The bulk crystal growth in binary supercooled melts with allowance for heat removal

Abstract

A mathematical model describing crystal growth in a supercooled binary melt in the presence of heat removal and withdrawal of product crystals from the working liquid of the crystallizer is formulated. An approximate analytical solution of the integro-differential system of the kinetic and balance equations is constructed using the separation of variables method and the saddle point technique to calculate the Laplace-type integral. The particle size distribution function and the dynamics of desupercooling in the metastable system taking into account the Meirs nucleation kinetics are found. It is shown that the distribution function increases with decreasing the impurity concentration.

Appendix: Expressions determining the analytical solution

$$\begin{aligned}&N\left( U,U^{'},U^{''},t \right) \\&\quad =\frac{R_{1}b_{2}\left( U^{'} \right) ^{2}I^{2}-R_{2}U^{'}I+R_{3}\left( U^{''}I+U^{'}B \right) }{\left( 1+b_{1}U^{'}A \right) U^{'}I+R_{3}U^{'}A}\, , \\&\quad U_{sd}=-MQ_{2}\left( 0 \right) -Q_{1}\left( 0 \right) -\left( Mb_{2}+b_{1} \right) I, \\&\quad R_{1}=Q_{1}+U^{''}+b_{1}U^{'}I,\, \,\\&\quad R_{2}=Q_{1}^{'}+b_{1}U^{''}I+b_{1}U^{'}B+MQ_{2}^{'}\, ,\\&\quad R_{3}=Q_{1}+b_{1}U^{'}I+U^{''}+MQ_{2},\, \\&\quad M=\frac{m\sigma _{0}}{\mathrm {\Delta }\mathrm {\Theta }_{0}},\, \, \, \, J=\left( U^{'} \right) ^{p-1}\, , \\&\quad I=\, \sum \limits _{k=0}^\infty \left( Z_{k}-u_{0}Y_{k} \right) \\&\quad \left[ F_{0k}\exp \left( \frac{-s_{k}}{4u_{0}} \right) +\frac{{4u_{0}\nu }_{k}}{s_{k}'} \right] +J\mathrm {\Psi }+\frac{u_{0}\delta J}{x_{0}+u_{0}}\, , \\&\quad \mathrm {\Psi }=\frac{\left( x_{0}+s_{*} \right) ^{4}-4s_{*}^{3}\left( x_{0}+s_{*} \right) +3s_{*}^{4}}{12\left( x_{0}+u_{0} \right) },\, \\&\quad \delta =\left[ \frac{\left( x_{0}+s_{*} \right) ^{3}}{3}-\frac{s_{*}^{3}}{3} \right] , \\&\quad A=-\sum \limits _{k=0}^\infty \left( Z_{k}-u_{0}Y_{k} \right) \frac{\nu _{k3}\left( U^{'} \right) ^{p-2}4u_{0}}{s_{k}'}\, , \\&\quad B=\sum \limits _{k=0}^\infty \left( Z_{k}-u_{0}Y_{k} \right) \\&\quad \left[ F_{0k}\exp {\left( \frac{-s_{k}}{4u_{0}} \right) \left( \frac{-s_{k}'}{4u_{0}} \right) +\frac{4u_{0}Es_{k}^{'}-4u_{0}\nu _{k}s_{k}^{''}}{\left( s_{k}^{'} \right) ^{2}}} \right] \\&\quad +\left( p-1 \right) \left( U^{'} \right) ^{p-2}U^{''}\mathrm {\Psi }+\frac{u_{0}\delta \left( p-1 \right) \left( U^{'} \right) ^{p-2}U^{''}}{x_{0}+u_{0}}\, , \\&\quad E=\nu _{k1} p\left( U^{'} \right) ^{p-1}U^{''}-\nu _{k2}\left( p-1 \right) \left( U^{'} \right) ^{p-2}U^{''}-\nu _{k3}\\&\quad \left( p-2 \right) \left( U^{'} \right) ^{p-3}\left( U^{''} \right) ^{2}\, ,\, \\&\quad Z_{k}={\int ^{{x}_{0}}_0} X_{k} \left( x \right) \left( x+s_{*} \right) ^{2}dx\, ,\, \, \, \, \\&\quad \mathrm {Y}_{\mathrm {k}}={\int ^{{x}_{0}}_0} \left( x+s_{*} \right) ^{2} \left( \frac{dX_{k}}{dx} \right) dx, \\&\quad \mathrm {\Phi }_{k}\left( U^{'} \right) = \left( 1+4u_{0}^{2}n_{k}^{2} \right) U^{'}\left( t \right) +4u_{0}\gamma , \\&\quad \nu _{k1}=\frac{1}{I_{k}}{\int ^{x_{0}}_0} \frac{X_{k}\left( x \right) \exp \left( -\frac{x}{u_{0}} \right) dx}{x_{0}+u_{0}} ,\, \,\\&\quad \nu _{k2}=\frac{\gamma }{I_{k}}{\int ^{{x}_{0}}_0} \frac{(x_{0}-x)X_{k}\left( x \right) \exp \left( -\frac{x}{u_{0}} \right) dx}{x_{0}+u_{0}} ,\,\\&\quad \nu _{k3}=\frac{\left( p-1 \right) }{I_{k}}{\int ^{{x}_{0}}_0} \frac{\left( x_{0}-x \right) X_{k}\left( x \right) \exp \left( -\frac{x}{u_{0}} \right) dx}{x_{0}+u_{0}} \, .\, \, \, \, \end{aligned}$$