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Inverse identification of interfacial heat transfer coefficient between the casting and metal mold using neural network
Energy Conversion and Management 51 (2010) 1898–1904
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Inverse identification of interfacial heat transfer coefficient between the casting and metal mold using neural network Liqiang Zhang a,b, Luoxing Li a,b,*, Hui Ju b, Biwu Zhu b a b
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082 Hunan, PR China College of Materials Science Engineering, Hunan University, Changsha, 410082 Hunan, PR China
a r t i c l e
i n f o
Article history: Received 30 November 2008 Received in revised form 23 July 2009 Accepted 27 February 2010 Available online 19 March 2010 Keywords: Neural network Finite element Interfacial heat transfer coefficient Inverse method
a b s t r a c t The effect of the heat transfer coefficient at the casting-mold interface is of prime importance to improve the casting quality, especially for castings in metal molds. However, it is difficult to determine the values of heat transfer coefficient from experiments due to the influence of various factors, such as contacting pressure, oxides on surfaces, roughness of surfaces, coating material, coating thickness and gap formation caused by the deformation of casting and mold, etc. In the present paper, the interfacial heat transfer coefficient (IHTC) between the casting and metal mold is identified by using the method of inverse analysis based on measured temperatures, neural network with back-propagation algorithm and numerical simulation. Then, by applying the identified IHTC in finite element analysis, the comparison between numerical calculated and experimental results is made to verify the correctness of method. The results show that the numerical calculated temperatures are in good agreement with experimental ones. These demonstrate that the method of inverse analysis is a feasible and effective tool for determination of the casting-mold IHTC. In addition, it is found that the identified IHTC varies with time during the casting solidification and varies in the range of about 100–3200 Wm2K1. The characteristics of the time-varying IHTC have also been discussed. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction With the rapid development of numerical simulation technology in the last two decades, the solidification simulation of casting has been taken as an effective tool for designing the casting process and improving the quality of casting [1,2]. However, some uncertainties must be eliminated before such simulations can be widely accepted as realistic descriptions of the process. The heat transfer at the casting-mold interface is one of these uncertainties. In general, it is difficult to determine the values of heat transfer coefficient from experiment due to the influence of various factors, such as contacting pressure, oxides on surfaces, roughness of surfaces, coating material, coating thickness and gap formation caused by the deformation of casting and mold [3–5], etc., especially for castings in metal molds. Fortunately, in recent years, the method of inverse analysis based on the field measurements has been rapidly developed and applied successfully to solve many complex engineering problems [6–8]. The essence of the method of inverse analysis is the realization of optimization techniques. Over the past few years, the interest in artificial neural networks has grown shar-
* Corresponding author. Address: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082 Hunan, PR China. Tel./fax: +86 7318821950. E-mail address: [email protected] (L. Li). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.02.020
ply for the more effective application of method of inverse analysis in complex problems. In the field of engineering science and computational mechanics, the research and applications of neural networks are very active and successful. The back-propagated multilayered network is one of the main types applied to engineering. The related works concern almost all topics of engineering science and mechanics [9–11], such as structural identification, parameter estimation for nonlinear finite element analyses, equation solver, identification of material characterization and fault diagnoses, etc. In the work, the casting-metal mold IHTC is identified by using the method of inverse analysis based on measured temperatures, neural network with back-propagation algorithm and numerical simulation. Then, applying the identified IHTC, the experimental and numerical calculated temperatures at various locations in the casting and mold are compared to verify the feasibility of the method for determination of the casting-mold IHTC. In addition, the characteristics of the time-varying IHTC have also been discussed in the view of thermal deformation, solidification contraction and influence of coating. 2. Experimental procedure For identifying the IHTC using the inverse method, two gravity die-casing experiments were selected to obtain the temperature
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data. All process conditions of the two experiments were the same except for the initial mold temperature. The initial mold temperature of the first experiment was 350 °C. The second one was 450 °C. The schematic diagram of experimental setup with the casting arrangement and the position of the thermocouples are shown in Fig. 1. In Fig. 1, T11, T12, T13 and T14 were the positions of the placed thermocouples of the first experiment. T21, T22, T23 and T24 were the measured positions of the second experiment. Four K-type thermocouples with the diameter of 0.2 mm protected by insulated bushing were inserted into the die cavity. For the inverse method to be applied successfully it is also necessary to optimize the location of the thermocouples. It should not be too close to the interface because the temperature may not be representative and the inverse method might amplify the noise present in the sampled data (stochastic response). It should also not be too far from the interface because the inverse method could become unstable [12]. Therefore, these thermocouples were located at 5 mm, 10 mm and 20 mm from the casting-mold interface respectively (see Fig. 1). The casting and mold temperatures were measured with a thermocouple at 0.3 s of time intervals. All thermocouples were connected by cables to a data acquisition system, which consisted of a MCT-2 data acquisition/switch unit and a computer. The experiments were performed with an A356 alloy (Al–7Si– 0.4Mg). It was melted in an electrically heated furnace using a graphite crucible under the protection of a cover nitrogen gas and pouring temperature was about 730 °C. The die material was low carbon steel. Its dimensions are shown in Fig. 1. The thermophysical parameters for each material are summarized in Table 1. Before the pouring begin, the die was imbedded in adiabatic wool of 5 mm thickness for improved surrounding heat isolation environment.
3. Inverse identification process for the casting-mold IHTC The inverse solving process for the casting-mold IHTC is shown in Fig. 2. It includes: (i) initial training of the NN model using the initial training data containing a set of assumed IHTC and their corresponding temperature distributions calculated based on numerical simulation, (ii) recognition of the casting-mold IHTC using the trained NN model by feeding in the measured temperatures varied with time, (iii) comparing the identified IHTC with assumed IHTC in initial training samples, if the identified IHTC is not in the content of the input training samples, the NN model shall be retrained by adjusting the training samples in order to obtain an improved IHTC and (iv) recognition of the transient IHTC between the casting and mold for the new time interval by repeating the above three processes. The detail of each part of this process will be given in the following sections. 3.1. Neural networks A NN model is referred to as a type of computational models that consists of hidden-layer neurons connected between the input and output neurons. The connections between the neurons are described by weights which are determined through training. The nonlinear hyperbolic functions are usually used as the activation functions to increase the modeling flexibility. In this work, the two-hidden layer NN model with three inputs and one output was adopted, as shown in Fig. 3. The back-propagation algorithm was employed as the learning algorithm. Using a BP neural network to solve the IHTC involves two basic stages, a learning stage and a recalling stage. The learning process is a supervised learning process which uses a training model and a set of target output values from the problem domain. In the recall-
Fig. 1. Schematic diagram of the experimental setup (all dimensions in mm).
Table 1 Thermal-physical parameters of alloy and die. Material
Density (kg m3)
Specific heat kJ (kg k)1
Latent heat (kJ kg1)
Thermal conductivity w (m k1)
Solidus temperature (k)
Liquidus temperature (k)
Casting (A356) Die (mild steel)
(2.42.7) 103 (6.97.8) 103
0.881.19 0.45–1.15
29.31294 53.71341
92.8184 27.260.2
829 –
889 –
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L. Zhang et al. / Energy Conversion and Management 51 (2010) 1898–1904
Fig. 2. Flow chart for the determination of casting-mold IHTC.
where g, a and r are defined as the learning rate, the momentum rate, and the iterative number, respectively. The derivatives in Eq. (3) can be written as:
@EðWÞ@ v ki @E ¼ dki ojk1 ; k @ v ki @wkij @wij
ð4Þ
where the wkij is the weight coefficient between the ith neuron in the kth layer and the jth neuron in the ðk þ 1Þth layer, v ki and okj is the input and output from the jth neuron of the kth layer respectively.
v ki ¼
X
wkij ok1 þ h; j
okj ¼ f
v ki
ð5Þ
j
dki can be expressed as: Fig. 3. A two-hidden layer NN model.
ing stage, the neural network can provide an appropriate output for any arbitrary unknown input. The goal of the training process is to modify the weights which characterize the BP neural network such that the actual output vector Yp approximates the target output vector Y t as closely as possible. A brief description of the formulation of the BP is given as follows. The error norm E between the determined output Yp vector and the targeted output vector Yt is defined as:
EðWÞ ¼ kY p Y t k2 :
ð1Þ
The operator kk2 represents the vector norm. The weight matrix is adjusted iteratively based on the following equations:
W rþ1 ¼ W r þ DW r ;
ð2Þ
@E @E DW r ¼ g þ ag ; @W W¼W r @W W¼W r1
ð3Þ
8 < fðyti yi Þf 0 ðv ki Þ; k di ¼ ðP dkþ1 wkþ1 Þf 0 ðv k Þ; : j i ji
k¼3 k–3
ð6Þ
j
In Eq. (5), h is the error derivative of back-propagation. The value should be between 0 and 0.15, and can be varied during the training process. The purpose of adding this small positive value is to prevent the weight matrix from stagnation. In Eq. (6) f0 (v ki ) is the first derivative of the activation function f() with respect to v ki . In this work, the activation function is given by the smooth sigmoid function.
f ðv i Þ ¼
1 1 þ ev i
ð7Þ
3.2. Training samples The samples for the initial training of the NN model consist of a number of sets of inputs and outputs. Firstly, in order to describe
L. Zhang et al. / Energy Conversion and Management 51 (2010) 1898–1904
the inverse characteristics of the IHTC calculation, these training samples including both the initial training and the retraining should be carefully selected. And these training samples should cover all possible values of IHTC. In this work, the training samples were obtained by using finite-element method instead of carrying out an actual experiment. Before obtaining the training samples, a function specification procedure proposed by Beck [4] was adopted to calculate time-varying IHTC at the interface between the casting and mold. The sequential function specification procedure is to assume temporarily that several future heat transfer coefficients are time invariant and the function can be a sequence of constant segments or of other forms such as straight line segments, parabolas, etc. According to Beck’s method, the time-varying IHTC was separated into a number of transient IHTC with a certain time interval in the inverse solver process. For more stable and faster results, the transient heat transfer coefficients were calculated for every 2 s interval from the time of pouring to 200 s after pouring. And then, a suitable initial value of transient IHTC was assumed and with this value, the temperature distribution of casting was obtained by using a commercial finite-element package, ProCAST to obtain the training samples. Compared with several other commercially available packages, ProCAST was chosen as the platform for the numerical simulation operation because it has the ability to develop user-programmable routines to describe dependent boundary conditions spatially and temporally [13]. Secondly, for successfully utilizing NN model, the inputs of the NN model should be carefully chosen so that the variation in the outputs can be truthfully reflected by the changes of these inputs. In addition, inputs for the NN model should be easily and accurately measured and be sensitive to the changes of the outputs. However, it is well known that the temperature changes are chiefly affected by the IHTC value during the heat transferring between two domains. Therefore, in the NN model, the temperatures at different time were expressed as x1 ; x2 and x3 , respectively and were used as the inputs for the NN model. The outputs of the NN model were the IHTC value, expressed as y in Fig. 3. 3.3. Normalization of training data sets The NN model requires the normalization of the input and the output data. As the sigmoid transfer function is used in the BP algorithm, the system cannot actually reach its extreme values of 0 and 1 without infinitely large weights. However, it is found better, in practice, to normalize the input patterns as well as output patterns to between 0.1 and 0.9 [7]. The inputs of the training samples are normalized linearly based on the following formulas:
v i ¼
vi vi min þ e1 vi max vi min þ e2
ð8Þ
where xi min and xi max are the minimal and maximal values of the ith input value xi, respectively in the sample data set; xi is the normalized value of parameter x ranging between 0.1 and 0.9. The e1 , e2 (0 6 e1 < e2 6 1) are the scaling factors for ensuring that the normalized values would not be close to 0 or 1. The outputs can be normalized in the exactly same way. 4. Results and discussions 4.1. Experimental cooling curve Fig. 4 shows the temperature variation with time measured at four locations in the casting with the initial mold temperature of 350 °C. The measured locations are described in Fig. 1. The distance to the interface between the casting and mold is all 10 mm at the location 1, 2 and 3 (T11, T12 and T13 as shown in Fig. 1). They are
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Fig. 4. Measured temperatures vs time curve with the initial mold temperature of 350 °C.
the same and close to the interface. The location 4 (T14 as shown in Fig. 1) is far away from the interface between the casting and mold, close to the center of casting and is 20 mm apart to the interface. In Fig. 4, it is obviously seen that the general shape of the 4 sets of cooling curve is similar. In the initial stage, the temperatures decrease rapidly. When they decrease down to about 550 °C, the temperatures are almost invariant for a short time with about 10 s. Then, the temperatures continue decreasing with a rapid velocity again. Whereas, the change of temperature at location four lag compared with the one in location 1, 2 and 3 and if the location 4 is removed, a better uniformity for the curves of temperature variation can be observed at location 1, 2 and 3. This demonstrates clearly that the experimental temperature differences perpendicular to the heat flow are very small with less than 5 °C and the temperature in casting change mainly along with the direction of heat flow in the improved surrounding heat isolation environment. It can be well known that such temperature distribution is greatly helpful to reduce the influence of noise and improve the accuracy of the inverse method. For identifying the IHTC, among the cooling curves obtained, the T11 data measured near to the interface was used as the known temperature history. The measured temperature curves at the other locations were used to test the accuracy of the identified IHTC. 4.2. Summary of the identified IHTC The identified IHTC between the casting and mold is shown in Fig. 5 together with the measured temperature change with time at location 1. In the inverse solving process as discussed above, the experimental temperature data at location 1 was used as the known temperature history for identifying the IHTC. As shown in Fig. 5, it can be clearly seen that the identified IHTC varies with time and can be divided into mainly three stages from the initial moment of the casting process until the completed solidification situation is reached. In the initial period of the casting solidification, the IHTC is about 250 Wm2K1 and is an ascending trend. The first stage lasts only up to 15 s. Then, the IHTC changes into the second stage. In this stage, the IHTC increases up to a maximum value of 3241 Wm2K1 in first and then decrease gradually. At about 43 s, the IHTC decreases rapidly down to a minimum value of 91.6 Wm2K1. As the temperature–time curve shown in Fig. 5, it can be known that the casting temperature is near solidus curve in the time which IHTC decreases sharply. In the third stage of the IHTC variation with time, The IHTC increases slowly up from the minimum value of 91.6 Wm2K1 to about 500 Wm2K1 and then stay constant. The variation trends and values of IHTC show
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Fig. 5. Variation of identified IHTC and measured temperature change with time.
good agreement with those in the literature [14,15]. For example, in Ref. [15], the authors reported the maximum value of IHTC for A356 alloy be nearly 2900 Wm2K1 and then decreasing fast down to around 200 Wm2K1. 4.2.1. First stage In the initial stage, the IHTC value between the casting and mold is low and increases slowly. The first stage last only about 15 s. It can be interpreted in two ways. One may be attributed to the thermal resistance of the coating and the other is the physically improving contact at the casting-mold interface. In the beginning of casting solidification, the heats transferred from the casting to the mold have to pass through the coating. Therefore, it needs some time to make the heat transfer process between the casting and the coating stable. According to Ref. [16,17], the ability of thermal resistance is different with the difference of coating materials or thickness and affects heavily the solidification time of the casting. On the other hand, the physical contact between the melt and mold would be improved as the melt level increases during pouring and the entrapped air in the cavity of mold escapes. After pouring, as the surface temperature of the mold increases the conductivity of air enclosed in the casting-mold interface also increases because the thermal conductivity of air varies with temperature, which increases finally the IHTC. It may be considered as a major reason for the increase of the IHTC in the initial period of casting solidification. 4.2.2. Second stage In the second stage of IHTC variation with time as shown in Fig. 5, it can be obviously seen that the casting-mold IHTC increases rapidly up to its maximum value firstly and then decreases gradually. Finally, when the temperatures of casting decrease down to solidus curve, the IHTC decreases sharply down to its minimum value. The rapid increase of IHTC is due to the formation of the perfect contact between the mold and the liquid metal. In the early stage of casting solidification, the liquid metal has a best contact with the mold surface and the cooling rate of casting is mainly affected by the contact type of casting-mold interface and the thermal conductivity of mold. Therefore, after the resistances of the coating and enclosed air are overcame in previous time, the IHTC reaches rapidly its maximum value. The gradual decreasing of IHTC can be attributed to the formation of shell in the interface between the casting and mold. During the liquid metal filling mold cavity, no shell formation of solid metal on the mold surface occurs due to the turbulences. When the filling is completed, the turbulences die out. A thin stable shell of solidified metal is formed on the mold because the mold extracts heat from the liquid metal and the
thickness of shell increases gradually with the casting heat extracted continuously. After the shell is formed, the interface is pressed against the mold by the hydrostatic pressure of the liquid metal and the hydrostatic pressure decreases with increasing of the solid shell thickness. This will result in the gradual decreasing of IHTC between the casting and mold with time. In addition, it can be clearly known that the temperature of casting is over solidus curve in the period of gradual decreasing of IHTC as shown in Fig. 5. When the temperature continues decreasing and reaches to the solidus curve, the casting-mold IHTC decreases rapidly. The rapid decreasing of IHTC is due to the formation of gap in the interface between the solidified casting and mold. After formation of an adequate solid metal on the chill surface, the perfect contact between the chill and the solidified casting no longer exists because of the contraction of the solidified casting. A thin gap is formed in the casting-mold interface. The heat transfers through the gap at interface by means of convection, radiation and gas conduction. This will have a great effect on the change of casting-mold IHTC. To sum up, the heat transfers at the casting-mold interface are complex in this stage. Many parameters, such as the wettability of the liquid metal on the mold surface, surface roughness of the mold, thermal conductivity of mold, hydrostatic pressure and turbulence of melt, etc., affect heavily the change of IHTC. 4.2.3. Third stage In the third stage, the casting-mold IHTC increases gradually up to a higher value and then remain at a constant value. Fig. 5 shows that the temperature of casting is under solidus curve in this stage. It can be well known that the solidification of casting in the mold has been completed here. The effect of the solidified casting contraction might be balanced by the thermal expansion of the mold. And that might produce a constant contact pressure between the casting and mold surfaces. A stable interface has possibly been formed at the surface between the casting and mold. This can contribute to a stable heat transfer through the interface from the casting to the mold. In addition, the thermal conductivity of the solid metal is invariant after the solidification of casting completed. This may be taken as a major factor for IHTC to remain a constant. 4.3. Verification of the identified heat transfer coefficients As mentioned above, the measured temperature curves at the location 2, 3 and 4 in the casting of the first experiment were used to test the accuracy of the identified IHTC. For the verification of the identified IHTC, the temperature distribution was calculated by feeding the identified IHTC into the ProCAST with the same boundary condition and then compared with the measured temperatures with the corresponding locations. The results are shown in Fig. 6. It can be obviously seen that a good agreement is obtained between the simulated and experimental temperatures. Fig. 7 shows that the comparison of simulated and measured temperatures with initial mold temperature of 450 °C. The numerical calculated temperatures are also obtained by feeding the identified IHTC into ProCAST with the same boundary condition as the experiment. TS1, TS2 , TS3 and TS4 is the calculated temperatures. Te1, Te2, Te3 and Te4 is the measured temperatures (T21, T22, T23 and T24 as shown in Fig. 1). Among them the TS4 and Te4 are the temperature curves of the mold, the others are the ones of the casting. It is clearly seen that the agreement of the calculations and measurements is also excellent. The maximal temperature differences are less than 5 °C and the mean temperature differences are less than 2 °C between the numerical calculation and measurements. The results further confirm that the proposed inverse method can be applied to identify the IHTC between the castings and mold accurately and reliably.
L. Zhang et al. / Energy Conversion and Management 51 (2010) 1898–1904
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Fig. 6. Comparison of simulated and measured temperatures with initial mold temperature of 350 °C: (a) location two, (b) location three and (c) location four.
Fig. 7. Comparison of simulated and measured temperatures with initial mold temperature of 450 °C.
5. Conclusions The calculated results have shown that the method of inverse analysis is applied successfully to determine the IHTC between the casting and metal mold. The major results of this paper can be summarized as follows: 1. The casting-metal mold IHTC has been successfully determined by using the method of inverse analysis based on measured temperatures at the various locations of casting, neural network with back-propagation algorithm and numerical simulation.
2. The correctness of method has been verified through the comparison between numerical calculated and experimental results based on the identified IHTC used in finite element analysis. The results show that the numerical calculated temperatures are in well agreement with experimental ones. It is adequately demonstrated that the method of inverse analysis is a feasible and effective tool for determination of the casting-mold IHTC. 3. The identified IHTC varies with time during the casting solidification and the values have varied in the range of about 100– 3200 Wm2K1. The variation trends and values of IHTC have shown good agreement with those in the literature. Moreover, the IHTC variation with time is complex when the casting temperature is over solidus curve. However, it is almost invariant and remains a constant under solidus curve. This may be attributed to the complex heat transfer during the casting solidification. During the entire period of casting solidification, many factors, such as the wettability of the liquid metal on the mold surface, surface roughness of the mold, thermal conductivity of mold, hydrostatic pressure and turbulence of melt, etc., affect heavily the heat transfer between the casting and mold and finally affect the change of IHTC.
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[3] Arunkumar S, Rao KVS, Kumar TSP. Spatial variation of heat flux at the metal mold interface due to mold filling effects in gravity die-casting. Int J Heat Mass Transfer 2008;51:2676–85. [4] Santos CA, Quaresma JMV, Garicia A. Determination of transient interfacial heat transfer coefficients in chill mold castings. J Alloy Compd 2001;319: 174–86. [5] Guo ZP, Xiong SM, Cho SH, Choi JK. Study on heat transfer behavior at metal/ die interface in aluminum alloy die casting process. Acta Metall Sinica 2007;11:1149–54. [6] Liu GR, Han X, Xu YG, Lam KY. Material characterization of functionally graded material by means of elastic waves and a progressive-learning neural network. Compos Sci Technol 2001;61:1401–11. [7] Han X, Xu D, Liu GR. A computational inverse technique for material characterization of a functionally graded cylinder using a progressive neural network. Neurocomputing 2003;51:341–60. [8] Liu GR, Lee JH, Patera AT, Yang ZL, Lam KY. Inverse identification of thermal parameters using reduced-basis method. Comput Methods Appl Mech Eng 2005;194:3090–107. [9] Deng S, Hwang Y. Applying neural networks to the solution of forward and inverse heat conduction problems. Int J Heat Mass Transfer 2006;49:4732–50. [10] Liang YC, Feng DP, Liu GR, Yang XW, Han X. Neural identification of rock parameters using fuzzy adaptive learning parameters. Comput Struct 2003;81:2373–82.
[11] Huber N, Tsakmakis Ch. A neural network tool for identifying the material parameters of a finite deformation viscoplasticity model. Comput Methods Appl Mech Eng 2001;191:353–84. [12] Dour G, Dargusch M, Davidson C, Nef A. Development of a non-intrusive heat transfer coefficient gauge and its application to high pressure die casting effect of the process parameters. J Mater Process Technol 2005;169:223–33. [13] Krimpenis A, Benardos PG, Vosniakos GC, Koukouvitaki A. Simulation-based selection of optimum pressure die-casting process parameters using neural nets and genetic algorithms. Int J Adv Manuf Technol 2006;27:509–17. [14] Sß ahin HM, Kocatepe K, Kayıkcı R, Akar N. Determination of unidirectional heat transfer coefficient during unsteady-state solidification at metal casting-chill interface. Energy Convers Manage 2006;47:19–34. [15] Kim TG, Lee ZH. Time-varying heat transfer coefficients between tube-shaped casting and metal mold. Int J Heat Mass Transfer 1997;40:3513–25. [16] Mirbagheri SMH, Shrinparvar M, Chirazi A. Modeling of metalo-static pressure on the metal-mould interface thermal resistance in the casting process. Mater Des 2007;28:2106–12. [17] Martorano MA, Capocchi JDT. Heat transfer coefficient at the metal-mould interface in the unidirectional solidification of Cu–8%Sn alloys. Int J Heat Mass Transfer 2000;43:2541–52.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Inverse identification of interfacial heat transfer coefficient between the casting and metal mold using neural network Liqiang Zhang a,b, Luoxing Li a,b,*, Hui Ju b, Biwu Zhu b a b
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082 Hunan, PR China College of Materials Science Engineering, Hunan University, Changsha, 410082 Hunan, PR China
a r t i c l e
i n f o
Article history: Received 30 November 2008 Received in revised form 23 July 2009 Accepted 27 February 2010 Available online 19 March 2010 Keywords: Neural network Finite element Interfacial heat transfer coefficient Inverse method
a b s t r a c t The effect of the heat transfer coefficient at the casting-mold interface is of prime importance to improve the casting quality, especially for castings in metal molds. However, it is difficult to determine the values of heat transfer coefficient from experiments due to the influence of various factors, such as contacting pressure, oxides on surfaces, roughness of surfaces, coating material, coating thickness and gap formation caused by the deformation of casting and mold, etc. In the present paper, the interfacial heat transfer coefficient (IHTC) between the casting and metal mold is identified by using the method of inverse analysis based on measured temperatures, neural network with back-propagation algorithm and numerical simulation. Then, by applying the identified IHTC in finite element analysis, the comparison between numerical calculated and experimental results is made to verify the correctness of method. The results show that the numerical calculated temperatures are in good agreement with experimental ones. These demonstrate that the method of inverse analysis is a feasible and effective tool for determination of the casting-mold IHTC. In addition, it is found that the identified IHTC varies with time during the casting solidification and varies in the range of about 100–3200 Wm2K1. The characteristics of the time-varying IHTC have also been discussed. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction With the rapid development of numerical simulation technology in the last two decades, the solidification simulation of casting has been taken as an effective tool for designing the casting process and improving the quality of casting [1,2]. However, some uncertainties must be eliminated before such simulations can be widely accepted as realistic descriptions of the process. The heat transfer at the casting-mold interface is one of these uncertainties. In general, it is difficult to determine the values of heat transfer coefficient from experiment due to the influence of various factors, such as contacting pressure, oxides on surfaces, roughness of surfaces, coating material, coating thickness and gap formation caused by the deformation of casting and mold [3–5], etc., especially for castings in metal molds. Fortunately, in recent years, the method of inverse analysis based on the field measurements has been rapidly developed and applied successfully to solve many complex engineering problems [6–8]. The essence of the method of inverse analysis is the realization of optimization techniques. Over the past few years, the interest in artificial neural networks has grown shar-
* Corresponding author. Address: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082 Hunan, PR China. Tel./fax: +86 7318821950. E-mail address: [email protected] (L. Li). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.02.020
ply for the more effective application of method of inverse analysis in complex problems. In the field of engineering science and computational mechanics, the research and applications of neural networks are very active and successful. The back-propagated multilayered network is one of the main types applied to engineering. The related works concern almost all topics of engineering science and mechanics [9–11], such as structural identification, parameter estimation for nonlinear finite element analyses, equation solver, identification of material characterization and fault diagnoses, etc. In the work, the casting-metal mold IHTC is identified by using the method of inverse analysis based on measured temperatures, neural network with back-propagation algorithm and numerical simulation. Then, applying the identified IHTC, the experimental and numerical calculated temperatures at various locations in the casting and mold are compared to verify the feasibility of the method for determination of the casting-mold IHTC. In addition, the characteristics of the time-varying IHTC have also been discussed in the view of thermal deformation, solidification contraction and influence of coating. 2. Experimental procedure For identifying the IHTC using the inverse method, two gravity die-casing experiments were selected to obtain the temperature
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data. All process conditions of the two experiments were the same except for the initial mold temperature. The initial mold temperature of the first experiment was 350 °C. The second one was 450 °C. The schematic diagram of experimental setup with the casting arrangement and the position of the thermocouples are shown in Fig. 1. In Fig. 1, T11, T12, T13 and T14 were the positions of the placed thermocouples of the first experiment. T21, T22, T23 and T24 were the measured positions of the second experiment. Four K-type thermocouples with the diameter of 0.2 mm protected by insulated bushing were inserted into the die cavity. For the inverse method to be applied successfully it is also necessary to optimize the location of the thermocouples. It should not be too close to the interface because the temperature may not be representative and the inverse method might amplify the noise present in the sampled data (stochastic response). It should also not be too far from the interface because the inverse method could become unstable [12]. Therefore, these thermocouples were located at 5 mm, 10 mm and 20 mm from the casting-mold interface respectively (see Fig. 1). The casting and mold temperatures were measured with a thermocouple at 0.3 s of time intervals. All thermocouples were connected by cables to a data acquisition system, which consisted of a MCT-2 data acquisition/switch unit and a computer. The experiments were performed with an A356 alloy (Al–7Si– 0.4Mg). It was melted in an electrically heated furnace using a graphite crucible under the protection of a cover nitrogen gas and pouring temperature was about 730 °C. The die material was low carbon steel. Its dimensions are shown in Fig. 1. The thermophysical parameters for each material are summarized in Table 1. Before the pouring begin, the die was imbedded in adiabatic wool of 5 mm thickness for improved surrounding heat isolation environment.
3. Inverse identification process for the casting-mold IHTC The inverse solving process for the casting-mold IHTC is shown in Fig. 2. It includes: (i) initial training of the NN model using the initial training data containing a set of assumed IHTC and their corresponding temperature distributions calculated based on numerical simulation, (ii) recognition of the casting-mold IHTC using the trained NN model by feeding in the measured temperatures varied with time, (iii) comparing the identified IHTC with assumed IHTC in initial training samples, if the identified IHTC is not in the content of the input training samples, the NN model shall be retrained by adjusting the training samples in order to obtain an improved IHTC and (iv) recognition of the transient IHTC between the casting and mold for the new time interval by repeating the above three processes. The detail of each part of this process will be given in the following sections. 3.1. Neural networks A NN model is referred to as a type of computational models that consists of hidden-layer neurons connected between the input and output neurons. The connections between the neurons are described by weights which are determined through training. The nonlinear hyperbolic functions are usually used as the activation functions to increase the modeling flexibility. In this work, the two-hidden layer NN model with three inputs and one output was adopted, as shown in Fig. 3. The back-propagation algorithm was employed as the learning algorithm. Using a BP neural network to solve the IHTC involves two basic stages, a learning stage and a recalling stage. The learning process is a supervised learning process which uses a training model and a set of target output values from the problem domain. In the recall-
Fig. 1. Schematic diagram of the experimental setup (all dimensions in mm).
Table 1 Thermal-physical parameters of alloy and die. Material
Density (kg m3)
Specific heat kJ (kg k)1
Latent heat (kJ kg1)
Thermal conductivity w (m k1)
Solidus temperature (k)
Liquidus temperature (k)
Casting (A356) Die (mild steel)
(2.42.7) 103 (6.97.8) 103
0.881.19 0.45–1.15
29.31294 53.71341
92.8184 27.260.2
829 –
889 –
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Fig. 2. Flow chart for the determination of casting-mold IHTC.
where g, a and r are defined as the learning rate, the momentum rate, and the iterative number, respectively. The derivatives in Eq. (3) can be written as:
@EðWÞ@ v ki @E ¼ dki ojk1 ; k @ v ki @wkij @wij
ð4Þ
where the wkij is the weight coefficient between the ith neuron in the kth layer and the jth neuron in the ðk þ 1Þth layer, v ki and okj is the input and output from the jth neuron of the kth layer respectively.
v ki ¼
X
wkij ok1 þ h; j
okj ¼ f
v ki
ð5Þ
j
dki can be expressed as: Fig. 3. A two-hidden layer NN model.
ing stage, the neural network can provide an appropriate output for any arbitrary unknown input. The goal of the training process is to modify the weights which characterize the BP neural network such that the actual output vector Yp approximates the target output vector Y t as closely as possible. A brief description of the formulation of the BP is given as follows. The error norm E between the determined output Yp vector and the targeted output vector Yt is defined as:
EðWÞ ¼ kY p Y t k2 :
ð1Þ
The operator kk2 represents the vector norm. The weight matrix is adjusted iteratively based on the following equations:
W rþ1 ¼ W r þ DW r ;
ð2Þ
@E @E DW r ¼ g þ ag ; @W W¼W r @W W¼W r1
ð3Þ
8 < fðyti yi Þf 0 ðv ki Þ; k di ¼ ðP dkþ1 wkþ1 Þf 0 ðv k Þ; : j i ji
k¼3 k–3
ð6Þ
j
In Eq. (5), h is the error derivative of back-propagation. The value should be between 0 and 0.15, and can be varied during the training process. The purpose of adding this small positive value is to prevent the weight matrix from stagnation. In Eq. (6) f0 (v ki ) is the first derivative of the activation function f() with respect to v ki . In this work, the activation function is given by the smooth sigmoid function.
f ðv i Þ ¼
1 1 þ ev i
ð7Þ
3.2. Training samples The samples for the initial training of the NN model consist of a number of sets of inputs and outputs. Firstly, in order to describe
L. Zhang et al. / Energy Conversion and Management 51 (2010) 1898–1904
the inverse characteristics of the IHTC calculation, these training samples including both the initial training and the retraining should be carefully selected. And these training samples should cover all possible values of IHTC. In this work, the training samples were obtained by using finite-element method instead of carrying out an actual experiment. Before obtaining the training samples, a function specification procedure proposed by Beck [4] was adopted to calculate time-varying IHTC at the interface between the casting and mold. The sequential function specification procedure is to assume temporarily that several future heat transfer coefficients are time invariant and the function can be a sequence of constant segments or of other forms such as straight line segments, parabolas, etc. According to Beck’s method, the time-varying IHTC was separated into a number of transient IHTC with a certain time interval in the inverse solver process. For more stable and faster results, the transient heat transfer coefficients were calculated for every 2 s interval from the time of pouring to 200 s after pouring. And then, a suitable initial value of transient IHTC was assumed and with this value, the temperature distribution of casting was obtained by using a commercial finite-element package, ProCAST to obtain the training samples. Compared with several other commercially available packages, ProCAST was chosen as the platform for the numerical simulation operation because it has the ability to develop user-programmable routines to describe dependent boundary conditions spatially and temporally [13]. Secondly, for successfully utilizing NN model, the inputs of the NN model should be carefully chosen so that the variation in the outputs can be truthfully reflected by the changes of these inputs. In addition, inputs for the NN model should be easily and accurately measured and be sensitive to the changes of the outputs. However, it is well known that the temperature changes are chiefly affected by the IHTC value during the heat transferring between two domains. Therefore, in the NN model, the temperatures at different time were expressed as x1 ; x2 and x3 , respectively and were used as the inputs for the NN model. The outputs of the NN model were the IHTC value, expressed as y in Fig. 3. 3.3. Normalization of training data sets The NN model requires the normalization of the input and the output data. As the sigmoid transfer function is used in the BP algorithm, the system cannot actually reach its extreme values of 0 and 1 without infinitely large weights. However, it is found better, in practice, to normalize the input patterns as well as output patterns to between 0.1 and 0.9 [7]. The inputs of the training samples are normalized linearly based on the following formulas:
v i ¼
vi vi min þ e1 vi max vi min þ e2
ð8Þ
where xi min and xi max are the minimal and maximal values of the ith input value xi, respectively in the sample data set; xi is the normalized value of parameter x ranging between 0.1 and 0.9. The e1 , e2 (0 6 e1 < e2 6 1) are the scaling factors for ensuring that the normalized values would not be close to 0 or 1. The outputs can be normalized in the exactly same way. 4. Results and discussions 4.1. Experimental cooling curve Fig. 4 shows the temperature variation with time measured at four locations in the casting with the initial mold temperature of 350 °C. The measured locations are described in Fig. 1. The distance to the interface between the casting and mold is all 10 mm at the location 1, 2 and 3 (T11, T12 and T13 as shown in Fig. 1). They are
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Fig. 4. Measured temperatures vs time curve with the initial mold temperature of 350 °C.
the same and close to the interface. The location 4 (T14 as shown in Fig. 1) is far away from the interface between the casting and mold, close to the center of casting and is 20 mm apart to the interface. In Fig. 4, it is obviously seen that the general shape of the 4 sets of cooling curve is similar. In the initial stage, the temperatures decrease rapidly. When they decrease down to about 550 °C, the temperatures are almost invariant for a short time with about 10 s. Then, the temperatures continue decreasing with a rapid velocity again. Whereas, the change of temperature at location four lag compared with the one in location 1, 2 and 3 and if the location 4 is removed, a better uniformity for the curves of temperature variation can be observed at location 1, 2 and 3. This demonstrates clearly that the experimental temperature differences perpendicular to the heat flow are very small with less than 5 °C and the temperature in casting change mainly along with the direction of heat flow in the improved surrounding heat isolation environment. It can be well known that such temperature distribution is greatly helpful to reduce the influence of noise and improve the accuracy of the inverse method. For identifying the IHTC, among the cooling curves obtained, the T11 data measured near to the interface was used as the known temperature history. The measured temperature curves at the other locations were used to test the accuracy of the identified IHTC. 4.2. Summary of the identified IHTC The identified IHTC between the casting and mold is shown in Fig. 5 together with the measured temperature change with time at location 1. In the inverse solving process as discussed above, the experimental temperature data at location 1 was used as the known temperature history for identifying the IHTC. As shown in Fig. 5, it can be clearly seen that the identified IHTC varies with time and can be divided into mainly three stages from the initial moment of the casting process until the completed solidification situation is reached. In the initial period of the casting solidification, the IHTC is about 250 Wm2K1 and is an ascending trend. The first stage lasts only up to 15 s. Then, the IHTC changes into the second stage. In this stage, the IHTC increases up to a maximum value of 3241 Wm2K1 in first and then decrease gradually. At about 43 s, the IHTC decreases rapidly down to a minimum value of 91.6 Wm2K1. As the temperature–time curve shown in Fig. 5, it can be known that the casting temperature is near solidus curve in the time which IHTC decreases sharply. In the third stage of the IHTC variation with time, The IHTC increases slowly up from the minimum value of 91.6 Wm2K1 to about 500 Wm2K1 and then stay constant. The variation trends and values of IHTC show
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Fig. 5. Variation of identified IHTC and measured temperature change with time.
good agreement with those in the literature [14,15]. For example, in Ref. [15], the authors reported the maximum value of IHTC for A356 alloy be nearly 2900 Wm2K1 and then decreasing fast down to around 200 Wm2K1. 4.2.1. First stage In the initial stage, the IHTC value between the casting and mold is low and increases slowly. The first stage last only about 15 s. It can be interpreted in two ways. One may be attributed to the thermal resistance of the coating and the other is the physically improving contact at the casting-mold interface. In the beginning of casting solidification, the heats transferred from the casting to the mold have to pass through the coating. Therefore, it needs some time to make the heat transfer process between the casting and the coating stable. According to Ref. [16,17], the ability of thermal resistance is different with the difference of coating materials or thickness and affects heavily the solidification time of the casting. On the other hand, the physical contact between the melt and mold would be improved as the melt level increases during pouring and the entrapped air in the cavity of mold escapes. After pouring, as the surface temperature of the mold increases the conductivity of air enclosed in the casting-mold interface also increases because the thermal conductivity of air varies with temperature, which increases finally the IHTC. It may be considered as a major reason for the increase of the IHTC in the initial period of casting solidification. 4.2.2. Second stage In the second stage of IHTC variation with time as shown in Fig. 5, it can be obviously seen that the casting-mold IHTC increases rapidly up to its maximum value firstly and then decreases gradually. Finally, when the temperatures of casting decrease down to solidus curve, the IHTC decreases sharply down to its minimum value. The rapid increase of IHTC is due to the formation of the perfect contact between the mold and the liquid metal. In the early stage of casting solidification, the liquid metal has a best contact with the mold surface and the cooling rate of casting is mainly affected by the contact type of casting-mold interface and the thermal conductivity of mold. Therefore, after the resistances of the coating and enclosed air are overcame in previous time, the IHTC reaches rapidly its maximum value. The gradual decreasing of IHTC can be attributed to the formation of shell in the interface between the casting and mold. During the liquid metal filling mold cavity, no shell formation of solid metal on the mold surface occurs due to the turbulences. When the filling is completed, the turbulences die out. A thin stable shell of solidified metal is formed on the mold because the mold extracts heat from the liquid metal and the
thickness of shell increases gradually with the casting heat extracted continuously. After the shell is formed, the interface is pressed against the mold by the hydrostatic pressure of the liquid metal and the hydrostatic pressure decreases with increasing of the solid shell thickness. This will result in the gradual decreasing of IHTC between the casting and mold with time. In addition, it can be clearly known that the temperature of casting is over solidus curve in the period of gradual decreasing of IHTC as shown in Fig. 5. When the temperature continues decreasing and reaches to the solidus curve, the casting-mold IHTC decreases rapidly. The rapid decreasing of IHTC is due to the formation of gap in the interface between the solidified casting and mold. After formation of an adequate solid metal on the chill surface, the perfect contact between the chill and the solidified casting no longer exists because of the contraction of the solidified casting. A thin gap is formed in the casting-mold interface. The heat transfers through the gap at interface by means of convection, radiation and gas conduction. This will have a great effect on the change of casting-mold IHTC. To sum up, the heat transfers at the casting-mold interface are complex in this stage. Many parameters, such as the wettability of the liquid metal on the mold surface, surface roughness of the mold, thermal conductivity of mold, hydrostatic pressure and turbulence of melt, etc., affect heavily the change of IHTC. 4.2.3. Third stage In the third stage, the casting-mold IHTC increases gradually up to a higher value and then remain at a constant value. Fig. 5 shows that the temperature of casting is under solidus curve in this stage. It can be well known that the solidification of casting in the mold has been completed here. The effect of the solidified casting contraction might be balanced by the thermal expansion of the mold. And that might produce a constant contact pressure between the casting and mold surfaces. A stable interface has possibly been formed at the surface between the casting and mold. This can contribute to a stable heat transfer through the interface from the casting to the mold. In addition, the thermal conductivity of the solid metal is invariant after the solidification of casting completed. This may be taken as a major factor for IHTC to remain a constant. 4.3. Verification of the identified heat transfer coefficients As mentioned above, the measured temperature curves at the location 2, 3 and 4 in the casting of the first experiment were used to test the accuracy of the identified IHTC. For the verification of the identified IHTC, the temperature distribution was calculated by feeding the identified IHTC into the ProCAST with the same boundary condition and then compared with the measured temperatures with the corresponding locations. The results are shown in Fig. 6. It can be obviously seen that a good agreement is obtained between the simulated and experimental temperatures. Fig. 7 shows that the comparison of simulated and measured temperatures with initial mold temperature of 450 °C. The numerical calculated temperatures are also obtained by feeding the identified IHTC into ProCAST with the same boundary condition as the experiment. TS1, TS2 , TS3 and TS4 is the calculated temperatures. Te1, Te2, Te3 and Te4 is the measured temperatures (T21, T22, T23 and T24 as shown in Fig. 1). Among them the TS4 and Te4 are the temperature curves of the mold, the others are the ones of the casting. It is clearly seen that the agreement of the calculations and measurements is also excellent. The maximal temperature differences are less than 5 °C and the mean temperature differences are less than 2 °C between the numerical calculation and measurements. The results further confirm that the proposed inverse method can be applied to identify the IHTC between the castings and mold accurately and reliably.
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Fig. 6. Comparison of simulated and measured temperatures with initial mold temperature of 350 °C: (a) location two, (b) location three and (c) location four.
Fig. 7. Comparison of simulated and measured temperatures with initial mold temperature of 450 °C.
5. Conclusions The calculated results have shown that the method of inverse analysis is applied successfully to determine the IHTC between the casting and metal mold. The major results of this paper can be summarized as follows: 1. The casting-metal mold IHTC has been successfully determined by using the method of inverse analysis based on measured temperatures at the various locations of casting, neural network with back-propagation algorithm and numerical simulation.
2. The correctness of method has been verified through the comparison between numerical calculated and experimental results based on the identified IHTC used in finite element analysis. The results show that the numerical calculated temperatures are in well agreement with experimental ones. It is adequately demonstrated that the method of inverse analysis is a feasible and effective tool for determination of the casting-mold IHTC. 3. The identified IHTC varies with time during the casting solidification and the values have varied in the range of about 100– 3200 Wm2K1. The variation trends and values of IHTC have shown good agreement with those in the literature. Moreover, the IHTC variation with time is complex when the casting temperature is over solidus curve. However, it is almost invariant and remains a constant under solidus curve. This may be attributed to the complex heat transfer during the casting solidification. During the entire period of casting solidification, many factors, such as the wettability of the liquid metal on the mold surface, surface roughness of the mold, thermal conductivity of mold, hydrostatic pressure and turbulence of melt, etc., affect heavily the heat transfer between the casting and mold and finally affect the change of IHTC.
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