Inverse Calculation of Time-Spatial Varying Mold Heat Flux During Continuous Casting from Fast Response Thermocouples

Abstract

This paper covers the determination of time-spatial varying mold heat flux using mold temperatures measured by fast response thermocouples at a frequency up to 60 Hz. A two-dimensional transient inverse heat conduction problem (2DIHCP) is established, where 2DIHCP is developed based on the sequential function specification method implemented with the spatial regularization terms to reduce the fluctuations in the estimated heat flux in both time and spatial domain. Then, the inverse problem was validated using a designed numeric test-problem. Finally, the inverse problem is applied to calculating the heat flux across the mold hot surface for a continuous casting trial using the mold simulator.

Introduction

Many surface defects in final rolled steel products originate from the initial solidification of molten steel inside continuous casting molds. This has been found to be associated with the heat transfer behaviors of the mold itself [1,2,3]. The mold heat flux, especially at the mold meniscus area, is extremely complex due to the transient nature of the infiltration of lubricant liquid mold flux, intensive fluid flow, and mold oscillation, such that it would be very difficult to get a clear comprehensive understanding of all dynamics within the system [4,5,6,7,8].

The heat flux of the mold is a space–time varying variable [9]. Usually, the mold heat transfer is monitored by temperature sensors [10]. Hundreds of thermocouples/sensors are employed to monitor the mold temperature during the continuous casting process, with a temperature sampling rate of 1–65 Hz. The mold heat flux cannot be calculated directly from the measured temperatures because of the lack of information on the boundary conditions. Mathematically, the determination of the mold heat flux from measured temperatures is an inverse problem that means the solution is usually unstable, not unique or does not exist, and a small measured error in the temperatures will result in a large error of heat flux [11, 12].

The inverse problem method has a broad application prospect in determining the heat flux from measured temperatures. The principle of inverse problems for restructuring the heat flux is to find out a heat flux with the maximum probability making the sum of squared deviations between the calculated temperatures T and the measured temperatures Y to be minimum [13].

The development of the use of inverse problem methods to calculate continuous casting mold heat flux from temperature measurements can be traced back to the works by Brimacomb et al. [14], followed by Thomas et al. [15], Wang et al. [16,17,18], Griffiths [19], Rajaraman [20], and Talukdar et al. [21]. Those inverse problem methods could be classified as the gradient-based type methods, such as the function specification method [9], the Tikhonov regularization [22] and the conjugate gradient method [23, 24], and the stochastic-based type methods, such as the genetic algorithm and the neural network algorithm [25]. However, the major weakness of the inverse problem persists in that it is extremely sensitive to temperature measurement errors, particularly as the time step is made smaller [12, 26]. As continuous casting technology progresses, fast mold thermal monitoring systems are adopted that could provide a more accurate detection precision supervising the fluctuation of mold temperature [5, 6]. However, the faster temperature sampling rate of mold thermal monitoring systems is inevitably accompanied by a higher intensity of the error/noise in the measured temperatures [12]. As a result of the increase of the temperature sampling rate, the use of small-time step frequently introduces instabilities in the solution of the inverse heat conduction problem.

In the engineering community, the inverse problem of Beck’s sequential function specification method is well-known and very successfully used in solving inverse heat conduction problems for over 50 years [27,28,29,30,31]. For Beck’s sequential function specification method, the heat flux form is assumed to be a constant function or a linear function over several future time steps due to the fact that the temperature response is lagging with respect to the boundary heat flux [23], then the stabilization of the solution in the time domain can be improved by choosing an appropriate number of future time steps [24, 32]. However, a common issue raised is how to improve the stabilization of the solution in the time and spatial domain for extending the sequential function specification method to the two-dimensional heat transfer problem [33, 34].

Therefore, the purpose of this work is to establish a two-dimensional transient inverse heat conduction problem (2DIHCP) for the estimation of the mold heat flux from fast-sampled temperature data. The inverse problem is developed where the function specification method with a spatial Tikhonov regularization [6, 35, 36] was used to improve the stabilization of the solution in time and spatial domains, respectively. Then, a numeric test-problem was designed to validate the inverse problem. Finally, the inverse problem is applied to calculating the mold heat flux during liquid steel casting.

The Experimental and Direct Problem Description

The continuous casting trial was conducted using a mold simulator. As Fig. 1 shows, the mold simulator applied to this study is an inverse type water-cooled copper mold (30 mm × 50 mm × 350 mm) with oscillation capability. A U-shaped type water-cooling groove with a diameter of 10 mm is manufactured along the center line of mold plate, 12.5 mm away from both ends and gets connected 20 mm above the bottom, where the water flows from one end to the other. The copper mold is equipped with an extractor that makes only one face of the mold exposed to the liquid melt. The temperatures in the mold are measured by T-type thermocouples at a frequency up to 60 Hz through a data acquisition system. The two columns of thermocouples are spaced 3 and 8 mm away from the mold surface, respectively, as shown in Fig. 1, the dots represent the locations of thermocouples. The arrangement allows the first column of thermocouples to catch the temperature history of the mold that will be used in the minimization of the objective function of the inverse problem. The second column of thermocouple inside the mold wall on ∂Ω₄ (CD) is to provide the temperature boundary condition f(t), where the temperature is interpolated linearly from two near measured temperatures for the nodes in between the two thermocouples at ∂Ω₄. This arrangement of those thermocouples in the mold wall was chosen based on the study of Badri et al. [37] so as to improve the signal-to-noise ratio for the temperature measurements during the experiment. Then, the rectangular area ABCD with the height (AB) of H (= 21 mm) and the width (BC) of W (8 mm) is set as the heat transfer computational domain, which consists of four boundary conditions ∂Ω₁ (DA), ∂Ω₂ (AB), ∂Ω₃ (BC), and ∂Ω₄ (CD).

Fig. 1

Computational area of the mold and locations of thermocouples for the casting experiment

The experiment runs as follows: Step I The oscillating water-cooling copper mold and the extractor are dipped into the hot melt of liquid steel. After the mold and extractor reached the target depth, the mold flux level and the liquid steel level would be located in the mold thermocouple-measuring zone. Step II The mold and extractor were held for several seconds to form an initial shell on the water-cooled copper mold to ensure the initial shell is strong enough to prevent tearing during extraction. Step III The extractor withdrew the solidifying shell downward at a constant speed to simulate continuous casting. The mold moved upward at a certain speed to compensate for the rise of the mold level, so that the liquid level could be kept in the same position with respect to the mold. Step IV When the casting was completed for the desired length, the mold and extractor were withdrawn out of the furnace and then cooled in air.

By assuming the mold heat transfer is two-dimensional in the vertical section ABCD perpendicular to the mold hot surface. The heat transfer within the mold vertical section ABCD is assumed to be governed by Fourier partial differential equations. The direct problem, namely the problem of 2D heat transfer within the rectangular area ABCD, as shown in Fig. 1, is governed by the following equations.

$$ c\rho \frac{\partial T}{{\partial t}} = \frac{\partial }{\partial x}\left( {\lambda \frac{\partial T}{{\partial x}}} \right) + \frac{\partial }{\partial y}\left( {\lambda \frac{\partial T}{{\partial y}}} \right),{\text{in}}\;{\text{domain}}\;\Omega = \left[ {0,W} \right] \times \left[ {0,H} \right] $$

(1a)

$$ - \lambda \frac{{\partial T(\partial \Omega_{1} \cup \partial \Omega_{2} \cup \partial \Omega_{3} ,t)}}{{\partial {\varvec{n}}}} = {\varvec{q}}\;{\text{to}}\;{\text{be}}\;{\text{estimated}}. $$

(1b)

$$ T(\partial \Omega_{4} ,t) = f\left( t \right) $$

(1c)

$$ T\left( {x,y,0} \right) = T_{{{\text{ini}}}} $$

(1d)

where the symbol c represents the specific heat in J kg⁻¹, ρ is the density in kg m⁻³, T is the temperature in K, t is the time in second, and λ is thermal conductivity in W m⁻¹ K⁻¹, q (W m⁻²) is the heat flux of boundary conditions ∂Ω₁, ∂Ω_2, and ∂Ω₃, n is the outer normal of boundary.

Inverse Problem Description

In this section, the inverse problem is developed based on the function specification method implemented by a spatial Tikhonov regularization [6, 35, 36]. The sequential function specification procedure is: (1) the heat fluxes are assumed to remain constant over the r future time steps, q^j = q^j+1 = ⋯ = q^j+r, owing to the temperature response is lagging with respect to the boundary heat flux, and (2) By knowing the heat fluxes for t < t_j, namely ${\tilde{\mathbf{q}}}^{1}$, ${\tilde{\mathbf{q}}}^{2}$, …, ${\tilde{\mathbf{q}}}^{j - 1}$ are known, estimate ${\mathbf{q}}^{j}$ so that makes the sum of the squares of the deviations between the calculated temperatures T and the measured temperatures Y in the time interval [t_j, t_j+r−1] to be minimized. Then, the inverse heat conduction problem could be converted into a problem that is to estimate ${\mathbf{q}}^{j}$ by making the following objective function to be minimum. Therefore, the definition of the inverse heat conduction problem is made as follows

$$ s({\varvec{q}}^{j} ) = \sum\limits_{k = 0}^{r - 1} {\sum\limits_{m = 1}^{M} {\left[ {Y_{m}^{j + k} - T_{m}^{j + k} ({\varvec{q}}^{j} )} \right]^{2} } } + \alpha R({\mathbf{q}}^{j} ) $$

(2)

where r is the number of future time steps. $Y_{m}^{j}$ and $T_{m}^{j}$ are the measured temperatures and the calculated temperatures at the time of t_j and the measured position m, respectively. M is the number of sensors. α is the spatial regularization parameter.

The finite difference method is used to solve the above direct problem, where the rectangular area ABCD (Ω) will be divided into grids, and the boundaries of ∂Ω₁, ∂Ω_2, and ∂Ω₃ will be split into $n_{1}$, $n_{2}$, and $n_{3}$ divisions, respectively. Then, q has N components, $N = n_{1} + n_{2} + n_{3}$, and $q_{k}^{j}$ represents the kth component of heat flux q at the time of t_j.

$$ \begin{aligned} {\varvec{q}}^{j} & = [{\varvec{q}}_{1}^{j} ,{\varvec{q}}_{2}^{j} ,{\varvec{q}}_{3}^{j} ]^{{\text{T}}} ,{\varvec{q}}_{1}^{j} = \left[ {q_{1}^{j} ,q_{2}^{j} , \ldots ,q_{{n_{1} }}^{j} } \right]^{{\text{T}}} , \\ {\varvec{q}}_{2}^{j} & = \left[ {q_{{n_{1} + 1}}^{j} ,q_{{n_{1} + 2}}^{j} , \ldots ,q_{{n_{1} + n_{2} }}^{j} } \right]^{{\text{T}}} \;{\text{and}}\; \\ {\varvec{q}}_{3}^{j} & = \left[ {q_{{n_{1} + n_{2} + 1}}^{j} ,q_{{n_{1} + n_{2} + 2}}^{j} , \ldots ,q_{N}^{j} } \right]^{{\text{T}}} . \\ \end{aligned} $$

Besides, a first-order of spatial Tikhonov regularization term is added to the objective function Eq. (2) so as to improve the spatial stabilization of the solution. It should be mentioned that heat flux q might be subjected to discontinuity at the intersection points between two adjacent boundaries of ∂Ω₁ and ∂Ω₂ (∂Ω₂ and ∂Ω₃). Thus, the spatial Tikhonov regularizations for the heat flux of the boundaries ∂Ω₁, ∂Ω_2, and ∂Ω₃ are described separately. That is

$$ R({\mathbf{q}}^{j} ) = \sum\limits_{n = 1}^{{n_{1} - 1}} {(q_{n}^{j} - q_{n + 1}^{j} )^{2} } + \sum\limits_{{n = n_{1} + 1}}^{{n_{1} + n_{2} - 1}} {(q_{n}^{j} - q_{n + 1}^{j} )^{2} + \sum\limits_{{n = n_{1} + n_{2} + 1}}^{N - 1} {(q_{n}^{j} - q_{n + 1}^{j} )^{2} } } $$

(3)

The estimated temperature ${\mathbf{T}}^{j}$ could be evaluated using a Taylor series expansion around the current solution ${\tilde{\mathbf{q}}}^{j}$.

$$ {\mathbf{T}}^{j} = {\tilde{\mathbf{T}}}({\tilde{\mathbf{q}}}^{j} ) + {\mathbf{X}}_{j} ({\mathbf{q}}^{j} - {\tilde{\mathbf{q}}}^{j} ). $$

(4)

${\tilde{\mathbf{T}}}({\tilde{\mathbf{q}}}^{j} )$ is the temperature calculated using the current solution ${\tilde{\mathbf{q}}}^{j}$. ${\mathbf{X}}_{j}$ is called as the M × N sensitivity coefficient matrix at time t_j and is defined as follows.

$$ \begin{aligned} {\mathbf{X}}_{j} & = \left[ {\frac{{\partial {\mathbf{T}}({\mathbf{q}}^{j} )}}{{\partial {\mathbf{q}}^{j} }}} \right]^{{\text{T}}} = \left( {\begin{array}{*{20}c} {X_{1,1}^{j} } & {X_{1,2}^{j} } & \cdots & {X_{1,N}^{j} } \\ {X_{2,1}^{j} } & {X_{2,2}^{j} } & \cdots & {X_{2,N}^{j} } \\ \vdots & \vdots & \ddots & \vdots \\ {X_{M,1}^{j} } & {X_{M,2}^{j} } & \cdots & {X_{M,N}^{j} } \\ \end{array} } \right),{\text{where}} \\ X_{m,n}^{j} & = \frac{{\partial T_{m}^{j} }}{{\partial q_{n}^{j} }}. \\ \end{aligned} $$

(5)

The least squares equation s(q^j) is minimized by differentiating it with respect to each component of unknown heat flux and setting the resulting expressions equal to zero, then a set of N equations is obtained for the estimation of heat flux.

$$ \frac{{\partial s({\varvec{q}}^{j} )}}{{\partial q_{1}^{j} }} = 0,\frac{{\partial s({\varvec{q}}^{j} )}}{{\partial q_{2}^{j} }} = 0, \ldots ,\frac{{\partial s({\varvec{q}}^{j} )}}{{\partial q_{N}^{j} }} = 0 $$

(6)

The above equations yield to a matrix system to estimate the increment for a new heat flux.

$$ \Delta {\mathbf{q}}^{j} = \left( {\sum\limits_{i = 1}^{r} {{\mathbf{X}}_{i}^{T} {\mathbf{X}}_{i} + \alpha {\varvec{H}}} } \right)^{ - 1} \left( {\sum\limits_{i = 1}^{r} {{\mathbf{X}}_{i}^{T} ({\mathbf{Y}}^{j + i - 1} - {\tilde{\mathbf{T}}}^{j + i - 1} )} } \right) $$

(7)

where H is the N × N block diagonal regularization matrix. Both Y^j and T^j are M × 1 vector, and M is the number of measurements.

$$ {\varvec{Y}}^{j} = [Y_{1}^{j} ,Y_{2}^{j} , \ldots ,Y_{M}^{j} ]^{{\text{T}}} $$

(8a)

$$ {\varvec{T}}^{j} = [T_{1}^{j} ,T_{2}^{j} , \ldots ,T_{M}^{j} ]^{{\text{T}}} $$

(8b)

$$ {\varvec{H}} = {\text{blkdiag}}({\varvec{H}}_{{n_{1} }} ,{\varvec{H}}_{{n_{2} }} ,{\varvec{H}}_{{n_{3} }} ) $$

(8c)

$$ {\varvec{H}}_{{n_{i} }} = \left[ {\begin{array}{*{20}c} 1 & { - 1} & {} & {} & {} \\ { - 1} & 2 & { - 1} & {} & {} \\ {} & \ddots & \ddots & \ddots & {} \\ {} & {} & { - 1} & 2 & { - 1} \\ {} & {} & {} & { - 1} & 1 \\ \end{array} } \right]_{{n_{i} \times n_{i} }} . $$

(8d)

Sensitivity Coefficient Matrix

${\text{X}}_{m,n}^{j}$ represents the temperature rise at the sensor location (x_m, y_m) for a unit step change in the surface heat flux at the point (x_n, y_n) of boundaries ∂Ω₁, ∂Ω_2, and ∂Ω₃ and the time t_j. According to the definition of sensitivity coefficient [Eq. (5)], the sensitivity coefficient problem for calculating sensitivity coefficient matrix is obtained by taking the derivative of Eqs. [1a through 1d] with respect to a heat flux component $q_{n}^{j}$ at the point (x_n, y_n) of boundaries ∂Ω₁, ∂Ω_2, and ∂Ω₃, and n is 1, 2, …, N. The sensitivity coefficient problem governing the sensitivity coefficients ${\text{X}}_{m,n}^{j}$ is

$$ \begin{aligned} & c\rho \frac{{\partial X_{,n} }}{\partial t} = \frac{\partial }{\partial x}\left( {\lambda \frac{{\partial X_{,n} }}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {\lambda \frac{{\partial X_{,n} }}{\partial y}} \right), \, 0 < t \le t_{r} \\ & \quad {\text{in}}\;{\text{domain }}\Omega = \left[ {0,W} \right] \times \left[ {0,H} \right] \\ \end{aligned} $$

(9a)

$$ - \lambda \frac{{\partial X_{,n} (\partial \Omega_{1} \cup \partial \Omega_{2} \cup \partial \Omega_{3} ,t)}}{{\partial {\varvec{n}}}} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {{(}x,y) = (x_{n} ,y_{n} {)}} \hfill \\ {0,} \hfill & {{\text{others}}} \hfill \\ \end{array} } \right. $$

(9b)

$$ X_{,n} (\partial \Omega_{4} ,t) = 0 $$

(9c)

$$ X_{,n} \left( {x,y,0} \right) = 0 $$

(9d)

The Determination of Regularization Parameter

The choice of the regularization parameter is required to balance the computational cost and the stability of the inverse problem solution algorithm. The L-curve method is adopted to optimize the regularization parameters α. L-curve criterion that plotted the curve of {log(||Y–T||), log(R(q^j))} often takes on a characteristic L shape, and the optimal regularization parameter was corresponding to the corner of maximum curvature in L-curve [38, 39]. However, many tests should be executed for the plot L-curve, which is very computationally expensive. Alternatively, we consider the use of Arcangeli’s discrepancy principle to significantly speed up the selection of the optimal regularization parameter [40], that is

$$ \left\| {{\mathbf{Y}} - {\mathbf{T}}} \right\| \approx \frac{\sigma }{\sqrt \alpha } $$

(10)

Stopping Criterion for the Iterative Procedure of Inverse Problem

The stopping criterion for Eq. (2) is given by

$$ s \le \varepsilon $$

(11)

As temperatures contain measurement error, the temperature residual could be approximated by

$$ \varepsilon = ({\mathbf{Y}} - {\mathbf{T}})^{{\text{T}}} ({\mathbf{Y}} - {\mathbf{T}}) \cong M\sigma^{2} $$

(12)

where σ is the standard deviation of temperature measurement error.

Algorithm of Inverse Problem

As shown in Fig. 2, specify the number of future time steps r. Step 1, 2DIHCP is running with an initial guess heat flux q^j and a guess regularization parameter α. Step 2, solve the direct problem given by Eq. (1) with the estimated heat flux q^j for T during the time from t_j to t_j+r. Step 3, solve the sensitivity coefficient problem given by Eq. (9a–d) for the sensitivity coefficient matrix X in the time interval from 0 to t_r. Step 4, check the stopping criterion given by Eq. (11). If the stopping criterion Eq. (11) is satisfied, let α_old = α and renew α using Eq. (10). Otherwise, the heat flux is updated using q^j = q^j + Δq and Eq. (7), and return to Step 2. Step 5, when |α − α_old|/α_old < 0.05 is satisfied, q^j could be regarded as the estimated heat flux at time t_j, then set j = j + l and return to Step 1 for the next time step calculation till the end of the time steps. Otherwise, the heat flux is updated using q^j = q^j + Δq and Eq. (7), and return to Step 2. The algorithm is achieved and programed using MATLAB^TM. All the partial differential equations are solved using the finite difference method.

Fig. 2

Algorithm of the two-dimensional inverse heat conduction problem (2DIHCP)

Model Verification

Validation for Solving Partial Differential Equations by Numeric Solution

Partial differential equations of heat conduction problems Eqs. [1a through 1d] and the sensitivity coefficient problem given by Eqs. [9a through 9d] are solved using the finite difference method (FDM) with the classical Crank-Nicolson (CN) semi-implicit scheme. The FDM code must be validated to be effective for solving partial differential equations before the run of the inverse calculation. Herein, the FDM code was also verified through comparison with the analytical solutions of the heat transfer in the solid from a classical textbook [41]. It was observed that the results of the FDM method were consistent with those analytic solutions, suggesting the FDM method can solve the heat conduction partial differential equation.

Validation for Algorithm of the Inverse Problem

The numeric test-problem is designed to verify the 2DIHCP. The direct problem is also a the rectangular area ABCD (made of copper, height H is 0.021 m and width W is 0.008 m, see Fig. 1) with the initial temperature of 273.15 K. The boundaries of ∂Ω₁, ∂Ω_3, and ∂Ω₄ are insulated boundaries, and the heat flux f(y, t) is applied to the ∂Ω₂. The f(y, t) is

$$ f(y,t) = \left\{ {\begin{array}{*{20}l} {1 \times 10^{6} ,} \hfill & {0 \le t < 5} \hfill \\ {1 \times 10^{6} \left( {1 - \frac{1000}{{21}}y} \right)\frac{t}{5} + 1 \times 10^{6} ,} \hfill & {5 \le t < 10} \hfill \\ {1 \times 10^{6} \left( {1 - \frac{1000}{{21}}y} \right) + 1 \times 10^{6} ,} \hfill & {{10} \le t < 15} \hfill \\ {1 \times 10^{6} \left( {1 - \frac{1000}{{21}}y} \right)\frac{15 - t}{5} + 1 \times 10^{6} ,} \hfill & {{15} \le t \le 20} \hfill \\ {1 \times 10^{6} ,} \hfill & {{2}0 \le t < 25} \hfill \\ \end{array} } \right.. $$

(13)

The first column of eight virtual thermocouples was set 3 mm apart vertically from each other and 3 mm away from the mold hot surface; while the second column of other eight virtual thermocouples was spaced 3 mm apart vertically and 8 mm horizontally away from the hot face of the mold. The temperature measurement is taken every 0.1 s during the test-problem runs. The Gaussian noise signals σω (T = T_ture + σω, σ is the standard deviation of noise and ω is a random variable and will be within −2.576 to 2.576 for the 99% confidence bounds) are added to the temperatures to mimic the thermocouple measurement error. Then, the measured temperatures are delivered to the 2DIHCP for the reconstruction of the heat flux in Ω₂. For the parameters used here, the number of future time steps is set as 15, the α is 1.71 × 10^–6, respectively. The estimation results of heat flux in Ω₂ with/without noise (Gaussian noise signals, σ = 0.1, to simulate the temperature measurement errors) and the exact value f(y, t) are shown in Fig. 3, where the heat fluxes at the locations of y at 0 mm, 3, 6, 9, 15, and 21 mm are listed, respectively. It was found that the heat fluxes calculated by 2DIHCP match well with the exact values, even for the situation that the measured temperature was contaminated by noise. This suggests that the 2DIHCP could reconstruct the boundary heat flux precisely and show the ability to resist measurement noises.

Fig. 3

Heat fluxes calculated by 2DIHCP and 1D-IHCP for the test-problem: a heat flux at y = 21 mm. b heat flux at y = 15 mm. c heat flux at y = 9 mm. d heat flux at y = 6 mm. e heat flux at y = 3 mm. f heat flux at y = 0 mm

The results are compared with those values calculated by a robust one-dimensional inverse heat conduction problem (1D-IHCP) developed by Beck et al. [12], where the future time step is 4. As shown in Fig. 3, although the heat fluxes obtained through 1D-IHCP show the same variation tendency with the exact value, the difference between the above two methods is obvious, especially when the measured temperature is contaminated with noise.

Application to Continuous Casting Mold

A continuous casting experiment of liquid steel ([C]: 0.200 wt%, [Si]: 0.230 wt%, [Mn]: 0.490 wt%, [P]: 0.012 wt%, and [S]: 0.03 wt%) was conducted, where the mold was oscillated with a frequency of 2.17 Hz that helps to separate the metal from the mold. With the progressive filling of the liquid steel (shown in Fig. 1), the responding copper mold wall temperatures were measured by thermocouples with a 60 Hz sampling rate, and the pouring temperature of steel was 1833 K. The calculation domain size and the distribution of the thermocouples are the same as the installation of the thermocouples in the numeric test-problem.

Figure 4 shows the measured in-mold temperature history of the first column thermocouples at different positions in the vertical. The responding temperature can be divided into two stages according to the process of the casting. In stage I (51.4–52.5 s), the responding temperatures increase quickly as the progressive filling of the liquid steel from the bottom to the top of the mold. After filling of the melt is completed, the liquid level corresponds to the location S9 with y is 9 mm, where the mold is exposed to air above this location (y > 9 mm). As it steps into stage II (52.5–58 s), the liquid steel is solidified against the surface of the water-cooled mold, the responding temperatures continue to increase and then keep relatively constant. It was found that the steady state temperature values for Y_tc16 (y is 0 mm) and Y_tc14 (y is 3 mm) are the highest, followed by Y_tc12 (y is 6 mm), Y_tc10 (y is 9 mm), Y_tc8 (y is 12 mm), Y_tc6 (y is 15 mm), Y_tc4 (y is 18 mm), and Y_tc2 (y is 21 mm). It is suggested that the thermocouples corresponding to mold surface exposed to the air are cooler than those of thermocouples below the melt level.

Fig. 4

Responding in-mold temperatures

Figure 5 shows the mold heat fluxes calculated by 2DIHCP from measured mold temperatures. For the 2DIHCP runs, the future time step is configured as 8, and α is 7 × 10^–9, respectively. During stage I of filling, it was found that the local heat fluxes increase rapidly and then reach their peak values and the highest value is around 1.03 MW/m², which is similar to industrial and other experimental results [3, 5,6,7,8]. During stage II, all the heat fluxes decrease rapidly at first due to the fast formation of the shell/mold air-gap, and then they continue to decrease gradually. From the above results, it could be suggested that during the initial filling of the liquid steel, the heat fluxes increased dramatically as the hot liquid directly contacted with the cold mold and the latent heat was released greatly with the increase of heat flux in stage I. Then, the initial shell is formed against the hot surface of the water-cooling mold; thus, the air gap in between the mold-metal interface is formed due to the solidification of liquid steel and consequently the interfacial thermal resistance increases dramatically, which explains the reason why the heat flux reduces intensively at the initial stage of stage II. With the further solidification of the liquid steel, the shell continues to grow and correspondingly the total thermal resistance across the solidified shell and mold-shell interface continues to increase. That is the reason why the heat fluxes were reduced gradually throughout the rest of stage II.

Fig. 5

Mold heat flux calculated by 2DIHCP

In the latter part of stage II, the other interesting phenomenon that was observed in this study is that the heat fluxes of locations at S3 (y is 3 mm) and S6 (y is 6 mm) are the highest, followed by S9 (y is 9 mm) and S0 (y is 0 mm), then S12 (y is 12 mm), S15 (y is 15 mm), S18 (y is 18 mm), and S21 (y is 21 mm). It might be explained that the location of S0 below the liquid steel level S9 (y is 9 mm) has a thicker shell thickness due to the longer solidification time, which resulted in a larger thermal resistance than the locations of S3 (y is 3 mm) and S6 (y is 6 mm), which the finding is consistent with other studies [5,6,7,8]. As for the position at the liquid level S9 (y is 9 mm), there exists significant vertical heat transfer upward to the top part of the mold, therefore, its heat flux is a little bit smaller than those of S3 and S6. As for the heat fluxes at S12 (y is 12 mm), S15 (y is 15 mm), S18 (y is 18 mm), and S21 (y is 21 mm), they are decreasing because they are far away from the liquid melt and the existing vertical heat transfer from the liquid to the upper mold. Therefore, the mold heat transfer is two-dimensional from the location 3 to 6 mm below the liquid melt vertically toward the upper part (may be a little downward) and horizontally to the inside of the mold.

The mold wall temperature evolution during the casting process is reconstructed by 2DIHCP and shown in Fig. 6. The temperatures of the lower part of the mold wall are increased (turns to red) at first with the initial filling of the liquid steel, and then the hot red area expands both horizontally and vertically with the progressive filling of the liquid steel, which indicates the significant 2D heat transfer inside the mold as suggested above. It was noticed that there was no significant evolution of the temperature inside the mold after 54 s, which is consistent with Fig. 4, and the maximum temperature of the mold wall reaches as high as 335 K.

Fig. 6

Temperature evolution of the mold wall during the casting

Conclusions

In this work, based on the function specification method with first-order spatial regularization, a two-dimensional transient inverse heat conduction problem (2DIHCP) is established for the determination of the mold heat flux using mold temperatures measured by fast response thermocouples at a frequency up to 60 Hz. The specific conclusions are summarized as:

• 1. The inverse problem of the function specification method implemented with the first-order spatial regularization could improve the temporal and spatial stabilization of recovered mold heat flux.

• 2. The built two-dimensional transient inverse heat conduction problem (2DIHCP) could reconstruct the boundary heat flux precisely and is capable of the ability to resist measurement noises. It could also provide more precise heat flux calculation than the one-dimensional inverse heat conduction problem (1D-IHCP).