Application of Heat Transfer Coefficient Estimation Using Data Assimilation and a 1-D Solidification Model to 3-D Solidification Simulation

Abstract

Solidification simulations are effective in designing a casting process to improve the quality of steel slabs and ingots. In this study, a new method was developed to efficiently estimate multiple heat transfer coefficients to improve the accuracy of three-dimensional (3-D) solidification simulation for a casting process. The heat transfer coefficients for the two heat transfer directions—side and bottom—of a prismatic mold were independently estimated by data assimilation using one-dimensional solidification simulations near the boundaries. The optimum values of the heat transfer coefficients at the side and bottom boundaries were elucidated by comparing the 3-D solidification simulation and experimental cooling curves. The maximum and average errors between the cooling curves of the 3-D solidification simulation with the optimum values and those of the experiment were less than 1.8% and 0.2%, respectively.

1. Introduction

An appropriate design of casting process is required to improve the quality of continuously-cast slabs and/or ingots of steels. Numerical simulations are methods of choice for effective designing of casting processes. There are various commercial software packages available to simulations of the solidification phenomenon during the casting process.¹⁾ Solidification simulation has become an important technique for predicting the temperature field to prevent delayed solidification in continuous casting and for designing the riser in ingot casting. Note that the accuracy of solidification simulations significantly depend on input parameters. Among them, the heat transfer coefficient between the cast and mold is one of the key parameters for accurate simulations. However, the accurate value of this parameter is generally not available because it depends on not only the types of cast and mold but also the type of process and the process conditions. Therefore, it is necessary to estimate the heat transfer coefficient for each process type and each process condition of interest in a trial-and-error way. The evaluation of the heat transfer coefficient corresponds to inverse analysis.^{2,3,4,5,6,7,8,9,10,11)} The heat transfer coefficient is often assumed a constant value in the inverse analysis by focusing on processes for a short time period. To improve the accuracy of the solidification simulation, however, the time dependence needs to be considered by assuming its function form such as the exponential function. Unfortunately, it is difficult to express the heat transfer coefficient with a simple function because, for instance, the air gap formed at the mold–alloy boundary due to solidification shrinkage causes complicated time-dependence of the heat transfer coefficient in some cases. Moreover, the trial-and-error approach requires significant time to estimate an appropriate heat transfer coefficient. Therefore, there is an urgent need for reliable methods for estimating the heat transfer coefficient.

Various inverse analysis methods^{12,13,14,15,16,17,18,19,20,21,22)} to estimate the heat transfer coefficient have been recently proposed using a neural network,^15,16) Bayesian approach,¹⁷⁾ and genetic algorithm.¹⁸⁾ However, the application of most inverse analysis methods are limits to special conditions such as steady state condition and no time dependency. Data assimilation is an interesting technique for overcoming these limitations, and can be used more generically than other methods for various casting conditions.^19,20,21,22) Data assimilation methods, including Kalman,²³⁾ ensemble Kalman,²⁴⁾ and particle²⁵⁾ filters, and four-dimensional variational methods,^26,27) are often used for parameter estimation. The particle filters are easy to implement in simulation models.²⁸⁾ Recently, Oka et al.²⁰⁾ proposed a method for simultaneously estimating the heat transfer coefficient and thermal conductivity in unidirectional solidification using a particle filter. Based on this method, we developed a method for estimating the time-dependent heat transfer coefficient in unidirectional casting experiments²¹⁾ and confirmed that the simulation with use of the heat transfer coefficient estimated by this method can accurately reproduce the cooling curves for the unidirectional casting of Al-1mass%Si alloy. We also observed that the position of the cooling curve(s) used in the estimation was the most important factor and that a cooling curve measured as close to the mold surface as possible should be employed for accurate estimation. Note that the heat transfer in region close to the mold surface is often one-dimensional (1-D) problem and therefore, it was possible to estimate an appropriate heat transfer coefficient using the local one-dimensional (L1D) heat transfer at the mold–alloy boundary.

Generally, there are multiple heat-removal directions and boundaries during the casting process. For example, in the continuous casting of slabs, the heat removal conditions (heat flux) on the wide and narrow surfaces of the mold must differ. It is therefore desirable to use different heat transfer coefficients when performing a solidification simulation. Thus, to perform the accurate solidification simulation of cast products, multiple heat transfer coefficients are required. However, the proper evaluation of multiple heat transfer coefficients, particularly, multiple time-dependent heat transfer coefficients, is difficult. Based on our previous study,²¹⁾ it is desirable to simultaneously estimate multiple time-dependent heat transfer coefficients by combining 3-D solidification simulation and data assimilation. However, because of the high calculation cost of estimation by 3-D simulation, it is necessary to devise a method with a low calculation cost. According to our previous study,²¹⁾ estimation of the heat transfer coefficient in the L1D region at the mold–alloy boundary is effective in overcoming the problem of multiple heat transfer coefficient evaluations. Additionally, we confirmed that the time-dependent heat transfer coefficient can be appropriately estimated by a particle filter with 1-D simulations. Therefore, in this study, we investigated the applicability of the heat transfer coefficient estimated for the L1D region using a particle filter—L1D estimation—for 3-D solidification simulation. The outline of the L1D estimation is as follows: First, a casting experiment was performed to obtain several cooling curves in the alloy. Subsequently, the heat transfer coefficients on the sides and bottom of the mold were independently estimated in the L1D region using data assimilation. Next, 3-D solidification simulations were performed using the heat transfer coefficient, estimated independently, and compared with the measured 3-D temperature distribution (cooling curves at several locations). The L1D estimation was evaluated based on the extent to which the cooling curve could be reproduced using the heat transfer coefficient obtained by L1D estimation.

2. Experiment and Model

2.1. Experimental Procedure

The casting experiment was performed using Al–Si alloy, with the mold schematically illustrated in Fig. 1. The employed mold and cap were made of heat-resistant brick. The Al-1mass%Si alloy was prepared from 99.99% pure Al and 99.999% pure Si. The alloy was melted in an electric furnace and poured into the mold at a casting temperature of 700°C, after which the cap was immediately placed on the mold. To measure the temperature change in the solidifying ingot and mold, four thermocouples were set in the alloy (positions i₁–i₄) and two thermocouples were set in the mold brick (m₁ and m₂). The cooling curves were obtained at one-second intervals. The ingot size obtained by the casting was 50 mm × 50 mm × 80 mm, and i₃ and i₄ were located 40 and 70 mm from the bottom of the mold, respectively.

Fig. 1.

Mold size and temperature measurement position.

2.2. Solidification Simulation Model

To simulate the 3-D temperature distribution in the alloy, we need the heat transfer coefficients at side and bottom boundaries. Such multiple heat transfer coefficients are estimated based on data assimilation using L1D simulation with the cooling curve closest to each mold wall. The heat transfer coefficients at the bottom (h₁) and side (h₂) of the mold were independently estimated by reproducing the cooling curves at i₁ and i₂, respectively. One-dimensional solidification analysis was performed to simulate the cooling curves at these positions. The scale of L1D calculation (L) was defined as the distance from the inner surface of the mold. The boundary condition at the mold surface was given by the heat flux h₁(T₁ – T_m1) for the bottom and h₂(T₂ – T_m2) for the side. Here, T_m1 and T_m2 are the measured temperatures at positions m₁ and m₂, respectively. The zero-Neumann condition was set as the boundary condition inside the alloy. The enthalpy method was adapted to calculate the heat flow including the release of latent heat during solidification and was used to simulate the L1D temperature distribution in the alloy. The heat conduction in the mold was not calculated, and the temperature changes at the bottom and sides of the mold followed the measured cooling curves of m₁ and m₂, respectively. The enthalpy method used in this study is explained in detail in our previous paper.²¹⁾

In the governing equation, the variation in enthalpy within each time step was calculated using Eq. (1):   

$$\rho \frac{\partial H}{\partial t}=\nabla \cdot \left(\kappa \nabla T\right)$$(1)

where H, T, t, κ, and ρ are the enthalpy, temperature, time, thermal conductivity, and density, respectively.

As the initial condition, the casting temperature was set to the initial temperature of all the calculation meshes in the 1-D calculation scale. Hence, the amount of heat removed from the mold increases with increasing L, and the estimated heat transfer coefficient depends on L. Therefore, the optimum value of L that yields the heat transfer coefficient used for accurate 3-D simulation cannot be known in advance. In this study, h₁ and h₂ were obtained for L = 6–15 mm at 1 mm intervals, and L₁ and L₂ are hereafter used to denote L for estimation of h₁ and h₂, respectively. Therefore, h₁ for L₁ = 6–15 mm and h₂ for L₂ = 6–15 mm (10 cases each) were independently estimated based on data assimilation with L1D simulation. Subsequently, 3-D solidification simulations were performed for all set of h₁ and h₂ to elucidate the optimal set that yields most accurate result of the cooling curves at the four points (i₁–i₄) in the casting. The mesh size was 1 mm in both the 1-D solidification simulations for estimating h₁ and h₂, and the 3-D solidification simulation. The κ and ρ values of the Al–Si alloy used in the simulations were 95 W K⁻¹ m⁻¹ and 2.39 × 10³ kg m⁻³, respectively.²⁹⁾ The temperature–enthalpy relationship for Al-1mass%Si alloy was obtained using Pandat software with the PanAl database. The liquidus and solidus temperatures were 654.6 and 607.0°C, respectively.

2.3. Estimation of Time-dependent Heat Transfer Coefficient Using Particle Filter

Data assimilation is employed to perform highly accurate estimation by combining observational data and a simulation model. Although it was initially developed for numerical weather prediction, it is also an effective and practical technique for estimating parameters or material properties that are difficult to measure experimentally. The particle filter is a data assimilation method that uses a simple algorithm and is easy to implement. Thus, we adopted a particle filter to estimate the time-dependent heat transfer coefficient, which is briefly described below. The method used in this study is the same as that used in a previous study.²¹⁾

Figure 2 is a schematic diagram explaining the estimation process of the time-dependent heat transfer coefficient (h) using a particle filter. In the initial state, arbitrary values of h were randomly assigned to each simulation called particle. In the present case, the particle corresponds to each L1D simulation. The heat flow was calculated using the assigned value of h for each particle. The value of likelihood for each particle was then calculated, followed by re-sampling. This operation is called “filtering.” The likelihood was determined by the difference between the measured and calculated cooling curves. The likelihood of particle i at t (λ_tⁱ) was given by the following equation:   

$${\lambda }_t^i=\frac{1}{\sqrt{{(2\pi )}^m\left|\mathrm{\Sigma }\right|}}exp\left[-\frac{1}{2}{\left({\mathbf{T}}_{meas}-{\mathbf{T}}_{cal}^i\right)}^T{\mathrm{\Sigma }}^{-1}\left({\mathbf{T}}_{meas}-{\mathbf{T}}_{cal}^i\right)\right]$$(2)

where Σ is a variance-covariance (m × m) matrix, m indicates the number of different positions of the thermocouples used for temperature measurement, and T_meas and Tⁱ_cal are vector notations of the measured and calculated temperatures at the different thermocouple positions, respectively. This work assumed Σ = σ_T²I, where the mean is zero, σ_T is the standard deviation of error in the temperature measurement, and I is the m × m unit matrix. For resampling after calculating the likelihood, the normalized likelihood of the ith particle (β_tⁱ) was calculated using Eq. (3):   

$${\beta }_t^i=\frac{{\lambda }_t^i}{\sum _{j=1}^N{\lambda }_t^j}$$(3)

where N is the total number of particles.

Fig. 2.

Schematic of the particle filter method used in this study.

In the subsequent resampling, particles with lower normalized likelihoods (smaller circles in Fig. 2), were randomly selected and removed, whereas those with higher normalized likelihoods (larger circles in Fig. 2) were duplicated. Note that the produced particles presented the same heat transfer coefficient as the original parent particle. Thus, system noise (v_t) was added to all the particles, i.e., h + v_t after updating. The values of h for each particle after resampling were used for the heat flow calculations in the next step. The system noise was taken as the product of h for the particles with the highest likelihood and a random number given by the Gaussian distribution, with the mean and a standard deviation are 0 and σ_h, respectively. The temperature profile calculated for the particles with the highest likelihood was updated as the initial temperature profile for the next step. In this study, h estimated at t was determined to be the one with the highest likelihood. By iterating these steps, we could estimate the time-dependent heat transfer coefficient. The values of σ_T and σ_h used in this study were 2.0 K and 0.2, respectively, while the number of particles N was 2048.

3. Results and Discussion

3.1. Experimental Cooling Curves

Figure 3 shows the cooling curve measured at each position in the casting experiment. Figure 3(a) shows the cooling curves at i₁, i₃ and i₄ in the alloy, corresponding to the temperature distribution in the centerline of the z-axis direction, while Fig. 3(b) shows the cooling curves at i₂ and i₃ in the alloy, corresponding to the temperature distribution in the centerline of the x-axis (or y-axis) direction. Figure 3(c) shows the cooling curves at m₁ and m₂ in the mold. From Figs. 3(a) and 3(b), the temperature gradient between i₁ and i₃ indicates that heat is removed from the bottom of the mold. In Fig. 3(b), it is understood that there is a temperature gradient between i₂ and i₃, the heat is removed from the side surface of the mold as shown in Fig. 3(c), the temperature at the bottom of the mold gradually increased to ~400°C, while it increased sharply to 500°C at its side. Therefore, we confirmed that the heat was removed from multiple (side and bottom) directions of the mold during solidification of the alloy.

Fig. 3.

Cooling curves for each temperature measurement position obtained in the casting experiment: (a) i₁, i₂, and i₄ and (b) i₂ and i₃ positions in the alloy; (c) m₁ and m₂ positions in the mold. (Online version in color.)

3.2. Estimation of Heat Transfer Coefficient Using L1D Solidification Simulation

As described in Section 1, the time-dependent heat transfer coefficients of the bottom and side of the mold were estimated from the particle filter with the L1D solidification simulation. Before showing the estimated heat transfer coefficients, we start the discussion from the comparison between the measured and simulated cooling curves. Figures 4(a) and 4(b) show the cooling curves at i₁ and i₂ obtained by L1D estimation for L₁ = 10 mm and L₂ = 10 mm, respectively. The measured cooling curves are shown in Fig. 4 for comparison. The simulated cooling curves at i₁ and i₂ well agreed with the measured curves. Such agreement was observed for all values of L₁ and L₂ tested in this study. The relative error (ΔE_t) was calculated at each time using Eq. (4), and the average of ΔE_t (ΔE^ave) was calculated using Eq. (5).   

$$\mathrm{\Delta }{E}_t[\%]=100\left|\frac{T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s},\mathrm{\hspace{0.17em} }t}^{}-T_{\mathrm{s}\mathrm{i}\mathrm{m},\mathrm{\hspace{0.17em} }t}^{}}{T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s},\mathrm{\hspace{0.17em} }t}^{}}\right|$$(4)

  

$$\mathrm{\Delta }{E}^{\mathrm{a}\mathrm{v}\mathrm{e}}[\%]=\frac{1}{M}\sum _{i=1}^M\mathrm{\Delta }{E}_i$$(5)

where T_meas,t and T_sim,t are the experimental and simulation temperatures at time t, respectively, and M is the number of time steps and M was set to 1000. Table 1 lists the maximum ΔE_t (ΔE^max) and ΔE^ave for all values of L₁ and L₂. In the h₁ estimation, ΔE^max and ΔE^ave were less than 0.210% and 0.005%, respectively, for all L₁ values, while in h₂ estimation, they were less than 0.536% and 0.008%, respectively, for all L₂ values. The errors in the measured and simulated cooling curves were very small under all conditions, indicating that the cooling curve can be accurately simulated from the time-dependent heat transfer coefficient estimated using data assimilation. Note that the temperature distribution due to undercooling before nucleation of the primary α-Al phase was also reproduced as seen in Fig. 4. The maximum error ΔE^max is actually associated with the undercooled region for all conditions. The enthalpy method employed in this study uses the relationship between the temperature and enthalpy in the equilibrium state. Thus, the temperature variation in the undercooling and recalescence due to the nucleation of the primary α-Al, in principle, cannot be reproduced in the present solidification model. This drawback of the simulation model is cured by employing the time-dependent heat transfer coefficient estimated by the particle filter. This is one of the advantageous features of particle filter.

Fig. 4.

Cooling curves of 1-D solidification simulation obtained using L1D estimation and the measured cooling curves. Cooling curves for the (a) i₁ position at L₁ = 10 mm and (b) i₂ position at L₂ = 10 mm. (Online version in color.)

Table 1. Maximum (ΔE^max) and average (ΔE^ave) errors between the cooling curves of the 1-D solidification simulation, obtained by L1D estimation and the measured cooling curve, and average heat transfer coefficient (h^ave).

	Scales for L1D calculation, L1, L2 [mm]
	6	7	8	9	10	11	12	13	14	15
ΔEmax [%]	0.210	0.196	0.176	0.192	0.156	0.149	0.149	0.152	0.103	0.084
ΔEave [%]	0.005	0.004	0.005	0.004	0.004	0.004	0.004	0.004	0.003	0.003
h1ave [Wm−2K−1]	19.30	22.70	26.20	29.86	33.58	37.48	41.40	45.52	49.53	53.61
ΔEmax [%]	0.536	0.409	0.411	0.350	0.329	0.224	0.292	0.207	0.177	0.195
ΔEave [%]	0.008	0.006	0.006	0.005	0.004	0.004	0.004	0.003	0.003	0.003
h2ave [Wm−2K−1]	33.26	38.67	43.99	49.42	55.22	61.45	68.32	75.76	82.55	91.71

Figures 5(a)–5(c) show the relationship between the time and h₁ obtained by L1D estimation for L₁ = 7, 11, and 15 mm, while Figs. 6(a)–6(c) show the relationship between time and h₂ obtained by L1D estimation for L₂ = 7, 11, and 15 mm. For all values of L₁ and L₂, the heat transfer coefficient increased sharply during undercooling and decreased during recalescence. Particularly, the heat transfer coefficient h₂, which comprised a high degree of undercooling, increases to more than 1000 W m⁻² K⁻¹. When the temperature increased due to recalescence, h₂ became negative. The negative value of h means that the heat transfer occurs from the mold to the alloy. However, this unphysical process does not occur in the present experiment. As mentioned above, the L1D simulation cannot reproduce the undercooling and thereby the subsequent recalescence, which usually causes the difference between the experimental and simulation results. However, the particle filter enables the accurate reproduction of the experimental results including the undercooling and recalescence even with the L1D estimation. Hence, the heat transfer coefficient exhibits the negative value during recalescence.

Fig. 5.

Time-dependent heat transfer coefficient of the bottom of the mold obtained by L1D estimation. L₁ = (a) 7, (b) 11, and (c) 15 mm. (Online version in color.)

Fig. 6.

Time-dependent heat transfer coefficient of the side of the mold obtained by L1D estimation. L₂ = (a) 7, (b) 11, and (c) 15 mm. (Online version in color.)

Excluding the time in the undercooled region, the estimated heat transfer coefficient was relatively close to a constant value for both h₁ and h₂. Table 1 lists the average values of the heat transfer coefficients for 100–1000 s, which increased with increasing calculation area (L₁ and L₂). This is because the longer is the 1-D area, the larger is the initial heat in the alloy. To remove a large amount of heat to the outside of the alloy while reproducing the measured cooling curve, the heat transfer coefficient must be large. Therefore, 10 different time-dependent heat transfer coefficients could be obtained for both h₁ and h₂, and we proceeded to determine the optimal set of h₁ and h₂ that yields the most accurate 3-D simulation result of temperature distribution in the alloy.

3.3. Applications to 3-D Simulation

To efficiently elucidate the optimum values of h₁ and h₂, we proceeded to perform a three-step evaluation: First, 3-D solidification simulations were performed using the heat transfer coefficients estimated for L₁ = L₂. Next, we compared the simulated and measured cooling curves at i₁ and i₂ and evaluated the degree of agreement using the solidification time. Finally, the optimum values of h₁ and h₂ were thoroughly investigated.

Figure 7 shows the cooling curves at i₁ and i₂ obtained by 3-D solidification simulation using the heat transfer coefficients estimated at L₁ = L₂ = 10, 11, and 12 mm. The accuracy of the simulation was evaluated by comparing the experimental and simulation solidification times. The solidification time is defined as the time when the temperature at the measurement position reaches the solidus temperature (607°C) of the Al-1mass%Si alloy. The simulated cooling curves at L₁ = L₂ = 11 mm were in relatively good agreement with the measured cooling curve at both i₁ and i₂. On the other hand, the solidification time of the simulation with L₁ = L₂ = 10 mm was longer than that of the experiment, while the one with L₁ = L₂ = 12 mm was shorter. Therefore, based on the conditions L₁ = L₂ = 11 mm, the optimal set was thoroughly investigated using h₁ and h₂ estimated for L₁ ≠ L₂ in the range 11±2 mm.

Fig. 7.

Cooling curves of 3-D solidification simulation using the set of h₁ and h₂ at L₁ = L₂ for the (a) i₁ and (b) i₂ positions. (Online version in color.)

Figure 8 shows the cooling curves at i₁ and i₂ obtained by 3-D solidification simulation using h₁ (L₁ = 9–11 mm) and h₂ (L₂ = 11 mm). Although the solidification time of the simulation with h₁ for L₁ = 9 mm was slightly longer than that of the experiment, the simulated and measured cooling curves were in good agreement. The agreement in the case of L₁ = 10 mm is better than the cases of L₁ = 9 and 11 mm. Figure 9 shows the cooling curves at i₁ and i₂ obtained by 3-D solidification simulation using h₁ (L₁ = 11 mm) and h₂ (L₂ = 10–12 mm). The solidification time in case of L₂ = 10 mm was longer than that of the experiment, while that of L₂ = 12 mm was shorter. Compared with the cooling curve of L₂ = 11 mm, these simulated cooling curves deviated significantly from the measured cooling curves. Table 2 shows the difference between the experimental and simulation solidification times for all values of L₁ and L₂ used in the simulated cooling curves (Figs. 7, 8, 9). At i₁–i₃, the difference in the simulation and experiment solidification times was the smallest for the heat transfer coefficient estimated for L₁ = 10 mm and L₂ = 11 mm, while at i₄, the difference was the smallest for the heat transfer coefficients for L₁ = L₂ = 11 mm. From a comprehensive evaluation of these results, we concluded that the optimal set is h₁ (L₁ = 10 mm) and h₂ (L₂ = 11 mm).

Fig. 8.

Cooling curves of 3-D solidification simulation using the set of h₁ and h₂ at L₂ = 11 mm for the (a) i₁ and (b) i₂ positions. (Online version in color.)

Fig. 9.

Cooling curves of 3-D solidification simulation using the set of h₁ and h₂ at L₁ = 11 mm for the (a) i₁ and (b) i₂ positions. (Online version in color.)

Table 2. Difference between the experimental and simulation solidification times at each position in the alloy.

L1 [mm]	L2 [mm]	Difference in solidification time
		i1	i2	i3	i4
10	10	142.6	144.4	144.0	154.4
11	11	−15.8	−12.8	−12.8	−1.4
12	12	−116.2	−112.4	−112.6	−101.4
9	11	23.4	26.0	25.8	37.0
10	11	0.8	2.6	2.8	15.2
11	10	121.2	124.4	123.8	134.4
11	12	−101.8	−99.8	−100.0	−89.0

-Positive and negative values indicate respective longer and shorter solidification times than those observed in the experiments.

Figure 10 shows the cooling curves at i₃ and i₄ obtained by 3-D solidification simulation using the optimal set of h₁ and h₂. Table 3 lists the average (ΔE^ave) and maximum (ΔE^max) errors of the 3-D simulation and experiment cooling curves at i₁–i₄. Similar to those observed for i₁ and i₂ (Figs. 8(a) and 8(b), respectively), the simulated and measured cooling curves at i₃ and i₄ were in good agreement, confirming that the 3-D temperature distribution in the alloy can be predicted using the heat transfer coefficients obtained by L1D estimation.

Fig. 10.

Cooling curves of 3-D solidification simulation using the set of h₁ and h₂ at L₁ = 10 mm and L₂ = 11 mm for the (a) i₃ and (b) i₄ positions. (Online version in color.)

Table 3. Maximum (ΔE^max) and average (ΔE^ave) errors between the cooling curves of the experiment and 3-D solidification simulation using the set of h₁ and h₂ at L₁ = 10 mm and L₂ = 11 mm.

	i1	i2	i3	i4
ΔEmax [%]	1.114	0.390	1.064	1.710
ΔEave [%]	0.154	0.047	0.150	0.199

Finally, the cooling behavior at each position was investigated in detail. As shown in Fig. 8(b), the simulated cooling curve at i₂ well agreed with the measured cooling curve, including the undercooled region. This cooling curve was accurately reproduced because the cooling curve at this position was used to estimate h₂. However, in the cooling curve at i₁ used in the estimation of h₁, the simulated temperature was slightly lower than the measured temperature at 100–350 s (Fig. 8(a)). Also, h₁ was slightly convex at 100–350 s (Fig. 5). In the L1D estimation, h₁ was estimated by removing all heat solely by 1-D heat transfer to the bottom of the mold, which might be overestimated as the heat transfer coefficient for 3-D simulation. In 3-D simulation, we believe that the heat removed from the bottom of the mold slightly decreased because the heat were removed from the side of the mold. Therefore, this excessive heat transfer to the bottom of the mold by the higher h₁ value estimated from 1-D condition was simulated as a temperature drop at 100–350 s. The same temperature drop seen at i₁ was also observed in the simulated cooling curve for i₃, which is located far from the bottom of the mold. The Al–Si alloy exhibits a high thermal conductivity and the size of the mold used in this study is relatively small. This temperature drop may have therefore been affected by heat removal from the bottom of the mold. Because steels with lower thermal conductivities than those of Al alloys tend to have a local 1-D temperature gradient near the mold wall, the heat transfer coefficient might be accurately obtained by L1D estimation. In addition, the calculation time required for L1D estimation approximated several tens of minutes, using a laptop computer for each L condition. Even if the optimal set of heat transfer coefficients is searched by the 3-D solidification simulation, the calculation cost of the present method is still lower than that of the simultaneous estimation of multiple heat transfer coefficients by the 3-D model.

4. Conclusions

In this study, we investigated whether a 3-D solidification simulation can be performed accurately using the heat transfer coefficient estimated in a L1D region. To estimate the heat transfer coefficient in this region, we used the particle filter with the L1D solidification simulation developed in our previous study.²¹⁾ The following conclusions were drawn from this study:

(1) Estimation of the heat transfer coefficient by the particle filter using L1D simulation enabled accurate simulation of the measured cooling curve (maximum error <1%).

(2) Data assimilation enables estimation of the time-dependent heat transfer coefficient that can accurately simulate the cooling curve including the undercooling and recalescence that the model does not explicitly consider.

(3) By estimating the heat transfer coefficient under various L1D conditions and selecting the optimum condition from the estimated heat transfer coefficients, the 3-D solidification simulation that requires multiple heat transfer conditions could be performed with high accuracy.

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP17K06874 and the 28th ISIJ Research Promotion Grant.

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