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Key words : IGT blade; Directional solidification; parallel conputing; numerical simulation
Blades with excellent high temperature performance are required for industrial gas turbines (IGT). However, defects such as stray grains are almost inevitable, which sharply decrease the properties. Liquid metal cooling (LMC) is used as a new process in directional solidification of large blade castings in recent years, and it still needs to be improved. In order to optimize the casting process, the mathematical models of solidification during investment casting of IGT blade were developed. In this model, the convection between the shell and the cooling liquid metal as well as the influence ofthe height of liquid metal surface was taken into account. Due to large size of IGT blade, the massive data in the model consumes very much memory and CPU time. A modified FD method is established to reduce memory usage; parallel computing is developed to speed up the calculation. Based on simulation, the temperature and mushy zone evolution could be studied. Validation experiments were carried out. The cooling curves either from experiment or simulation corresponded well with each other.
The properties of turbine increase with the temperature and pressure of the burning gas. The Hotter the En gine, the Better [1] . Thus, the mechanical property, such as fatigue life, creep resistance, etc, at high temperature of key parts, especially the blades, is very important. Directional solidified (DS) turbine blades are widely used in advanced gas turbines. A very strict process control is needed to prevent from defects occurring. However, defects easily occur during solidification.
In the past 10 years, Gas turbine blade casting is mainly manufactured by Bridgman directional solidification technology [2] . However, during Bridgman process, the mushy zone is sometimes very wide, and usually concave, which will easily result in stray grain. Some times Bridgman process cannot provide enough temperature gradient, especially for the blades with large transverse cross-section. A case in point is the heavy-duty gas turbine blade. In a word, Bridgman method is not appropriate to IGT blades.
In recent years, liquid metal cooling (LMC) is used as a new process in large blade casting [3] , but it still needs to be improved. Fig. 1 is the schematic of a LMC furnace. In this process. low melting-point metal is used as cooling liquid metal. Instead of radiation, heat convection happens between the cooling liquid metal and the shell below the liquid surface. A new way is needed to find more optimized process by LMC without too many experiments.
With the development of solidification simulation technology, Numerical methods provide effective waysto simulate the directional solidification process, and then to optimize the process [4] .studying the directional process and predicting the formation of casting defects are helpful to optimize technical parameters, avoid casting defects and reduce the development cycle and cost, Finite element (FE) and finite difference (FD) are most widely used discrimination methods in computer aided engineering. FE relatively reduces the memory cost. In the other side, FD can solve the microstructure coupled with temperature field, without the complex FE shape function, Computational efficiency is core problem of numerical simulation. With the popularity of computer aided engineering, the numerical models become larger and larger, consuming more and more memory and CPU time. Due to large size of IGT blade, the numerical model of temperature and grain growth of the casting is particularly huge;and it is difficult to be solved by ordinary serial program with tranditional FD method. Recently, with the constant advancement in computer manufacture and technology, computers are developing towards multi-core and multi-processor. Together with multi-core and multi-processor computers, parallel computation can greatly fasten the calculation.
Fig. 1 Schematic of a LMC funace
Some experimental researches has been done by Giamei A F [3] , Elliott A J [5] , Liu C [6] and Zhang J [7] , revealing that higher thermo gradient can be obtained by LMC, which get finer dendrite and precipitation as well as less defect. Haipeng J [8] and Dong H B [9] simulated the directional solidification of aero turbine blade by Bridgman process. They calculated temperature by FE and microstructure by FD. Their Microstructure calculation is limited in the start block and the grain selector because of too massive data, let alone the IGT blade has much larger volume than aero turbine blade. Simulation of temperature in LMC process is attempted only by Elliott A J [10] , Kermanpur A [11] .Simulation of real IGT blade higher than 200mm is not reported.
In this paper, a model of casting, shell and furnace is established. The convection between the cooling liquid metal and the shell as well as the influence of the height of liquid metal surface is taken into account. To cope with the large model and to increase the computational efficiency, a quasi FE method is proposed and applied, as well as multithreading. The temperature evolution and solidification is calculated and the temperature result is validated by experiment.
Heat transferring equation: The non-linear conduction equation is described as follows:
Where T is the temperature, t is the time, ρ is the density, L is the latent heat, c is the specific 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 radiaion exchange volume.
The type of heat transfer at the surface is determined by the value of z .Suppose the height of the cooling liquid metal surface is h , when z > h , the surface element radiates with the furnace wall, a Monte Carlo algorithm [12] is applied to calculate the view factor varying with withdrawal distance. When z < h , the following equation is used to calculate the heat transfer:
Where α is the convection coefficient, T l is the temperature of liquid metal.
h is updated by:
Where h c is the withdrawal distance, S c the area of the horizontal section of mould and casting at the liquid surface level, S p is the area of the horizontal section of the liquid metal container.
FD method is applied to solve the temperature field. In traditional ordinary FD method, the 3D model is discretized into a 3D matrix whose size depends on the smallest circumscribed box of all the geometry. Fig. 2 schematically shows that the vacuum elements in white occupy memory as well, but they are never used in the solution.
Fig. 2 Geometry discretization in traditional FD
To reduce memory, the box method is abandoned. Similar to that of FE, only the solid elements are recorded, ignoring the vacuum elements. The topology relationship is stored in the variable of each element. This algorithm is not used only in temperature calculation;it will be even more helpful in microstructure calculation in further works.
In spite of the reduced memory, the element number is still too large to calculate by single processor, it takes months to solve a casting, which delays the research period. Multi thread program with parallel algorism is constructed to save time (Fig. 3). Except for the temperature, microstructure calculation in future works will also be based upon parallel algorithm.
Fig. 3 Schematic of Parallel computation
The acceleration efficiency is defined as:
Where T s is the calculation time by single thread, T m is the calculation time by multi thread, m is the thread number.
In parallel program with ideal efficiency, this ratio equals to 1, e.g.the speed is strictly in direct proportion to the number of cores. In practical parallel program, it will be less than 1 due to efficiency losses.
Practical process is done in a LMC furnace (Fig. 4). W-Re thermocouples are installed in the casting, protected by quartz tube with 1mm thickness. Slurry is smeared on the thermocouple wires outside the thermocouple hole (Fig. 5). The geometry of the solidified slurry is also measured and meshed in the FD model. Four holes are prepared for thermocouples.
Fig. 4 Practical sample
Fig. 5 Shell ready to be filled in the experiment
Modified FD sharply reduced the memory consumption (Table 1)
Fig. 6 Casting and shell used for memory test
Table 1 Memory consumption of twomethods by the model in Fig. 6
By parallel computing, The speedup of a blade with 12.8×10 4 elements is show in Fig. 7 The calculating speed is defined as the iterative times per millisecond in calculation.
It is revealed that the acceleration efficiency is better with 8 or 4 threads except for single thread. 16 threads got the highest speed, but the acceleration efficiency is lowered by communication between threads.
The speedup of blades in different sizes is shown in Fig. 8.The heights of them vary from 200mm to 600mm. For comparison, the calculation time is normalized as time in hour divided by element number. The speed up ratio deceases sharply with the increase of element number. It has something to do with the Memory access optimization.
Fig. 7 Speedup of a blade
Fig. 8 Speedup of blades in different mesh sizes
As same as speedup, accuracy is also a core problem in calculation. Speedup often leads to Accuracy losses. The results by single and multi thread are compared, indicating the results are hearly identical (Fig. 9). That's to say, multi thread cause no more error than single thread.
A blade in real practical process is calculated (Fig. 10,Fig. 11). With the withdrawing mould, the mushy zone is convex at first. It becomes flat in the middle level, and get concave finally in the upper end.
The calculated temperature curves agree well with the experimental, shown in Fig. 12.The max error is approximately 6%.
Fig. 9 Comparison of results by single and multi thread
Fig. 9 Comparison of results by single and multi thread (Continued)
Fig. 10 Calculated result oftemperature field
Fig. 11 Calculated result of mushy zone
Fig. 12 Comparison of the calculated and experimental results
1) Numerical simulation can be used in directorial solidification of IGT by LMC to predict the temperature Field.
2) In FD modeling, data store and access by quasi FE method largely decreased the cost of computer memory and thus increased the utilization rate of memory.
3) Parallel computing is applied in this calculation;the solution speed is multiplied without significant loss of accuracy.
4) The calculated temperature curves agree well with the experimental. The simulation is demonstrated to be reliable.
The work is financially supporied by National Basic Research Program of China (No. 2005CB724105, 2011CB706801), National Natural Science Foundation of China (No. 10477010), National High Technology Research and Development Program of China (No. 2007AA04Z141), and Important National Science &Technology Specific Projects (2009ZX04006-041-04).
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