z @r 2 r @r r 2 @z2

For a length L of the pipe the equation may be written as Qˆ (2)

for axial motion. The continuity equation is @u @w ‡ ˆ0 @r @z

(3)

These equations are very complex, but can be simpli®ed for a particular application. From Eqs. (1) and (2), it can be seen that acceleration terms on the left-hand side and viscous terms on the right-hand side are considered in the ¯uid ¯ow system. The acceleration terms in the equations are important when the ¯uid has a low viscosity. However, for steady viscous ¯ow, the viscous terms in the equation are important, whereas for accelerating ¯ow (for example due to section changes, junctions, bends, etc.) and unsteady ¯ow, the acceleration terms are important, particularly in the case of high density ¯uids. Generally for low in a pipe, one direction of ¯ow will dominate and the Navier effects will be very small in the pressure generation due to viscosity and high speed ¯ow, therefore they can be neglected. From these assumptions, simpli®cation of Eqs. (1) and (2) follows and a network approach may be used in the metal ¯ow analysis to solve the unidimensional form. Eqs. (1) and (2) becomes   @w @w @p @ 2 w m @w ‡w (4) ˆÿ ‡m 2 ‡ r @t @z @z @r r @r or inertia force ˆ pressure force ‡ viscous force. By considering viscid ¯ow in a pipe of constant section, the force balance gives ÿ

@p @ 2 w m @w ‡m 2 ‡ ˆ0 @z @r r @r r 2 @p ‡ Br ‡ C 4m @z

r 2 @p 2 …R ÿ r 2 † 4m @z

pR4 @p 8m @z

An expression in terms of diameter D rather than radius R is often more suitable, since it may be extended to the hydraulic diameter for non-circular pipe sections, and is given by Qˆ

pD4 DP 128mL

(10)

Eq. (10) can be applied to the laminar ¯ow of molten metal in the ®lling system. Since the metal injection period is short, the ¯ow in the ®lling system may be expected to be turbulent and therefore highly complex [6,7]. Random ¯uctuating components are superimposed on the main ¯ow and as these non-uniform movements are unpredictable, no complete theory has been developed for the microscopic analysis of turbulent ¯ow. Although Darcy's law [6] suggests an equation for turbulent ¯ow in pipes, it is suitable for plain pipe ¯ow only. Where branched ¯ow systems are used the `¯ow squared' term has been found to give problems with the conservation of mass at the junction. Therefore, for pipes with branches, an alternative method is to devise an effective viscosity to be introduced in Eq. (10). Therefore, for turbulent ¯ow, the rate is given as Qˆ

pD4 DP 128meff L

(11)

and peff is de®ned as follows: meff ˆ 0:01317

r wR Re0:25

(12)

(6)

Pe ˆ r

(7)

For the discharge through the entire cross-section Qˆ

(9)

(5)

As explained by Massey [6] the constants of integration B and C are determined from the boundary conditions. As there is no slip at the wall of pipe, w ˆ 0 where r ˆ R and there is centreline symmetry. Consequently B ˆ 0 and C must have the value ÿ(R2/4m)(dp/dz), and so the velocity at any radial location is given by wˆÿ

pR4 DP 8m

In the runner and gating system, the volume of ¯uid is accelerated by the application of external forces. This arises from section and directional changes and also if temporal changes are applied to the ¯uid by a piston or plunger. Pressure due to temporal changes or the inertial pressure is given by

By integrating this equation, it becomes wˆ

225

(8)

L dQ (13) A dt Pressure arises from section and direction changes (as in bends and junction) sometimes termed as the minor head losses and can be incorporated in the Bernoulli's equation for points 1 and 2 in the ¯ow domain    21  22 ÿ W W (14) ‡ rghL P1 ÿ P 2 ˆ r 2 pressure drop ˆ acceleration ‡ losses component. Component losses or minor losses are given by the following equation: hL ˆ

4fL v2 v2 ˆk 2g D 2g

(15)

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S. Sulaiman et al. / Journal of Materials Processing Technology 100 (2000) 224±229

By integrating the volume ¯ow along the metal feed path the extent of metal progression may be mapped. Also the ¯ow rate enables the determination of velocities within different branches of the feeding system and consequently the extent of turbulent action.

or Lˆ

kD 4f

where f is the friction factor de®ned as f ˆ

16 Re

for laminar flow

0:079 for turbulent flow Re…1=4† The loss coef®cient k depends on the nature of the ¯ow changes as de®ned in [6,8]. f ˆ

3. Matrix representation One-dimensional modelling allows a network-type approach where the runner and gating system may be broken down into short straight pipe elements used to represent it and its minor losses. Taking two ends of a pipe duct as having node i and j and by using a linear one-dimensional beam type element, Eqs. (11) and (13) can be represented in a matrix form.From Eq. (11)      128meff L Qi Pvi 1 ÿ1 ˆ (16) Pvj Qj ÿ1 1 pD4 From Eq. (13) 9 8 dQi > >    = < rL Pei 1 ÿ1 dt ˆ Pej ÿ1 1 A > ; : dQj > dt

(17)

With the incorporation of the inertia due to ¯ow acceleration, Eq. (17) requires the inclusion of a further `load vector', i.e. 9 8 dQi > >     = rw <  2i rL Pei 1 ÿ1 dt ‡ ˆ (18) Pej ÿ1 1  2j A > ; 2 w : dQj > dt This is found to be the most straight-forward method of incorporating the inertia term, requiring a calculation of metal velocities in the runner and gating system at each time step based on ¯ow rate and sectional area. From these equations, the effective pressure in the runner and gating system is given by the total of viscous pressure and acceleration pressure, i.e. P t ˆ P v ‡ Pe

(19)

Advancement of metal in the runner and gating will affect the pressure drop. Volumetric ¯ow rate can be calculated as follows: Q ˆ AV

(20)

For multiple impressions, the metal volume can be calculated as follows: Z Z Z (21) Vm ˆ Q1 …t† dt ‡ Q2 …t† dt ‡


‡ Qn …t† dt

4. Model implementation technique for feeding system The model developed for ¯uid feeding system consists of six steps as follows: 1. 2. 3. 4. 5. 6.

conceptual design; initial geometric model; mesh generation; element volume and material properties speci®cation; loss coef®cient determination and boundary condition application.

There are various steps of product models and structures, as explained in [9]. Conceptual design is the earliest design phase and consists of two steps. The product modelling system used in conceptual design must support both the production of the design speci®cation and the construction of the initial geometric model. In the following sections, modelling tools are proposed for these two steps, which are combined together as the product modelling system in the conceptual design phase. The product model is a widely recognised concept to present the solution in a computer system instead of drawings. However, the most current CAD systems adopt a geometric model as the product model, where only ideal geometric shapes are treated and technological information, including dimensions, tolerances and materials properties are incorporated. By taking the importance of technological information into account, it should be connected with geometric shapes. Mesh generation involves ®tting a network of volume over the feeding system, as shown in Fig. 1. In order to deal with pressure losses, two types of elements are used, these being physical element and loss elements. 5. Fluid network program This approach is based on the set of equations and their matrix representations mentioned above. Programs are written using FORTRAN 77 language, which consists of several subroutines applicable for laminar and turbulent ¯ow conditions. Each segment of the program deals with different module. The program has been developed to solve problems for single and multiple impression dies. Generally, the element ®lling conditions can be calculated in two ways, based on metal volume progresses [10] and on ®xed elements [11]. For a ¯uid network program, the ®xed element method is considered, as the metal volume progress method is complex

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227

Fig. 2. Liquid metal progress for the ¯ow rate of 0.009 m3/s at time t ˆ 0.0040 s. Fig. 1. Model mesh including loss elements.

and requires signi®cant computing effort. In this method three types of elements, namely, empty element, mixed element (comprising metal and air) and metal element can be found by involving the physical properties of the air and metal in the element in calculation. 5.1. Veri®cation The results of the ¯uid network program have been veri®ed with the results of experiment, as explained in [14]. With regard to the nature of ®lling process, qualitative agreement between measurement and prediction has been achieved. 6. Results and conclusions In this simulation, the material for die carbon steel with 1% of carbon and for the ®lling material duralumin with 96% A1, 4% Cu and trace Mg is used. Some of the physical properties of the materials used are shown in Table 1.

elements 9 and 22 at a time step of 0.0040 s. Fig. 3 shows that the ®lling material has already ®lled elements 1, 3, 5, 7, 9, 11, 13, 20, 22, 24, 26 and 33 and moving towards elements number 15, 28 and 35 at a time of 0.010 s. Fig. 4 shows that the liquid metal already ®lled the elements up to 43 and is moving toward element 45 at a time of 0.016 s. Fig. 5 shows that at time of 0.022 s, the ¯uid is moving to element number 63. Fig. 6 shows all element of the die cavity are ®lled by the liquid at a time of 0.024 s. Fig. 7 shows the pressure variation with time for the ¯ow rate of 0.009 m3/s. The pro®le of the curve is almost synchronised to the pro®le of the time±pressure curve of previous work [12]. 6.2. Comparison between different draft angles of the runner system and ¯ow rate In expandable mould casting, the purpose of the draft is to facilitate removal of the pattern from the die. In die-casting

6.1. Fluid feeding process Fig. 2 shows the element number 1 and element number 3 already ®lled by the liquid ¯uid. The whole elements 5, 7, 20 are partially ®lled by the ¯uid and ¯uid tends to move to Table 1 Physical properties for die and cast materials [15]

Density (kg/m3) Conductivity (W/m8C) Specific heat capacity (J/kg8C)

Carbon steel (C  1.0%)

Aluminium (duralumin) (96% Al, 4% Cu, trace Mg)

7790 43 470

2790 207 881

Fig. 3. Liquid metal progress for the ¯ow rate of 0.009 m3/s at time t ˆ 0.010 s.

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S. Sulaiman et al. / Journal of Materials Processing Technology 100 (2000) 224±229

Fig. 7. Pressure variation at node 1 with time for the ¯ow rate 0.009 m3/s.

Fig. 4. Liquid metal progress for the ¯ow rate of 0.009 m3/s at time t ˆ 0.016 s.

its purpose is to aid in removal of the part from the mould/ die. Similar tapers should be allowed if solid cores are used in the casting process. The required draft needs to be only about 18 for sand casting and 2±38 for die-casting [13]. Simulation has been carried out for draft angles of 18, 28 and 38 by taking different ¯ow rates. Fig. 8 shows a comparison of the ¯ow rate vs. pressure at the different draft angles. The trend of the curve for the draft angle is almost similar to the exponential manner, so it can be concluded that the pressure is almost proportional to the ¯ow rate and that the pressure varies with the ¯ow rate in exponential manner. Fig. 8 also

Fig. 5. Liquid metal progress for the ¯ow rate of 0.009 m3/s at time t ˆ 0.022 s. Fig. 8. Comparison of ¯ow rate vs. pressure at node 1 for different draft angles.

Fig. 6. Liquid metal progress for the ¯ow rate of 0.009 m3/s at time t ˆ 0.024 s.

Fig. 9. Comparison of ¯ow rate vs. time for different draft angles.

S. Sulaiman et al. / Journal of Materials Processing Technology 100 (2000) 224±229

illustrates that as the draft angle increases the pressure also increases at a particular ¯ow rate in an exponential manner. Fig. 9 shows the comparison of ¯ow rate vs. time for different draft angles. The trend of the curve shows that the time required for the ®lling is almost inversely proportional to the ¯ow rate. The curve pro®le is almost a logarithmic function and it also shows that draft angle 18 needs less time to ®ll the casting cavity compared to the draft angles of 28 and 38. Acknowledgements The project is ®nancially supported by Intensi®cation of Research in Priority Area (IRPA), under the Ministry of Science Technology and the Environment Malaysia. References [1] J. Campbell, Casting, Butterworths, London, 1993. [2] A.J. Murphy, Non-Ferrous Foundry Metallurgy, Pergamon Press, London, 1954, p. 207. [3] R.A. Stoehr, W.S. Howang, Modelling the ¯ow of molten metal having a free surface during entry into moulds, modelling and casting and welding processes II, Engineering Foundation Conference, 1983, pp. 47±58.

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[4] D.M. Gao, G. Dhatt, J. Belanger, A.B. Cheikh, A ®nite element simulation of metal ¯ow in moulds, in: R.W. Lewis, K. Morgan (Eds.), Numerical Methods in Thermal Problems, Proceedings of the Sixth International Conference, Part 1, Vol. VI, 3±7 July, Swansea, UK, 1989, pp. 421±431. [5] H. Schlichting, Boundary-Layer Theory, 6th Edition, McGraw-Hill, New York, August 1966. [6] B.S. Massey, Mechanics of Fluids, 2nd Edition, Van Nostrand Reinhold, New York, 1972. [7] A.J. Ward-Smith, Internal Fluid Flow Ð The Fluid Dynamics of Flow in Pipes and Ducts, Clarendon Press, Oxford, 1980. [8] ASHRAE Handbook and Product Directory Fundamentals, ASHRAE, NY, 1977. [9] Arai, Eiji, Kazuaki Iwata, Product modelling system in conceptual design of mechanical products, Robotics Computer-Integrated manuf. 9(4/5) (1989) 327±334. [10] A.N. Alexandrou, N.R. Anturkar, T.C. Papanastasiou, An inverse ®nite element method with an application to extrusion with solidi®cation, IJNMF 9 (1989) 541±555. [11] E.G. Morgan Thermal variables and their control in diecasting, Foundry Trade J., 18 November 1982, pp. 745±750. [12] D.J. Beadle, Castings, Macmillan, UK, 1971. [13] P. Groover, P. Mikell, Zimmers Jr., W. Emory, CAD/CAM: Computer Aided Design and Manufacturing, 12th Edition, Prentice-Hall of India Private Ltd., 1995. [14] S.B. Sulaiman, D.T. Gethin, A network technique for metal ¯ow analysis in ®lling system of pressure diecasting and its experimental veri®cation on cold chamber machine, Proc. IMechE 206 (1992) 261±275. [15] H.Y. Wong, Handbook of essential formulas and data on heat transfer for engineers, Langman, UK, 1976.