Thermoelastic Stability Analysis of Solidification of Pure Metals on a Coated Planar Mold of Finite Thickness

Abstract

A theoretical model for investigating the thermoelastic instability/mechanism during pure metal solidification on a coated mold of finite thickness is developed. This study extends the previous theoretical works on growth instability during solidification process by investigating the effects of an added coating layer. Mold coating is one of the most important factors controlling the heat transfer rate, and hence it has a very important role on the solidification rate and the development of microstructure. In this model, thermal and mechanical problems are coupled through the pressure-dependent contact resistances at mold/coating and coating/shell interfaces. The thermal diffusivities of solidified shell, coating, and mold materials are assumed to be zero. This assumption provides us to solve heat transfer problem analytically. A linear perturbation method is used to simplify complexity of the modeled solidification problem, and governing equations are solved numerically using a variable step variable order predictor–corrector algorithm. The effects of coating layer thickness and coupling rates at shell/coating and coating/mold interfaces are investigated in detail. The results show that coating thickness has destabilizing effect on the growth instability when the coupling rates are small. However, when these coupling rates are increased individually or together, the destabilizing effect of coating thickness turns to be stabilizing. On the other hand, coupling rates have generally destabilizing effects on the process, but an increase in the thickness of coating leads to diminishing coupling rates effect in some cases.

Appendix

The time-dependent coefficients in Eqs. [33] through [35] are found as follows:

$$ c_{1} \left( t \right) = \frac{{c_{3} \left( t \right)K^{b} }}{{K^{c} }};\;c_{2} \left( t \right) = - \left[ {s_{1} \left( t \right)\frac{{Q_{0} \left( t \right)}}{{K^{c} \cosh \left( {ms_{0} } \right)}} + c_{1} \left( t \right){ \tanh }(ms_{0} )} \right] $$

(A1)

$$ \begin{aligned} c_{3} \left( t \right) = \frac{{Q_{1} \left( t \right)}}{{mK^{b} }}\left( {\cosh \left( {mu + mh} \right)\cosh^{2} \left( {mu} \right) - \sinh \left( {mu + mh} \right)\sinh \left( {mu} \right)\cosh \left( {mu} \right)} \right) \hfill \\ + \left( {\frac{{Q_{1} \left( t \right)}}{{Q_{0} \left( t \right)}} + \frac{{R^{\prime}_{\text{m}} P_{1} \left( t \right)}}{{R_{\text{m}} }}} \right)\left( {T_{\text{m}} + Q_{0} \left( t \right)\left( { - \frac{u}{{K^{b} }} - \frac{{s_{0} \left( t \right)}}{{K^{c} }} - R - \frac{h}{{K^{d} }}} \right)} \right){ \sinh }(mu) \hfill \\ + \frac{{Q_{1} \left( t \right)}}{{mK^{d} }}\left( { - \cosh \left( {mu + mh} \right)\sinh^{2} \left( {mu} \right) + \sinh \left( {mu + mh} \right)\cosh \left( {mu} \right){ \sinh }(mu)} \right) \hfill \\ \end{aligned} $$

(A2)

$$ \begin{aligned} c_{4} \left( t \right) = \left( {\frac{{Q_{1} \left( t \right)}}{{Q_{0} \left( t \right)}} + \frac{{R^{\prime}_{\text{m}} P_{1} \left( t \right)}}{{R_{\text{m}} }}} \right)\left( {T_{\text{m}} + Q_{0} \left( t \right)\left( { - \frac{u}{{K^{b} }} - \frac{{s_{0} \left( t \right)}}{{K^{c} }} - R - \frac{h}{{K^{d} }}} \right)} \right){ \cosh }(mu) \hfill \\ + \frac{{Q_{1} \left( t \right)}}{{mK^{d} }}\left( { - \cosh \left( {mu + mh} \right)\cosh \left( {mu} \right){ \sinh }(mu) + \sinh \left( {mu + mh} \right)\cosh^{2} \left( {mu} \right)} \right) \hfill \\ \frac{{Q_{1} \left( t \right)}}{{mK^{b} }}\left( {\cosh \left( {mu + mh} \right)\cosh \left( {mu} \right)\sinh \left( {mu} \right) - \sinh \left( {mu + mh} \right)\sinh^{2} \left( {mu} \right)} \right) \hfill \\ \end{aligned} $$

(A3)

$$ c_{5} \left( t \right) = \frac{{Q_{1} \left( t \right)}}{{mK^{d} }}\cosh \left( {mu + mh} \right);\;c_{6} \left( t \right) = \frac{{Q_{1} \left( t \right)}}{{mK^{d} }}{ \sinh }(mu + mh) $$

(A4)

The coefficients of Eq. [36], which denotes the amplitude of the added perturbation on the extracted heat flux at the bottom of the mold, are

$$ d_{1} = L^{c} \rho^{c} \frac{{{\text{d}}s_{1} \left( t \right)}}{{{\text{d}}t}} + ms_{1} \left( t \right)Q_{0} \left( t \right)\tanh \left( {ms_{0} \left( t \right)} \right) - \frac{{R^{\prime}_{\text{m}} \left( {P_{o} } \right)P_{1} \left( t \right)mK^{b} E{ \sinh }(mu)}}{{\cosh \left( {ms_{0} \left( t \right)} \right)R_{\text{m}} }} $$

$$ d_{2} = \frac{{\cosh \left( {mu + mh} \right)\cosh^{2} (mu) - \sinh \left( {mu + mh} \right)\sinh \left( {mu} \right){ \cosh }(mu)}}{{{ \cosh }(ms_{0} \left( t \right))}} + \frac{{E\sinh \left( {mu} \right)mK^{b} }}{{\cosh \left( {ms_{0} \left( t \right)} \right)Q_{0} (t)}} + \frac{{K^{b} }}{{K^{d} }}\left( {\frac{{ - \cosh \left( {mu + mh} \right)\sinh^{2} \left( {mu} \right) + \sinh \left( {mu + mh} \right)\cosh \left( {mu} \right){ \sinh }(mu)}}{{{ \cosh }\left( {ms_{0} \left( t \right)} \right)}}} \right) $$

$$ E = T_{\text{m}} + Q_{o} (t)\left( { - \frac{u}{{K^{b} }} - \frac{{s_{0} \left( t \right)}}{{K^{c} }} - R - \frac{h}{{K^{d} }}} \right) $$

The coefficients of the first coupled equation in Eq. [37] which obtained at the end of the thermal problem are

$$ \begin{aligned} d_{3} = \hfill \\ \left( {L^{c} \rho^{c} (\cosh \left( {mh} \right)\cosh \left( {ms_{0} \left( t \right)} \right)\sinh \left( {mu} \right)K^{c} K^{d} } \right) \hfill \\ + \sinh \left( {mu} \right)\sinh \left( {ms_{0} \left( t \right)} \right)K^{{b^{2} }} \left( {\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right) \hfill \\ + K^{b} (\cosh \left( {mh} \right)\cosh \left( {mu} \right)\sinh \left( {ms_{0} \left( t \right)} \right)K^{d} \hfill \\ + K^{b} (\cosh \left( {mh} \right)\cosh \left( {mu} \right)\sinh \left( {ms_{0} \left( t \right)} \right)K^{d} \hfill \\ + \left. {\left. {\cosh \left( {ms_{0} \left( t \right)} \right)K^{c} \left( {\cosh \left( {mu} \right)\left( {\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right) + mK^{d} R} \right)} \right)} \right) \hfill \\ {{\left( {uK^{c} K^{d} + K^{b} \left( {K^{c} \left( {h + K^{d} \left( {R_{\text{m}} + R} \right) + K^{d} s_{0} \left( t \right)} \right)} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {uK^{c} K^{d} + K^{b} \left( {K^{c} \left( {h + K^{d} \left( {R_{\text{m}} + R} \right) + K^{d} s_{0} \left( t \right)} \right)} \right)} \right)} {\left. {\left. {\left( {mT_{\text{m}} K^{{b^{2} }} K^{{c^{2} }} K^{d} (\cosh \left( {mh} \right)\cosh \left( {mu} \right)K^{d} + \sinh \left( {mu} \right)K^{b} (\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right)} \right)R} \right)}}} \right. \kern-0pt} {\left. {\left. {\left( {mT_{\text{m}} K^{{b^{2} }} K^{{c^{2} }} K^{d} (\cosh \left( {mh} \right)\cosh \left( {mu} \right)K^{d} + \sinh \left( {mu} \right)K^{b} (\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right)} \right)R} \right)}} \hfill \\ \end{aligned} $$

(A5)

$$ \begin{aligned} d_{4} = \hfill \\ \left( {mT_{\text{m}} \cosh \left( {ms_{0} \left( t \right)} \right)\sinh \left( {mh} \right)\sinh \left( {mu} \right)K^{{b^{3} }} K^{c} K^{d} + mT_{\text{m}} \cosh \left( {mu} \right)\sinh \left( {mh} \right)\sinh \left( {ms_{0} \left( t \right)} \right)K^{{b^{2} }} K^{{c^{2} }} K^{d} + } \right. \hfill \\ mT_{\text{m}} \cosh \left( {mh} \right)\cosh \left( {mu} \right)\cosh \left( {ms_{0} \left( t \right)} \right)K^{{b^{2} }} K^{c} K^{{d^{2} }} + mT_{\text{m}} \cosh \left( {mh} \right)\sinh \left( {mu} \right)\sinh \left( {ms_{0} \left( t \right)} \right)K^{b} K^{{c^{2} }} K^{{d^{2} }} + \hfill \\ m^{2} T_{\text{m}} \cosh \left( {ms_{0} \left( t \right)} \right)\sinh \left( {mu} \right)K^{{b^{3} }} K^{c} K^{{d^{2} }} R_{m} + m^{2} T_{\text{m}} \cosh \left( {mu} \right)\sinh \left( {ms_{0} \left( t \right)} \right)K^{{b^{2} }} K^{{c^{2} }} K^{{d^{2} }} R_{\text{m}} + \hfill \\ \left. {m^{2} T_{\text{m}} \sinh \left( {ms_{0} \left( t \right)} \right)K^{{b^{2} }} K^{{c^{2} }} K^{{d^{2} }} R} \right)/ \hfill \\ \left( {mT_{\text{m}} K^{{b^{2} }} K^{{c^{2} }} K^{d} \left( {\cosh \left( {mh} \right)\cosh \left( {mu} \right)K^{d} + \sinh \left( {mu} \right)K^{b} \left( {\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right)} \right)R} \right) \hfill \\ \end{aligned} $$

(A6)

$$ \begin{aligned} d_{5} = \hfill \\ \left( {mT_{\text{m}} \cosh \left( {mh} \right)K^{{b^{2} }} K^{{c^{2} }} K^{{d^{2} }} R^{\prime}_{\text{m}} - m^{2} T_{\text{m}} \sinh \left( {mu} \right)K^{{b^{3} }} K^{{c^{2} }} K^{{d^{2} }} R^{\prime}_{\text{m}} R + } \right. \hfill \\ mT_{\text{m}} \sinh \left( {mh} \right)\sinh \left( {mu} \right)K^{{b^{3} }} K^{{c^{2} }} K^{d} R^{\prime} + mT_{\text{m}} \cosh \left( {mh} \right)\cosh \left( {mu} \right)K^{{b^{2} }} K^{{c^{2} }} K^{{d^{2} }} R^{\prime} + \hfill \\ {{\left. {m^{2} T_{\text{m}} \sinh \left( {mu} \right)K^{{b^{3} }} K^{{c^{2} }} K^{{d^{2} }} R_{\text{m}} R^{\prime}} \right)} \mathord{\left/ {\vphantom {{\left. {m^{2} T_{\text{m}} \sinh \left( {mu} \right)K^{{b^{3} }} K^{{c^{2} }} K^{{d^{2} }} R_{\text{m}} R^{\prime}} \right)} {\left( {mT_{\text{m}} K^{{b^{2} }} K^{{c^{2} }} K^{d} \left( {\cosh \left( {mh} \right)\cosh \left( {mu} \right)K^{d} + \sinh \left( {mu} \right)K^{b} \left( {\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right)} \right)R} \right)}}} \right. \kern-0pt} {\left( {mT_{\text{m}} K^{{b^{2} }} K^{{c^{2} }} K^{d} \left( {\cosh \left( {mh} \right)\cosh \left( {mu} \right)K^{d} + \sinh \left( {mu} \right)K^{b} \left( {\sinh \left( {mh} \right) + mK^{d} R_{\text{m}} } \right)} \right)R} \right)}} \hfill \\ \end{aligned} $$

(A7)