Progress on modeling and simulation of directional solidification of superalloy turbine blade casting. *Xu Qingyan, Liu Baicheng, Pan Dong, Yu Jing

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1 Progress on modeling and simulation of directional solidification of superalloy turbine blade casting *Xu Qingyan, Liu Baicheng, Pan Dong, Yu Jing (Key Laboratory for Advanced Materials Processing Technology, MOE, Department of Mechanical Engineering, Tsinghua University, Beijing , China) Abstract: Directional solidified turbine blades of Ni-based superalloy are widely used as key parts of the gas turbine engines. The mechanical properties of the blade are greatly influenced by the final microstructure and the grain orientation determined directly by the grain selector geometry of the casting. In this paper, mathematical models were proposed for three dimensional simulation of the grain growth and microstructure evolution in directional solidification of turbine blade casting. Ray-tracing method was applied to calculate the temperature variation of the blade. Based on the thermo model of heat transfer, the competitive grain growth within the starter block and the spiral of the grain selector, the grain growth in the blade and the microstructure evolution were simulated via a modified Cellular Automaton method. Validation experiments were carried out, and the measured results were compared quantitatively with the predicted results. The simulated cooling curves and microstructures corresponded well with the experimental results. The proposed models could be used to predict the grain morphology and the competitive grain evolution during directional solidification. Key words: Ni-based superalloy; microstructure; directional solidification; modeling CLC numbers: TG /TP391.9 Document code: A Article ID: (2012) Nowadays, Ni-based superalloy turbine blade castings produced by Bridgman directional solidification technology have been widely used in advanced gas turbine for aeronautic industry and energy industry as well [1, 2]. The single crystal turbine blade casting s properties are quite sensitive to the grain orientation, which determined directly by the grain selector geometry of the casting [3, 4]. The final microstructure consisting of large columnar grains of the blade directly determines the casting s mechanical properties. As a powerful tool, numerical simulation technology could be used to study the directional solidification process, predict the final microstructure and optimize the process parameters [5-6]. Great efforts have been put into the study on directional solidification technology by numerical simulation in the last two decades. Earliest researches focused on one-dimensional * Xu Qingyan Male, born in In 1998 he obtained his Ph.D. degree in Materials Science and Engineering from Harbin Institute of Technology. In the same year, he became a postdoctoral research fellow in Tsinghua University and then works there till now. His research interest mainly focuses on the multi-scale modeling and simulation of casting processes. Currently he is paying much more attention on the microstructure simulation of superalloy in directional solidification. scjxqy@tsinghua.edu.cn Received: ; Accepted: model [7]. Yu [8, 9] studied the directional solidification process through commercial FEM software and developed the defects map which represented the correlation between the thermal gradient and casting defects. A two-dimensional model was developed by Saitou and Hirata [10] to analyze the shape of liquid-solid interface in Bridgman directional solidification. Galantucci et al. [11] investigated the directional solidification process of turbine blades through 2D FEM method with comparison of experimental results. In consideration of heat conduction and radiation, Wang and Overfelt [12] presented a new two-dimensional model which considered the variable radiation view factor with the withdrawal process. Zhu and Kimatsuka et al. [13] focused on the boundary of heat radiation in directional solidification and developed the software package using regular and irregular mixed grids. Li et al. [14, 15] investigated the solidification process of single crystal investment castings by ProCast, a commercial software. Our research group did a lot of work on the microstructure modeling of the directional solidified turbine blades [16-18]. In this paper, a mathematical model based on the modified CA-FD method was developed for the three dimensional simulation of the microstructure evolution in the directional solidification of Ni-based superalloy turbine blades. The model was used to investigate the temperature distribution and grain growth of the casting during directional solidification. 69

2 CHINA FOUNDRY Vol.9 No.1 Validation experiments were carried out, and particular attention was paid to the grain morphology and grain orientation of primary <001> dendrite, with respect to the longitudinal axis of the turbine blade casting. The simulated cooling curves and final microstructures were compared quantitatively with the experimental results. 1 Physical and mathematical models 1.1 Physical model The simplified schematic of the directional solidification process for turbine blade casting is shown in Fig. 1. In this model, the inner geometry of the furnace, the pattern of the blades on the chill plate and the withdrawal rate are most important. calculated as follows: Q net, i = Ai qout, i j= i j where, q out,i is outgoing radiation energy flux of surface i. The incoming radiation energy is a combination of the outgoing radiant energy from all the surfaces being visible to surface i. The view factor φ i-j is the ratio of radiant energy leaving from surface j to surface i to all the radiant energy leaving surface j. Then the Q net,i can be regarded as sum of net heat radiation exchange between surface i and each domain, seen in Eq. (4). 1 out, j i 1 Qnet,i domain = Ai ( Ti Tdomain) + q i i, domain (2) (3) Q net = Q, i n net, i domain domain= 1 (4) In this paper, the n domains are labeled by n rays from center of surface i to outer space respectively. Figure 2 shows the definition of ray. The rays are scattered regularly in the space 1 - heat radiation, 2 - blade (covered by shell), 3 - baffle, 4 - chill plate, 5 - cycling water, 6 - furnace, 7 - heating zone, 8 - cooling zone, 9 - withdrawal unit Fig. 1: Schematic of directional solidification process of superalloy turbine blade 1.2 Mathematical models Macro heat transfer There exists complex heat transfer during the directional solidification progress. The macro temperature distribution within the casting and shell was calculated according to the transient non-linear heat conduction equation as follows: T T T T fs c = + + (1) L + QR t x y z t where T represents the temperature, t is the time, ρ is the density, c is the specific heat, L is the latent heat, λ is the heat conductivity, x, y and z are the coordinates; f S is the mass fraction of solid phase, Q R is the heat exchange between the casting surface and the ambient environment by radiation. It is difficult to calculate the heat radiation in directional solidification because of the memory required to store the view factors for each surface cell against others. In this paper, Ray Tracing model derived from Monte Carlo method [19] was adopted to deal with the heat radiation between every two discrete surfaces. For the discrete surface i, such as a surface of a finite difference element, the net heat exchange, Q net, i, can be Fig. 2: Definition of the ray in three dimensional space according to the angles φ and θ Microstructure nucleation and growth The microstructure simulation was based on the modified CA method. A continuous nucleation model [20] was employed to calculate the nucleus number in the undercooled melts as follows: n( T ) = dn = d( T ') T 0 [1 f N max S 2π T dn ( T ')] d( T ') d( T ') σ 1 T ' T exp[ ( 2 T where ΔT is the undercooling, n(δt) is the nucleus density, N max is the maximum nucleus density, ΔT σ is the standard deviation of the distribution, ΔT N is the average nucleation undercooling, f S (ΔT ') is the fraction of solid phase. The grain growth was calculated based on the KGT model [21]. The growth speed of dendrite tip was calculated as follows: v( T) = α 1 T + α 2 T where α 1 and α 2 are the coefficients. σ 2 3 N ) 2 ] (5) (6) (7) 70

3 1.2.3 Competitive growth of grains In modified CA model [17-18], each grain s orientation was randomly determined. Nevertheless, the growth velocity of the grain tip was equally calculated by the temperature gradient in front of the dendrite tip. Therefore, the grains which were not well-aligned with respect to the maximum gradient of the temperature field would grow at a much slower speed than those which were best aligned, which made them grow behind the growth front. This made it possible for those well-aligned grains to have their secondary and tertiary side arms grow out and occupy the space just in front of the less well-aligned grains, as shown in Fig. 3. This is known as the grain s competitive growth. Fig. 3: Schematic of grain competitive growth Layer by layer method Figure 4 shows the schematic of simulation for microstructure in directional solidification process. At every macro time step, we could pick up the mushy zone, as shown in the left of Fig. 4. The mushy zone included several layers in longitudinal direction in macro scale. We divided every layer into many cells further at micro-scale. For example we could subdivide each macro layer into eight sublayers. At the same time the sublayers were divided into micro cells by the same micro step in x and y directions. These sublayers were connected in longitudinal direction. We combined some sublayers in the neighboring and the local layers to form the micro computational domain, shown in Fig. 4. The accessorial sublayers were added to remain the continuum of grains. The micro cells in the sublayers have some properties inherit from the macro cells, such as temperature, concentration. At a macro time step, we calculated the microstructure evolution in the micro computational domain. The micro computational domain stored in memory is smaller than the mushy zone because it only included one macro layer. So we have to replace the data in computational domain one macro layer by one macro layer Rule of cell s capture In the program we define a new capture rule to deal with the grain growth in random preferential directions in the regular finite element grids. First, we divide the interface micro cell into two groups. One are those which become interface cell from liquid cell by nucleation, the other are those captured by neighbor solid cell. The cell of former group has six preferential growth directions which are set randomly, such as [100], [010], [001] and their negative directions. When the solid fraction of the cell of the former group increases from 0 to 1, it will capture the surrounding liquid cells in the six directions which are called child directions. The cell of latter group also has six preferential growth directions, but one direction is set as the same with that captured cell. When the solid fraction increases to 1, it will only capture liquid cells in five directions. In the five directions, the one which is the same as the capturing cell is called father direction. The other four directions are called child directions. That means that one cell could have six children directions or one father direction and four children directions alternatively. If a liquid cell is captured by an interface cell in the child direction of the interface cell, the children direction of the interface cell is passed to the liquid cell as its father direction, and the interface cell is regarded as the father cell of the liquid cell. If a liquid cell is captured by an interface cell in its father direction, the father direction and the father cell are passed to the liquid cell together. The offset caused by finite difference cells during capturing process should be considered when the cell captures neighbor cell in its father direction. Figure 5 is 2D schematic of grain growth from nucleation to capturing other cells step by step. 2 Microstructure simulation of turbine blade in directional solidification process Fig. 4: Schematic of layer by layer computational method 2.1 Numerical modeling of directional solidified (DS) turbine blade Figure 6 shows the manufacturing process of directional solidified (DS) turbine blades. A group of twelve turbine blade castings were produced together. The 3D geometry of the blade is shown in Fig. 7. Six thermocouples were placed on the 71

4 CHINA FOUNDRY Vol.9 No.1 (c). (d) (e) (f) Liquidus cell Captured cell Additionally captured cell for keeping the dendrite arm continuous Fig. 5: 2D schematics of capturing rules of grain growth, ->->(c)->(d)->(e)->(f), and,, (c), (d), (e), (f) represent original solidification moments respectively P1 P3 P5 P2 P4 P6 Fig. 6: Manufacturing process of DS turbine blades surface of the casting at different heights for the temperature measurement in the experiment, see also in Fig. 7 for the thermocouple locations. Figure 8 shows the simulated and experimental cooling curves at those positions indicated in Fig. 7. The comparison of the simulated and experimental results shows a favorable agreement. That indicates the model proposed for the Fig. 7: Casting shape and thermocouple locations temperature simulation is reliable enough to serve as the foundation of microstructure simulation. Figure 9 shows the simulated grain evolution progress during solidification at different times, in which the liquid is not displayed so that the solid-liquid interface could be clearly shown. Figure 10 shows the simulated and experimental microstructure morphologies of the turbine blade. 72

5 Fig. 8: Simulated and experimental cooling curves at different points of DS blade casting It is shown that the simulated microstructure of the turbine blade has similar texture and grain size with the experimental, which shows the existence of dozen of large columnar grains in the blade which lie along the direction of the withdrawal (also the axes of the blade). The simulated results show that a great number of tiny grains appear at the bottom of the blade shortly after the beginning of the withdrawal progress. When growing upwards, those grains with a more favorable crystal growth direction, which had a small angle with the heat flux direction, get more chances to grow bigger and finally grew into several dozen of large columnar grains. This shows that the competitive grain growth progresses during directional solidification, and reproduces the formation of the typical large columnar grains in the directional solidified turbine blade castings. t = 8 min t = 20 min (c) t = 36 min Fig. 9: Simulated grain evolution progress Fig. 10: Simulated and experimental microstructure morphology of the blade 2.2 Numerical modeling of single crystal (SX) turbine blades The alloy DD6, a second generation single crystal superalloy, was used for the production of sample blades. Chemical composition and thermophysical data of the alloy were referred to Ref. [22] The three dimensional sample blade model was shown in Fig. 11. The locations of thermocouples and crosssections for microstructure observation were also shown in Fig. 11. Some simulated and experimental cooling curves at special positions in Fig. 11 are shown in Fig. 12 and agree well. From the comparison of simulated and experimental cooling curves, it could be seen that the ray tracing model was efficient. Figure 13 is the temperature distribution of SX model in solidification process at the withdrawal rate of 200 mm h -1. Similar with the DS blade, the solid/liquid interface is oblique. The outside of the interface is higher than the inside. And the outside temperature in front of the interface is lower than the inside, which agreed well with the experimental cooling curves. Figure 14 shows the microstructure in mushy zone at different withdrawal time in two groups of experiments. In the first group we could see stray grains formed at platform in simulation results and real blades. However, in the second group, the simulation results were no longer stable because of the randomness of nucleation. Figure 14 also shows the appearance 73

6 CHINA FOUNDRY Vol.9 No.1 of the platform in the blade without stray grains. The final microstructure of the total blade was shown in Fig. 15. Figure 15 shows the picture of real blade, from that we could observe the stray grain in the corner of platform directly. The simulated and experimental results were coincident. Fig. 11: Schematic of locations of thermocouples 2.3 Numerical modeling of grain selection of SX turbine blades Figure 16 shows the manufacturing process of DS turbine blades. A group of twelve turbine blade castings were produced together. The 3D geometry of the blade is shown in Fig. 17. Two thermocouples were placed on the surface of the casting at different heights for the temperature measurement in the experiment. Figure 18 shows the simulated and experimental cooling curves at Points 1 and 2 during the solidification. The simulated cooling curves at both points match the experimental results well in general. Figures 19 and show the simulated and Fig. 12: Comparison of experimental and simulated cooling curves of SX turbine blade ( ) ( ) ( ) 15 min 30 min (c) 45 min Fig. 13: Temperature distribution of SX model at different solidification time 74

7 (d) (c) (e) Fig. 14: Experimental and simulated defects at platform:, (d) simulated results;, (e) experimental results; (c) positions of simulation domain Fig. 15: Microstructure of blade: simulation experiment experimental microstructure of the starter block. The simulated microstructure has similar texture with the experimental, and the sizes of the column grain are also similar to the experimental. Figure 20 shows the grain evolution at different solidification time. Different colours represent different grains. The liquid is not displayed in the picture, so that the growth interface is clearly shown. The results show that a great many small grains appear at the bottom of the starter block and grow upwards, and only a few grains survives in the starter block and grow to a certain size. At the top of the starter block, several grains are allowed to grow into the pigtail. In the pigtail, the grains growth is limited and interpreted by the shape of the pigtail, and just one of them survives in this selection in the middle height of the pigtail, which grows bigger and finally occupies Fig. 16: Schematic of Bridgman directional solidification process Fig. 17: Schematic of casting shape and thermocouple locations 75

8 CHINA FOUNDRY Vol.9 No.1 Fig. 18: Simulated and experimental cooling curves at different points Figure 21 shows the simulated and measured grain number at different heights of the grain selector, in which the Electron BackScatter Diffraction (EBSD) method was used for the experimental measurement of grain numbers at each transverse section. Good agreement was observed between the simulated and experimental results. It is indicated that a great number of tiny equiaxed grains emerge at the bottom surface of the starter block, and transfer into a much smaller number of columnar grains when growing upwards, which is well known as the grain competition process determined by the heat flux direction and the grain s fastest growth direction. The grain number decreases quickly in the starter block, leaving only less than 10 columnar grains growing into the spiral. The grains in the spiral continue to decrease when growing upwards, and only one of them survives from this grain selection, which finally grows into a whole single crystal casting. Fig. 19: Simulated and experimental microstructure of the starter block t =10.0 min t =12.0 min (c) t =14.5 min (d) t =22.0 min Fig. 20: Simulated grain evolution of Ni based DD6 superalloy the whole casting at the top, known as single crystal. This result coincides with the experimental. This indicates that the geometry of that grain selector is efficient to produce a single crystal casting for the Ni based DD6 superalloy. Fig. 21: Predicted and measured grain number at different positions of the grain selector 3 Conclusions (1) Mathematical models based on CA-FD method were developed for three dimensional simulation of grain selection and grain evolution in directional solidification process of Ni-based superalloy turbine blade casting. Based on heat transfer modeling of the directional withdrawing process, the competitive grain growth within the starter block and the spiral were simulated by using Cellular Automaton (CA) method. (2) The simulated cooling curves and microstructures agree well with the experimental validation results. It indicated that the model is able to quantitatively reproduce the temperature distribution and microstructure morphology of the casting during directional solidification. References [1] Reed R C. The Superlloys Fundamentals and Applications. USA: Cambridge University Press, [2] Schafrik R E, Scott Walston. Challenges for high temperature materials in the new millennium. In: Proceedings of the 11th International Symposium on Superalloys, 2008:

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