Stability of Interfacial Phase Growth in a Slab with Convective Boundary Conditions

Abstract

The mass transport and energy equations for a semi-infinite porous slab are solved using similarity variables and closed form functions to describe freezing with remelt at the interface. Heat and mass balance analyses give a transcendental equation for the unknown interfacial freezing velocity for solving on the computer. The solutions for the temperature and mass concentration are decoupled and solved analytically. The solution for convective boundary conditions is compared with that for Dirichlet boundary conditions. The progressive development of the solution with material thickness and change of functional time dependence and effect on the stability of nucleation is outlined. A discussion with biological adaptation to extreme cold and possible evolution of molecules in heat transfer regimes is included in light of the above.

Appendices

Appendix 1

$$ \begin{aligned} &\frac{\partial }{\partial t}\left( {erfc\left( {\frac{x}{{2\sqrt {at} }} + h\sqrt {at} } \right)\exp \left( {hx + h^{2} at} \right) - {\text{erfc}}\left( {\frac{x}{{2\sqrt {at} }}} \right)} \right) \hfill \\ &\quad = ah^{2} e^{{ah^{2} t + hx}} {\text{erfc}}\left( {h\sqrt {at} + \frac{x}{{2\sqrt {at} }}} \right) \quad - \frac{{2e^{{ah^{2} t - \left( {h\sqrt {at} + \frac{x}{{2\sqrt {at} }}} \right)^{2} + hx}} \left( {\frac{ah}{{2\sqrt {at} }} - \frac{ax}{{4(at)^{3/2} }}} \right)}}{\sqrt \pi } - \frac{{axe^{{ - \frac{{x^{2} }}{4at}}} }}{{2\sqrt {\pi \left( {at} \right)^{3/2} } }} \hfill \\ \end{aligned} $$

$$ \begin{aligned} &\frac{{\partial^{2} }}{{\partial x^{2} }}\left( {{\text{erfc}}\left( {\frac{x}{{2\sqrt {at} }} + h\sqrt {at} } \right)\exp \left( {hx + h^{2} at} \right) - {\text{erfc}}\left( {\frac{x}{{2\sqrt {at} }}} \right)} \right) \hfill \\ & \quad = h^{2} e^{{ah^{2} t + hx}} {\text{erfc}}\left( {h\sqrt {at} + \frac{x}{{2\sqrt {at} }}} \right) - \frac{{2he^{{ah^{2} t - \left( {h\sqrt {at} + \frac{x}{{2\sqrt {at} }}} \right)^{2} }} + hx}}{{\sqrt \pi \sqrt {at} }} \hfill \\ &\qquad + \,\frac{{e^{{ah^{2} t - \left( {h\sqrt {at} + \frac{x}{{2\sqrt {at} }}} \right)^{2} + hx}} \left( {h\sqrt {at} + \frac{x}{{2\sqrt {at} }}} \right)}}{\sqrt \pi at} - \frac{{xe^{{ - \frac{{x^{2} }}{4at}}} }}{{2\sqrt \pi at\sqrt {at} }} \hfill \\ \end{aligned} $$

Appendix 2

$$ \begin{aligned} &he\frac{{ah^{2} t}}{{k^{2} }}{\text{erfc}}\left( {\frac{{h\sqrt {at} }}{k}} \right) + hx\left( {\frac{{he\frac{{ah^{2} t}}{{k^{2} }}{\text{erfc}}\left( {\frac{{h\sqrt {at} }}{k}} \right)}}{k} - \frac{{\sqrt {at} }}{\sqrt \pi at}} \right) \hfill \\ &\quad +\, hx^{2} \left( {\frac{{h^{2} e\frac{{ah^{2} t}}{{k^{2} }}{\text{erfc}}\left( {\frac{{h\sqrt {at} }}{k}} \right)}}{{2k^{2} }} - \frac{{h\sqrt {at} }}{2\sqrt \pi akt}} \right) + hx^{3} \left( {\frac{{h^{3} e\frac{{ah^{2} t}}{{k^{2} }}{\text{erfc}}\left( {\frac{{h\sqrt {at} }}{k}} \right)}}{{6k^{3} }} - \frac{{\sqrt {at} \left( {\frac{{4ah^{2} t}}{{k^{2} }} - 2} \right)}}{{24\sqrt \pi a^{2} t^{2} }}} \right) \hfill \\ &\quad +\, h^{2} x^{4} \frac{{\left( {2\sqrt \pi a^{2} h^{3} t{}^{2}e\frac{{ah^{2} t}}{{k^{2} }}} \right){\text{erfc}}\left( {\frac{{h\sqrt {at} }}{k}} \right) - 2ah^{2} kt\sqrt {at} + k^{3} \sqrt {at} }}{{48\sqrt \pi a^{2} k^{4} t^{2} }} + O\left( {x^{5} } \right) \hfill \\ \end{aligned} $$