The Estimation and Control of the Electroslag Remelting Melt Rate by Mechanism-Based Modeling

The process control of industrial electroslag remelting production is addressed in this work. This article proposes a mechanism-based model using electrode displacement to estimate the melt rate, designs the remelting process control system, and uses practical application data to verify the validity of the model. The soft measurement of the melt rate based on mechanism modeling is proved to be an economical and reliable solution to the online melt rate estimation and control for large industrial electroslag remelting furnaces.

Appendix A

According to Figure 4, the conservation of mass is stated as

$$ \rho_{e} \frac{{\pi d_{t}^{2} }}{4}\left( {L_{t - 1} - L_{t} } \right)\kappa = \tilde{\rho }\frac{{\pi \tilde{D}_{t}^{2} }}{4}\left( {y_{t - 1} - y_{t} } \right) $$

(A1)

Assuming that the electrode immersion is constant, the melted length of the electrode will be obtained in terms of the downward displacement of electrode Δx as follows:

$$ L_{t - 1} - L_{t} = {{\Updelta}}x + y_{t} - y_{t - 1} - {{\Updelta}}h $$

(A2)

When the slag pool height increases by Δh, the following equation must hold according to the conservation of mass:

$$ \hat{\rho }\frac{{\pi D_{t}^{2} }}{4}\Updelta h = \eta \rho_{e} \frac{{\pi d_{t}^{2} }}{4}\left( {L_{t - 1} - L_{t} } \right) - \rho_{t} \frac{\pi }{4}\left( {D_{t}^{2} - \tilde{D}_{t}^{2} } \right)\left( {y_{t} - y_{t - 1} } \right) $$

(A3)

Furthermore, applying the conservation of mass to electrode consumption, there is

$$ \kappa + \eta = 1 $$

(A4)

Because both the electrode and the crucible have the shape of the frustum of a cone, the time-varying diameters of these objects are computed by

$$ D_{t} = \frac{{h_{c} - y_{t} }}{{h_{c} }}\left( {d_{\text{bot}} - d_{\text{top}} } \right) + d_{\text{top}} $$

(A5)

$$ d_{t} = \frac{{h_{\text{trod}} - L_{t} }}{{h_{\text{trod}} }}\left( {\varphi_{\text{bot}} - \varphi_{\text{top}} } \right) + \varphi_{\text{top}} $$

(A6)

Based on the preceding equations, the length of the melted portion of the electrode L _t−1 − L _t can be computed, from which the melted weight the melt rate V _m are obtained.

Because the slag skin is usually thin (less than 2 mm), it can be neglected in computation. Thus, the crucible’s inner diameter is approximately equal to the ingot diameter, i.e., $D_{t} = \tilde{D}_{t}$. Using Eqs. [A1], [A2], and [A3], the length of the melted part of the electrode is given by

$$ L_{t - 1} - L_{t} = \frac{1}{{1 - \left( {\frac{\eta }{{\hat{\rho }}} + \frac{\kappa }{{\tilde{\rho }}}} \right)\rho_{e} \varsigma_{t}^{2} }}\Updelta x $$

(A7)

The ingot height is

$$ y_{t + 1} = y_{t} + \frac{1}{{\frac{1}{{\mu \varsigma_{t}^{2} \kappa }} - \left( {1 + \frac{{\eta \tilde{\rho }}}{{\kappa \hat{\rho }}}} \right)}}\Updelta x $$

(A8)

The melt rate is

$$ V_{m} = \frac{{\left( {L_{t - 1} - L_{t} } \right)\pi d_{t}^{2} \rho_{e} }}{{4T_{0} }} $$

(A9)