IOP Conference Series: Materials Science and Engineering Simplified pressure model for quantitative shrinkage porosity prediction in steel castings To cite this article: A V Catalina and C A Monroe 01 IOP Conf. Ser.: Mater. Sci. Eng. 33 01067 View the article online for updates and enhancements. Related content - Simulation-based prediction of microshrinkage porosity in aluminum casting: Fully-coupled numerical calculation vs. criteria functions J Jakumeit, S Jana, B Böttger et al. - Fractal characteristics of dendrite in aluminum alloys K Ohsasa, T Katsumi, R Sugawara et al. - A combined enthalpy / front tracking method for modelling melting and solidification in laser welding G Duggan, W U Mirihanage, M Tong et al. This content was downloaded from IP address 148.51.3.83 on 17/04/019 at 0:01
Simplified pressure model for quantitative shrinkage porosity prediction in steel castings A V Catalina 1 and C A Monroe 1, 1 Caterpillar Inc., 14009 Old Galena Road, Mossville, IL 6155, USA University of Alabama at Birmingham, 1075 13th Street South, Birmingham, AL 3594-4440, USA Email: Catalina_Adrian_V@cat.com Phone: +1 309 578 6314 Abstract. In order to reduce the computational effort, a simplified pressure model is proposed for predicting shrinkage-related defects in steel castings. Based on the specific conditions developing during casting solidification, the governing fluid flow equations (i.e., Stokes and Darcy) have been reformulated such as the pressure field could be calculated only based on volume (density) changes occurring during the solidification process. Velocity is thus eliminated from the governing equations of fluid flow therefore simplifying the method of solution for pressure and tremendously reducing the computational effort. Model development and simulation results are presented and discussed. 1. Introduction Metals and alloys typically undergo a negative volume change during the liquid-to-solid transformation. This volume change is responsible for a series of casting defects known as shrinkage cavities and shrinkage porosities. Because shrinkage porosity is one of the major defects encountered in foundries, with consequences ranging from high rework cost to casting rejection, development of predictive mathematical models has been an important topic of research for many years. The number of models produced to describe the occurrence of shrinkage porosity in castings is nothing short of impressive (see for example the reviews in Reference 1 and ). Perhaps the most simple of them is the model proposed by Niyama et al. [3], also known as Niyama criterion, which relates the occurrence of microporosity in steel castings to local thermal parameters (e.g., cooling rate and temperature gradient). While widely used in metal casting, the Niyama criterion is, however, only a qualitative indicator. For quantitative predictions more complex models have been developed. In addition to the heat transfer and phase change problem, these models also account for the liquid flow through the interdendritic channels as well as the precipitation out of the melt and growth of gas bubbles [4, 5, 6, 7, 8]. Because the occurrence of porosity is strongly related to the evolution of local pressure of the melt in the interdendritic regions, metal pressure has to be calculated by solving the governing fluid flow equations in the fully liquid regions (Navier-Stokes) and the mushy zone (Darcy's Law) as well. This can be a very laborious task as the numerical schemes for solving the Navier-Stokes equations are complex and, for the problem at hand, the solution convergence may be difficult to obtain. In order to circumvent this problem, Pequet et al. [6] proposed a model that solves the governing equations only Published under licence by Ltd 1
in the mushy zone, but establishing the appropriate boundary conditions around this zone could be by itself a laborious task and source of inaccuracies. A simplified model for calculating the liquid alloy pressure during casting solidification is proposed in this paper. By making use of the continuity condition and taking advantage of specific conditions at the solid boundaries during the solidification process, the governing equations of flow are rewritten in a form that can be integrated for pressure without much effort. Examples of model application to shrinkage porosity prediction in steel castings will be presented. Gas precipitation out of solution and pore growth are not considered at this stage of development and therefore porosity is assumed to occur only because of insufficient feeding with liquid in the interdendritic channels.. Model Description.1. Governing Equations The cooling of metal from its pouring to solidification temperature is accompanied by density changes that induce motion (flow) of the liquid metal in the volume of the casting. Under a pressure gradient, the metal flows toward the area where shrinkage is occurring until the flow is blocked either by the precipitating solid phase, by non-metallic inclusions, or by gas bubbles [9]. By assuming that at a certain moment in time a volume element of the casting is composed of a mixture of solid (s) and liquid (liq) phases as well as of some amount of porosity (p), then the volume fractions (f) of the constituents of the mixture satisfy the relationship: f s + fliq + f p = 1 (1) while the density of the mixture, ρ, is given by: ρ = f s ρ s + fliq ρliq + f p ρ p () The rate of density change and the resulting flow can be calculated from the equations of energy, mass/continuity, and liquid momentum conservation..1.1. Energy Conservation. By neglecting the advection of heat due to the feeding flow, the energy conservation equation can be written as: ( k T ) + ρ H f t ρ C T t = (3) p s f s where C p is specific heat, T is temperature, t is time, k is thermal conductivity, H f is latent heat of fusion, and f s is the volume fraction of solid. For the solid fraction evolution of multicomponent alloys solidifying as solid solutions the model described in Reference 8 has been employed in this work..1.. Continuity. 1 ρliq ρ + V r t where V r is the superficial liquid velocity vector. = 0 (4).1.3. Liquid Momentum Conservation. By assuming that the flow occurs at low Reynolds number (Re << 1), then the Stokes equations can be used in the liquid regions as well as in those solidifying regions that did not reach the point of dendrites coherency (i.e., the liquid + solid mixture can flow together). Beyond the point of dendrites coherency the solid phase becomes entirely immobile and Darcy Law of flow through porous media can be applied. These equations are, respectively: r r P = η V + ρliq g (5) η r r P = V + ρliq g (6) K mz where P is metal pressure, η is the dynamic viscosity of the liquid, g r is the gravity vector, and K mz is permeability of the mushy zone that can be expressed as:
3 ( f ) 1 s K mz s where f s is the solid fraction, λ is the secondary dendrite arm spacing (SDAS), and C λ is an adjustable parameter. For C λ = 1 Equation 7 becomes identical to the expression used for K mz in References 6 and 10. Other expressions for K mz can also be found in the literature, such as in References 4 and 11 for instance. Because they all give quite different values for K mz, the use of the adjustable parameter C λ was chosen in Equation 7. λ = Cλ (7) 180 f Equation 5 and 6 along with the continuity condition in Equation 4 allow for calculating the pressure and flow velocity fields in the solidifying casting. However, as mentioned earlier, the numerical procedure for obtaining the solution could require a high computational effort and therefore makes this type of models unattractive for practical purposes. Consequently, a simplified model is proposed in the next section. It allows one to calculate only the pressure field, but this is not necessarily a shortcoming as all the current models aiming at predicting porosity occurrence in castings are based mainly on the metal pressure and not on fluid velocity... Simplified Model for Metal Pressure The elimination of fluid velocity, V r, from Equation 5 and 6 is at the core of the model. This can be achieved in a straightforward manner by rewriting Equation 5 and 6, respectively, as: r r P = η ( V) + ( ρ liq g) (8) K r ρliqk mz mz r P = V + g (9) η η Both Equation 8 and 9 contain in their right hand side the term V r which can be replaced by the density change term from the continuity equation as seen in Equation 4. Thus, by using the simplifying notation: 1 ρ S = = V r h (10) ρliq t the equations for pressure become: in the liquid: P = η S ( liq gr h + ρ ) (11) K ρliq K mz mz in the mushy zone: P = S + g r h (1) η η In principle, Equation 11 and 1 can be solved numerically without difficulty provided that appropriate boundary conditions are available. For the flow occurring in the mushy zone or Equation 1, the boundary condition at solid boundaries is given by: P = ρ g r liq (13) which can be readily derived from Equation 6 by setting the velocity equal to zero at the solid boundary (mold wall or dendrite root). Because Equation 1 was obtained by applying the divergence operator to Equation 6, it follows that the boundary condition in Equation 13 applies to Equation 1 as well. However, the condition in Equation 13 does not apply to the flow occurring in the Stokes regime in Equation 11 except for the limit case when the liquid is still. Therefore, in order to establish the solid boundary conditions for Equation 11, one has to make some simplifying yet reasonable assumptions related to the specifics of the solidification process and formation of shrinkage porosity. A first observation related to the solidification process is that it begins at the contact with the mould walls and then proceeds inwards toward the thermal center. In addition to that, shrinkage porosity can develop only within the mushy zone and not into the free liquid. Therefore, except for a relatively short early stage, it can be safely assumed that the liquid regions where flow occurs in the Stokes regime are always bounded by a layer of mushy zone where the flow is governed by Darcy's Law. 3
Consequently, solid boundary conditions for Stokes flow, or Equation 11, are not necessary for the problem at hand. The remaining boundary condition is that at the free surface of the melt, which usually is the atmospheric pressure, i.e: P fs = P atm (14) where P fs is the applied pressure on the free surface of the melt, such as the top surface of the downsprue or of the open risers. 3. Numerical Implementation And Computed Results The simplified pressure model for shrinkage porosity prediction was implemented into the Finite- Volume (FV) solidification code SolCAT3D, which is an in-house developed code of Caterpillar Inc. The solidification model employed by the code is a refined version of that presented in Reference 8. Equation 11 and 1 along with the boundary conditions shown in Equation 13 and 14 are used to calculate the pressure field for shrinkage porosity predictions. It is assumed that Stokes type flow occurs in those regions of the casting characterized by an average solid fraction f s < f cr, where f cr is a critical solid fraction at which the transition between the Stokes and Darcy type of flow occurs. The average solid fraction f s is calculated as: f s = N f s, i i= 1 N i= 1 Vol where N is the number of the first nearest neighbours also including the central FV cell, f s,i is the solid fraction of the cell i, and Vol i is the volume of the cell i. Fully solidified metal cells or mould cells are excluded, i.e., Vol i = 0 for such cells. This averaging is used in order to ensure a smooth transition between regions characterized by Stokes' and Darcy's flow regimes, respectively. The pressure field is computed at every time-step during solidification although the solution can be completely unrealistic before a mushy zone of f s f cr develops at the contact with the mould walls. In this work, shrinkage porosity is considered to occur in those regions where metal pressure drops to negative values. The reason of choosing this criterion is because at this stage of development a model for gas porosity was not yet implemented. Similar to the approach in Reference 4, once the porosity begins developing in a certain volume element (i.e., P < P cr = 0) the pressure is no longer calculated and all the subsequent volume changes of the metal will contribute toward the development of shrinkage porosity. Because in this work flow velocity is not explicitly calculated, the pipe shrinkage is evaluated according to the procedure described by Pequet et al. [6]. In order to assess the capability of the proposed model, the solidification of steel castings of chemical composition shown in Table 1 was performed and is presented in this paper. Table 1 also shows the liquidus (T liq ) and solidus (T sol ) temperatures as well as the total shrinkage of the alloy in the solidification interval. T liq and T sol were calculated according to the model presented in Reference 8 where all the thermodynamic parameters employed by the solidification model are also presented. The density change occurring during solidification was calculated based on the relationships provided by Miettinen in Reference 1. The relationship used to calculate SDAS of the alloy was also taken from Reference 1, i.e.: Vol i i (15) 6 0.385 λ, [m] (16) = 150 10 T & where T & is the cooling rate at the liquidus temperature. The value of λ was kept constant for the entire duration of solidification. The critical solid fraction, f cr, at which transition from Stokes to 4
Darcy flow occurs was taken as f cr = 0.75 while for the coefficient C λ used to calculate the permeability of the mushy zone (see Equation 7) a value C λ = 3 was chosen in this work. Table 1. Chemical composition and characteristics for the steel used in this work. C Si Mn P S Cr T liq C T sol C Solidification Shrinkage % 0.19 0.40 1.5 0.0 0.0 0. 1511 1408 4.45 The castings chosen for simulations are presented in Fig.1. The casting presented in Figure 1a is an end-risered plate (5.4 x 140 x 483 mm) also used in the study of Carlson and Beckermann [13], while in Figure 1b and Figure 1c a casting used on Caterpillar's machinery is presented. The difference between Figure 1b and Figure 1c is the number of risers applied to the same casting, i.e., two additional risers applied to the casting in Figure 1c. The four chills shown in Figure 1b were also considered for the configuration shown in Figure 1c, although they cannot be seen on that figure. Mould filling was not simulated and therefore the gating system was not included. The initial melt temperature was set to 1570 C and furan-bonded sand mold was considered for all simulations. Figure 1. The casting geometries used in the simulations (dimensions are shown) The envelope of the calculated shrinkage porosity in the plate casting is shown in Figure, while Figure 3 shows the calculated porosity in the mid-thickness plane of the plate. In Figure 3 the radiograph of a real plate [13] is also presented for comparison. Because the evolution of gas porosity was not simulated, a precise quantitative comparison between the radiograph in Figure 3a and simulated results of Figure 3b is not attempted at this time. However, it can be observed that the extent of the porosity regions in Figure 3a and Figure 3b are very similar. Figure 4 shows calculated pressure and solid fraction distributions in the cross-section of the plate+riser casting just before the shrinkage porosity begins to develop. It can be observed on Figure 4a that at the time when porosity begins to develop the entire liquid alloy is bounded by a solid or mushy zone layer therefore making applicable the boundary condition expressed in Equation 13. The shape exhibited by the solid Figure. The envelope of calculated shrinkage porosity in the plate + riser casting 5
fraction contour near the top of the riser in Figure 4b is because of the piping that develops in that region. Also, the reason the pressure does not exhibit any value in that region (see Figure 4a) is because all the FV cells are either empty or fully solid. (a) (b) Figure 3. Porosity in the mid-thickness plane of the plate casting: (a) - radiograph of the real plate [13]; (b) - calculated porosity Figure 4. The calculated pressure (a) and solid fraction distribution (b) in the cross-section of the plate + riser casting just before the shrinkage porosity begins to develop Figure 5 shows the envelope of the predicted shrinkage porosity for the casting presented in Figure 1b. Various porosity regions, away from the riser, can be observed on this figure. Perhaps the most deleterious for the casting is the porosity predicted in the hot spot marked with A, where it can reach a value as high as 3.5-3.7%. This is not unexpected, given the thinner walls that separates the hot spot from the riser. The region marked as B is also a hot spot but much smaller than that of region A and the predicted porosity there is therefore in the range of 0.9-1.0%. The soundness of the casting can be much improved by using the two additional risers shown in Figure 1c. The predicted outcome for this case is presented in Figure 6 where it can be observed that the porosity of region A has completely disappeared. Also, the porosity region at the horizontal bottom wall of the casting decreased its size considerably. However, region B is still too far from the 6
risers for a proper feeding and therefore the porosity predicted here is practically unchanged when compared to the results shown in Figure 5. A A (a) B (b) Figure 5. The predicted shrinkage porosity in the casting shown in Figure 1b; a)-the envelope of shrinkage porosity; b)-a cross section through the region containing the highest amount of porosity B (a) (b) B Figure 6. The predicted shrinkage porosity in the casting shown in Figure 1c; a)-the envelope of shrinkage porosity; b)-a cross-section view in a plane similar to that shown in Figure 5b. 4. Conclusions In order to reduce the computational effort for predicting the occurrence of shrinkage porosity in castings, a simplified model for calculating the metal pressure during casting solidification has been developed and implemented in Caterpillar's casting/solidification code SolCAT3D. Based on the specific conditions developing during solidification of castings (e.g., small Re number, liquid alloy bounded by a layer of mushy zone), the proposed model eliminates the velocity from the governing equations of fluid flow thus allowing for a straightforward computation of the pressure field. Examples of model application toward predicting shrinkage porosity in a simple end-risered plate as well as on a more complicated casting have been presented. While the location and size of the predicted porosity regions seem reasonable, a quantitative comparison between the predictions and amount of porosity in the real castings has not been attempted at this time. Such a comparison would be appropriate only when the gas precipitating out of the melt is also considered. Further work to include the gas effect is underway. 7
Acknowledgements This work was supported by Caterpillar Inc. Responsibility for opinions and statements contained in this paper are that of the authors, not of Caterpillar Inc. References [1] Lee P D, Chirazi A and See D 001 J. Light Metals 1 15-30 [] Stefanescu D M 005 Int. J. Cast Met. Res. 18 (3) 19-143 [3] Niyama E, Uchida T, Morikawa M and Saito S 198 AFS Cast Met. Res. J. 7 5-63 [4] Sabau A S and Viswanathan S 00 Metall. Mater. Trans. B 33B 43-55 [5] Sung P K, Poirier D R and Felicelli S D 00 Modelling Simul. Mater. Sci. Eng. 10 551-568 [6] Pequet C, Gremaud M and Rappaz M 00 Metall. Mater. Trans. A 33A 095-106 [7] Carlson K D, Lin Z, Beckermann C, Mazurkevich G and Schneider M 006 in 'Modeling of Casting, Welding and Advanced Solidification Processes-XI' ed. Gandin C and Bellet M 67-634 [8] Catalina A V, Leon-Torres J F, Stefanescu D M and Johnson M L 007 in 'Solidification Processing 007' ed. H. Johnes 699-703 [9] Stefanescu D M 00 in 'Science and Engineering of Casting Solidification' 1st ed, Kluwer Academic, New York 41 [10] Kubo K and Pehlke R D 1985 Metall. Mater. Trans. 16B 359-366 [11] Khajeh E and Maier D M 010 Acta Mater. 58 6334-6344 [1] Miettinen J 1997 Metall. Mater. Trans. B 8B 81-97 [13] Carlson K D and Beckermann C 009 Metall. Mater. Trans. A 40A 163-175 8