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

Abstract

To ensure accurate estimation of mold heat flux, this study investigated the impacts of the thermocouples placement inserted in the mold wall, the temperature sampling rate (f_s), and the noise level of temperature data on the precision of two-dimensional Inverse Heat Conduction Problem (2DIHCP). The results showed the accuracy of heat flux estimations decreases as the distance between the thermocouple and the mold surface increases, and it is recommended that the distance should not exceed 3 mm. The accuracy of the heat flux initially increases as f_s increases from 5 to 10 Hz, reaches a relatively stable state as f_s increases from 10 to 60 Hz, and eventually decreases as f_s increases from 60 to 100 Hz. Additionally, higher temperature measurement errors typically lead to decreased accuracy in inverse analysis. 2DIHCP was employed to compute the heat flux for a mold simulator experiment, and the results demonstrated its effectiveness in reconstructing the mold heat flux at the meniscus level during the time lapse of a mold oscillation cycle.

Appendix A: Sensitivity Coefficient Matrix

${\mathbf{J}}_{{\text{j}}}$ is called as the M × N sensitivity coefficient matrix at time t_j and is defined as follows:

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

(A1)

$J_{m,n}^{j}$ represents the temperature rise at the sensor location (x_m, y_m) in response to a unit step change in the heat flux at point (x_n, y_n) on boundaries Γ₁, Γ₂, and Γ₃, at time t_j. To obtain the sensitivity coefficient problem, the partial derivative of Eqs. [2b] through [2e] is taken with respect to a heat flux component $q_{n}^{j}$, (Eq. [A1]), which yields the governing sensitivity coefficient problem.

$$\frac{\partial J}{\partial t}=\frac{\partial^2J}{\partial x^2}+\frac{\partial^2J}{\partial y^2},0<t\leq t_r\,\text{in}\,\Omega=[0,W]\times[0,H]$$

(A2a)

$$ \left. { - \frac{{\partial J}}{{\partial \varvec{n}}}} \right|_{{\Gamma _{{1,}} \cup \Gamma _{2} \cup \Gamma _{3} }} = \left\{ \begin{gathered} \begin{array}{*{20}l} {1,} & {{\text{(}}x,y) = (x_{n} ,y_{n} {\text{)}}} \\ \end{array} \hfill \\ \begin{array}{*{20}l} {0,} & {{\text{others}}} \\ \end{array} \hfill \\ \end{gathered} \right. $$

(A2b)

$$ \left. J \right|_{{_{{\Gamma_{4} }} }} = 0 $$

(A2c)

$$ J(x,y,0) = 0 $$

(A2d)