铸造技术动态建模过程和模具设计中英文对照外文翻译文献

中英文对照外文翻译(文档含英文原文和中文翻译)原文：《Modelling the dynamics of the tilt-casting process and the effect of the mould design on the casting quality》AbstractAll titanium alloys are highly reactive in the molten condition and so are usually melted in a water-cooled copper crucible to avoid contamination using processes such as Induction Skull Melting (ISM). These provide only limited superheat which, coupled with the surface turbulence inherent in most conventional mould filling processes, results in entrainment defects such as bubbles in the castings. To overcome these problems, a novel tilt-casting process has been developed in which the mould is attached directly to the ISM crucible holding the melt and the two are then rotated together to achieve a tranquil transfer of the metal into the mould. From the modelling point of view, this process involves complex three-phase flow, heat transfer and solidification. In this paper, the development of a numerical model of the tilt-casting process is presented featuring several novel algorithm developments introduced into a general CFD package (PHYSICA) to model the complex dynamic interaction of the liquid metal and melting atmosphere. These developments relate to the front tracking and heat transfer representations and to a casting-specific adaptation of the turbulence model to account for an advancing solid front. Calculations have been performed for a 0.4 m long turbine blade cast in a titanium aluminide alloy using different mould designs. It is shown that the feeder/basin configuration has a crucial influence on the casting quality. The computational results are validated against actual castings and are used to support an experimental programme. Although fluid flow and heat transfer are inseparable in a casting, the emphasis in this paper will be on the fluid dynamics of mould filling and its influence on cast quality rather than heat transfer and solidification which has been reported elsewhere.KeywordsTilt-casting; Mould design; 3-D computational model; Casting process；1. IntroductionThe casting process is already many centuries old, yet many researchers are still devoted to its study. Net shape casting is very attractive from the cost point of view compared to alternative component manufacturing methods such as forging or machining. However, reproducible quality is still an issue; the elimination of defects and control of microstructure drive research. Casting involves first the filling of the mould and subsequently the solidification of the melt. From the numerical modelling point of view, this simple sequence results in a very complex three-phase problem to simulate. A range of interactions of physical phenomena are involved including free surface fluid flow as the mould fills, heterogeneous heat transfer from the metal to the mould, solidification of the molten metal as it cools, and the development of residual stresses and deformation of the solidified component.In industry there are many variants of the casting process such as sand casting, investment casting, gravity, and low and high pressure die casting. In this study, the investment casting process, also called lost-wax casting, has been investigated. One of the advantages of this process is that it is capable of producing (near) net shape parts, which is particularly important for geometrically complex and difficult-to-machine components. This process starts with making a ceramic mould which involves three main steps: injecting wax into a die to make a replica of the component and attaching this to a pouring basin and running system; building a ceramic shell by the application of several layers of a ceramic slurry and ceramic stucco to the wax assembly; de-waxing and mould firing. The pouring of the casting is performed either simply under gravity (no control), or using a rapid centrifugal action [1] (danger of macro-segregation plus highly turbulent filling), or bysuction as in counter-gravity casting (e.g. the Hitchiner process[2]), or by tilt-casting. In this study, tilt-casting was chosen in an attempt to achieve tranquil mould filling. Tilt-casting was patented in 1919 by Durville [3] and has been successfully used with sand castings[4] and aluminium die castings[5]. In the IMPRESS project [6], a novel process has been proposed and successfully developed to combine Induction Skull Melting (ISM) of reactive alloys with tilt-casting[7], [8], [9] and [10], with a particular application to the production of turbine blades in titanium aluminide alloys. As shown in Fig. 1, this is carried out inside a vacuum chamber and the mould is pre-heated in situ to avoid misruns (incomplete mould filling due to premature solidification) and mould cracking due to thermal shock.Tilt-casting process: (a) experimental equipment; (b) schematic view of the ISM crucible and mould, showing the domed shape acquired by the molten metal; (c) different stages of mould filling showing the progressive replacement of gas by the metal.The component(s) to be cast are attached to a pouring basin which also doubles as a source of metal to feed the solidification shrinkage. The components are angled on the basin to promote the progressive uni-directional flow of metal into the mould. As the metal enters the mould it displaces the gas and an escape route has to be included in the design so that the two counter-flowing streams are not mixed leading to bubbles trapped in the metal. Vents are also used to enable any trapped gas to escape. The ‘feeder’ used to connect the mould to the crucible is normally in any casting the last port ion of metal to solidify, so supplying metal to the mould to counter the effects of solidification shrinkage. In tilt-casting, the feeder is also the conduit for the tranquil flow of metal into the mould and also for the unhindered escape of gas. For this reason, the fluid dynamics of the mould feeder interface merit detailed study.As well as the mould/feeder design, the production of castings involves several other key parameters, such as the metal pouring temperature, initial mould temperature, selective mould insulation and the tilt cycle timing. All these parameters have an influence on the eventual quality of the casting leading to a very large matrix of experiments. Modelling (once validated) is crucial in reducing the amount of physical experiments required. As mentioned above, the mathematical models are complex due to the fact that this is a three-phase problem with two rapidly developing phase fronts (liquid/gas and solid/liquid). In this paper, a 3-D computational model is used to simulate the tilt-casting process and to investigate the effect of the design of the basin/feeder on the flow dynamics during mould filling and eventually on casting quality.2. Experimental descriptionDetails of the experimental setup have been published elsewhere [11], but for completeness a summary description is given here. Fig. 1a shows an overall view of the equipment used to perform the casting. The Induction Skull Melting (ISM) copper crucible is installed inside a vacuum chamber. To enable rotation, it is attached to a co-axial power feed, which also allows cooling water containing ethylene glycol to be supplied to the ISM crucible and the induction coil. The coil supplies a maximum of 8 kA at a frequency of ∼6 kHz. The crucible wall is segmented, so that the induction field penetrates through the slots (by inducing eddy currents into each finger segment) to melt the charge and at the same time repel the liquid metal away from the side wall to minimise the loss of superheat. A billet of TiAl alloy is loaded into the crucible before clamping on the ceramic shell mould. The mould is surrounded by a low thermal mass split-mould heater.After evacuating the vacuum chamber, the mould is heated to the required temperature (1200 °C maximum) and the vessel back-filled with argon to a partial pressure of 20 kPa prior to melting. This pressure significantly reduces the evaporative loss of the volatile aluminium contained in the alloy. The power applied to the induction coil is increased according to a pre-determined power vs. time schedule so that a reproducible final metal temperature is achieved. At the end of melting (7–8 min), the mould heater is opened and moved away. The induction melting power is ramped down and, simultaneously, the ISM crucible and mould are rotated by 180° using a programmable controller to transfer the metal into the mould. The mould containing the casting is held vertically as the metal solidifies and cools down.3. Mathematical model3.1. Fluid flow equationsThe modelling of the castingprocess has involved a number of complex computational techniques since there are a range of physical interactions to account for: free surface fluid flow, turbulence, heat transfer and solidification, and so on. The fluid flow dynamics of the molten metal and the gas filling the rest of the space are governed by the Navier–Stokes equations, and a 3D model is used to solve the incompressible time-dependent flow:(1)(2)where u is the fluid velocity vector; ρ is the density; μ is the fluid viscosity; Su is a source term which contains body forces (such as gravitational force, a resistive force (Darcy term) [12]) and the influence of boundaries. There is a sharp, rapidly evolving, property interface separating metal and gas regions in these equations as explained below.3.2. Free surface: counter diffusion method (CDM)One of the difficulties of the simulation arises from the fact that two fluid media are present during filling: liquid metal and resident gas and their density ratio is as high as 10,000:1. Not only does the fluid flow problem need to be solved over the domain, but the model also has to track the evolution of the interface of the two media with time. A scalar fluid marker Φ was introduced to represent the metal volume fraction in a control volume and used to track the interface of the two fluids, called the Scalar Equation Algorithm (SEA) by Pericleous et al. [14]. In a gas cell, Φ = 0; in a m etal cell, Φ = 1; for a partially filled cell Φ takes on an intermediate value which the interface of the two media crosses through. The dynamics of the interface are governed by the advection equation:(3)The interface then represents a moving property discontinuity in the domain, which has to be handled carefully to avoid numerical smearing. As in [14], an accurate explicit time stepping scheme such as that by Van Leer [15] may be used to prevent smearing. However, the scheme is then limited to extremely small time steps for stability, leading to very lengthy computations. To overcome this problem, a new tracking method, the counter diffusion method (CDM) [11] and [16], was developed as a corrective mechanism to counter this ‘numerical diffusion’. This discretizes the free surface equation in a stable, fully implicit scheme which makes the computations an order of magnitude faster. The implementation assumes that an interface-normal counter diffusion flux can be defined for each internal face of the computational mesh and applied with opposite signs to elements straddling the interface as source terms for the marker variable. The equation for the flux per unit area F can be written as:(4)where C is a scaling factor, a free parameter in CDM allowing the strength of the counter diffusion action to be adjusted, and n is the unit normal vector to the face in the mesh. Of the two cells either side o f the face, the one with the lower value of the marker ΦD becomes the donor cell while the ‘richer’ cell ΦA is the acceptor (in order to achieve the counter diffusion action). The proposed formula makes the counter diffusion action self-limiting as it is reduced to zero where the donor approaches zero (gas) and where the acceptor reaches unity (liquid). In this form, the adjustment remains conservative. Quantitative validation of CDM against other VOF type techniques is given in a later section of the paper for accuracy and efficiency.3.3. Heat transfer and solidificationHeat transfer takes place between the metal, mould and gas, and between cold and hot metal regions as the mould filling is carried out. The heat flow is computed by a transient energy conservation equation:(5)where T is the temperature; k is the thermal conductivity; cp is the specific heat (properties can be functions of the local temperature or other variables); ST is the source term which represents viscous dissipation, boundary heat transfer and latent heat contributions when a phase change occurs. For the latter, a new marker variable fL is used to represent the liquid fraction of the metal with (1 − fL) being the volume fraction of solidified metal. V oller et al. [13] used a non-linear temperature function to calculate the liquid fraction. In this study, the liquid fraction is assumed to be a linear function of the metal temperature:(6)TL is the liquidus temperature and TS is the solidus temperature.3.4. LVEL turbulence model (applied to solid moving boundaries)Even at low filling speeds, the Reynolds number is such that the flow is turbulent. The LVEL method of Spalding [17] is chosen to compute the turbulence because of its mixing-length simplicity and robustness. LVEL is an abbreviation of a distance from the nearest wall (L) and the local velocity (VEL). The approximate wall distance is solved by the Eqs. (7) and (8):(7)∇·(∇W)=-1where W is an auxiliary variable in the regions occupied by the moving fluid with boundary conditions W = 0 on all solid walls.(8)This distance and the local velocity are used in the calculation of the local Reynolds number from which the local value of the turbulent viscosity νt is obtained using a universal non-dimensional velocity profile away from the wall. The effective turbulent viscosity is then computed from the following equation:(9)where κ = 0.417 is the von Karman constant, E = 8.6 is the logarithmic law constant [17] and u+ is determined implicitly from the local Reynolds number Reloc = uL/ν with the magnitude of the local velocity u and the laminar k inematic viscosity ν[17]. The LVEL method was extended to moving solid boundaries and in particular to solidifying regions by setting W = 0 in every region that is no longer fluid and then solving Eqs. (7) and (8) at each time step.In simulating the tilt-casting process, the geometry is kept stationary and the gravitational force vector is rotated to numerically model the tilt instead of varying the coordinates of the geometry. The rotating gravitational force vector appears in the source term of Eq. (1) for the tilt-casting process. A mathematical expression relating the tilting speed to the tilting angle θ has be en used. Since θ is a function of time, the variable rotation speed is adjustable to achieve tranquil filling. This technique neglects rotational forces within the fluid (centrifugal, Coriolis) since they are negligible at the slow rotation rates encountered in tilt-casting. Finally, the numerical model of the tilt-casting process and the new algorithm developments were implemented in the general CFD package (PHYSICA).4. Description of simulations4.1. Geometry, mould design and computational meshThe casting is a generic 0.4 m-long turbine blade typical of that used in an Industrial Gas Turbine. Fig. 2 shows three mould designs which comprise the blade, a feeder/basin and a cylindrical crucible. Fig. 2a incorporates a separate cube-shaped feeder that partially links the root of theblade and the basin. Fig. 2b is a variant in which the plane of the blade is rotated through 90°. In both cases, the computational mesh contains 31,535 elements and 38,718 points. Six vents are located on the platform and the shroud of the blade, as seen in Fig. 2a and b. Fig. 2c is an optimised design where the feeder and basin are combined to provide a smooth connection between the blade and the crucible. Two vents are located in the last areas to be filled to help entrapped gas to escape from the mould. Mesh of the crucible-mould assembly for the three cases investigated.The mesh for the last case contains 30,185 elements and 37,680 vertices. As in all the cases presented, numerical accuracy depends on mesh fineness and also the degree of orthogonality. To ensure a mostly orthogonal mesh the various components of the assembly were created separately using a structured body-fitted mesh generator and then joined using a mixture of hexahedral and tetrahedral cells. The mesh was refined as necessary in thin sections (such as the blade itself or the shroud and base plates), but not necessarily to be fine enough to resolve boundary layer details. For this reason the LVEL turbulence model was used rather than a more usual two-equation model of turbulence that relies on accurate wall function representation. The practical necessity to run in parallel with the experimental programme also limited the size of the mesh used. As with all free surface tracking algorithms, the minimum cell size determines the time step size for the stable simulations. Although the CDM method is implicit, allowing the time step to exceed the cell CFL limit, accuracy is then affected. With these restrictions, turnaround time for a complete tilt-casting cycle was possible within 24 h.As stated earlier, the feeder is necessary to minimise the solidification shrinkage porosity in the blade root. Two alternative designs have been considered: a cubic feeder with a volume to cooling surface area ratio of 14.5 mm, and a cylindrical feeder designed with better consideration of fluid dynamics during mould filling and which had a slightly lower volume to area ratio of 13.8 mm.4.2. Initial and boundary conditionsThe choice of parameters for the calculations was based on the experiments [16]. The properties of the materials used in the calculations are listed in Table 1. The initial conditions (the same as in the trials) and boundary conditions of the calculations are shown in Table 2.Table 1.Properties of the materials in this study.Ti–46Al–8Ta alloy MouldDensity (kg/m3) 5000 2200Thermal conductivity (W/(m K)) 21.6 1.6Specific heat (J/(kg K)) 1000 1000Viscosity (kg/(m s)) 0.5 ×10−60.1Liquidus temperature (°C) 1612 –Solidus temperature (°C) 1537 –Latent heat (J/kg) 355,000 100,0004.3. Tilt cycleThe molten metal in the ISM crucible is poured via the basin/feeder into the mould by rotating the assembly. A parabolic programmed cycle [16] is employed to complete the castingprocess with a total filling time of 6 s. The carefully designed cycle includes a fast rotation speed at the earlystage of the mould filling to transfer the molten metal into the basin/feeder, a subsequent deceleration to a nearly zero velocity to allow most of the metal to fill the mould horizontally and to avoid forming a back wave and surface turbulence, and then the rapid completion of the filling to reduce the heat loss to the mould wall.5. Computing requirementsThe results presented here have been obtained using an Inter (R) Xeon (R) CPU E5520 2.27 GHz, 23.9 GB of RAM. For a typical mesh of 30,000 finite volume cells, each full tilt-casting simulation (real time 6 s) took approximately 15 h and 1200 time steps to complete. The CDM algorithm uses a fixed time step of 0.005 s which is at least five times larger than that used in conventional methods such as Van Leer or Donor–Acceptor. Similar computations carried out with the alternative Donor–Acceptor algorithm took typically one week to complete.The speed of execution and stability of the CDM method does not necessarily compromise accuracy. This can be demonstrated in the classic collapsing column benchmark experiment of Martin and Moyce [18] shown schematically in Fig. 3. A rectangular water column with a height of 2 m and a width of 1 m is initially confined between two vertical walls in hydrostatic equilibrium. Air is present as the outer medium. Once the confining wall is removed, the water column collapses on to the plane y = 0 under gravity and spreads out along the x direction.Fig. 3. Configuration of water column collapsing experiment.View thumbnail images The experiment was designed specifically so that it could be modelled computationally in two dimensions. Therefore, a 2D domain was used meshed into 880 cells (40 × 22).The comparison between the numerical result with CDM, the Van Leer and the popular Donor–Acceptor algorithm against the experimental data is presented in Fig. 4, where the horizontal extent of the water front and the residual height of the water column are plotted as functions of elapsed time. It can be seen that there is generally good agreement between the numerical results and the experimental data. However, although the three numerical methods match each other perfectly, there is some disagreement against the experiment when the non-dimensional time t* is greater than 1.4. It is concluded that in terms of accuracy, CDM is at least as good as the alternative explicit techniques which have been in widespread use for many years.Fig. 4. Validation of the CDM method and comparisons of the CDM against Van Leer, and donor acceptor for (a) the front position and (b) the residual height of the collapsing water column experiment of Martin and Moyce [18].As mentioned above, a feature of the CDM method is that the discretization of the free surface equation is made in a stable, fully implicit scheme which makes the computations an order of magnitude faster. Table 3 presents a comparison of CDM against the other two methods investigated, in terms of the computational efficiency. It is shown that CDM can be applied with a bigger time step than the other methods since CDM it is not limited by the Courant–Friedrichs–Levy (CFL) criterion. Furthermore, due to greater numerical stability, the number of iterations per time step is also reduced which makes the CDM simulation even faster. The first two columns in the table show that the time step for CDM can be ten times bigger than the others. The running time with the Van Leer total variation diminishing (TVD) scheme is 1.3 times longer than with CDM for the same time step, but the Van Leer scheme suffers from interface smearing. The running time of the most popular scheme for casting simulations, the donor acceptor method, is almost four times longer than that with CDM when the same time step is used. CDM is up to eight times faster (16 s vs. 132 s as shown underlined in Table 3) when the optimal time step for CDM is used.Table 3. Comparisons of the efficiency of CDM with others numerical methods.Δt1 = 0.1 s Δt1 = 0.05 s Δt1 = 0.01 sMethodN t (s) N t (s) N t (s)Van Leer Error Exceeds CFL limit 10 47Donor Acceptor Error Exceeds CFL limit 40 132CDM 20 16 15 17 5 34Notes: Δt = time step; t = running time; N = average number of iterations per time step.6. Simulations – results and discussion6.1. Effect of mould orientationCalculations with two orientations (Fig. 2a and b) for the assembly with the cubic feeder have been performed. Fig. 5 shows the mould filling progression as iso-surface plots of the free surface marker, at Ф = 0.5, at a filling time of 3.2 s. It is seen that in a design without consideration for flow behaviour, the metal is thrown into the cubic feeder in both cases in a turbulent state, becauseof the sudden change in cross-section. At any given time during filling, more metal enters the cubic feeder and less enters the blade in orientation 2, Fig. 5b, compared with orientation 1, Fig. 5a, leading to a restricted exit path for the escaping gas. For both orientations, the sudden drop at the connection between the feeder and the root of the blade leads to jetting and turbulence at the point where the metal flows from the feeder into the blade cavity.Comparison of mould filling with two orientations in contour plots of the free surface marker Ф = 0.5 at the interface, time = 3.2 s for a cubic feeder: (a) orientation 1: mould oriented at 30° to tilt axis; (b) orientation 2: long axis of the root perpendicular to the tilt plane.A later stage in the filling process is presented in Fig. 6 for the same two orientations, with the blades now filled with metal. Although both orientations display the same problems of gas mixing and turbulence caused by the two sudden steps in the feeder, it seems that orientation 1 leads to less gas mixing than orientation 2. Fig. 7 shows the 0.4 m-long turbine blade castings produced by the process. There is surface evidence of porosity at the connection between the feeder and the root of the blade on the concave sides, and this is worse for orientation 2 than for orientation 1. Radiography indicates the internal extent of this porosity. Although several factors are responsible for its formation, including the presence of a hot spot leading to an isolated liquid pool during solidification and subsequent shrinkage, the presence of trapped gas is a major contributorComparison of mould filling with two orientations in contour plots of the free surface marker Ф = 0.5 at the interface, time = 5.2 s for a cubic feeder: (a) orientation 1: mould oriented at 30° to tilt axis; (b) orientation 2: long axis of the root perpendicular to the tilt plane.Comparisons of the experimental results with two orientations: (a) orientation 1: mould oriented at 30° to tilt axis; (b) orientation 2: root axis perpendicular to the tilt plane.6.2. Effect of the mould design: cubic vs. cylindrical feederIn the above discussion, it was shown that the orientation of the blade relative to the tilt axis in Fig.2 is important, and that the sudden changes in cross-section with a cubic feeder lead to turbulent mixing of gas and liquid metal. In the following section, the effect of the feeder design on casting quality will be studied comparing two mould designs: one with a cylindrical feeder (Fig. 2c) and the other with a cubic feeder with the preferred orientation (Fig. 2a).Fig. 8 shows a comparison of the instantaneous free surface location at a filling time of 3.0 s. As can be seen, the metal is smoothly entering the blade cavity in the case of the cylindrical feeder. In contrast the metal is thrown into the cubic feeder because of the sudden change in the cross-section. The sudden drop at the connection between the feeder and the root of the bladeleads to jetting and turbulence when the metal flows from the feeder into the blade cavity. The comparison also shows that the filling of the blade with the cylindrical feeder is faster than with the cubic feeder. This phenomenon is demonstrated in Fig. 9 as well.The comparison of the mould filling with the two designs of feeder: iso-surface plots of the free surface marker Ф = 0.5 at time = 3.0 s: (a) cube feeder; (b) cylindrical feeder.Comparison of the mould filling with the two feeders: contour plots with the free surface marker Ф = 0.5 at the interface, time = 4.6 s: (a) cubic feeder; (b) cylindrical feeder.9 shows the flow progress at a later stage of the mould filling (rotation time of 4.6 s) for the two competing designs. It can be seen that the design with the cylindrical feeder and with the vertical orientation of the blade provides a better gas escape route back to the crucible (in addition to gas escaping through the vents in the mould) than the design with the cubic feeder. There are two flow restrictions in the cubic feeder design: one is the connection between the basin and the feeder and the other is the connection between the feeder and the root of the blade, both leading to a step change in cross-section. This geometric feature of the assembly causes the gas to be easily trapped in the upper corner of the root.Fig. 10 highlights the velocity vector field as the metal enters the mould in the cubic feeder design, Fig. 2a. It is seen that the metal is pushed back from the root of the blade (zoomed). The metal and the gas re-circulate in the cavity of the root. This recirculation will result in mixing of gas with the metal which presents a high risk of forming casting defects such as bubblesFig. 10. The computed velocity field and iso-surface (free surface marker Ф = 0.5 at the interface) time = 3.1 s for the cubic feeder.The computed velocity field in Fig. 11a illustrates that the gas is trapped and gas recirculation takes place in the cube feeder although some gas in the aerofoil and in the platform is slowly evacuated by the vents at the platform of the blade (zoomed). Gas recirculation leads to gas–metal mixing. This introduces a high risk of the formation of gas bubbles which are then blocked inside the casting if the superheat is not high enough to allow them time to float up before the casting solidifies. In Fig. 11b, it is shown that the cross-section at the connection of the basin with the cubic feeder is fully blocked by the metal coming from the crucible at a certain moment during the mould filling. This is the reason that gas recirculation appears in the cube feeder and the root of the blade. For the cylindrical feeder, the gas evacuation path is clear (Fig. 11c and d) and there is no danger of the gas being trapped in the upper corner of the root, especially since a vent is located at the top of the platform (see Fig. 2). Comparison of the computed velocity field and iso-surface (free surface marker Ф = 0.5 at the interface) time = 4.8 s。