Symbols	Nomenclature
${C_0}^s$	The reference concentration value for species s
$D$	Solute diffusion coefficient in the liquid phase
$E$	Elastic modulus using multiple regression analysis to predict [MPa]
$H$	Enthalpy [$\mathrm{k}\mathrm{J}·{\mathrm{k}\mathrm{g}}^{-1}$]
$K$	Heat transfer coefficient [$\mathrm{W}·{\mathrm{m}}^{-1}·{°\mathrm{C}}^{-1}$], the material Constant
$K_P$	The permeability coefficient
$L$	Latent heat of solidification [$\mathrm{J}·{\mathrm{k}\mathrm{g}}^{-1}$]
$R$	Radius of a dendrite tip, billet
$T,T_{water},T_{saturation}$	Temperature, Initial, Saturation temperature of cooling water [°C]
Y	Direction of pull
$a_2,a_3$	Dynamic coefficients of dendrite growth
$c$	Specific heat [J$·{\mathrm{k}\mathrm{g}}^{-1}·{°\mathrm{C}}^{-1}$]
${f,f}_s,{f_s}^{th}$	Yield stress, solid fraction, packing limit
$h_{con}$	Heat transfer coefficient of the melt contact to the mold [$\mathrm{W}·{\mathrm{m}}^{-2}·{\mathrm{K}}^{-1}$]
$h_{air}$	Heat transfer coefficient at air gap [$\mathrm{W}·{\mathrm{m}}^{-2}·{\mathrm{K}}^{-1}$], $h_{air}=150[\mathrm{W}·{\mathrm{m}}^{-2}·{\mathrm{K}}^{-1}]$
$k$	The partition coefficient
$m$	The slope of the Al-X liquidus, the strain rate sensitivity Coefficient
$n$	Nucleation density [${\mathrm{m}}^{-3}$], the strain hardening coefficient
$n_{max}$	Maximum nucleation density [${\mathrm{m}}^{-3}$]
$y,\tilde{y},{\widehat{y}}_i$	Truth value, predicted value, the overall mean of truth value
$\rho$	Density [$\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}$]
$p$	Pressure [$\mathrm{P}\mathrm{a}$]
$t$	Time [$\mathrm{s}]$
$u$	Velocity [$\mathrm{m}/\mathrm{s}]$
$\mu ,{\mu }_t$	The dynamic viscosity, turbulent viscosity
${\alpha }_0,{\alpha }_1,{\alpha }_2\dots {\alpha }_n$	Regression coefficient, ${\alpha }_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$
$\sigma$	Node stress [MPa]
${{\epsilon }_s}^T,{{\epsilon }_s}^{\epsilon },{{\epsilon }_s}^p$	Thermal, elastic, plastic strain
${\epsilon }_p,{\dot{\epsilon }}_p,{{\epsilon }_p}^0$	Total plastic strain, equivalent plastic strain rate [${\mathrm{s}}^{-1}$], initial strain
$\delta$	Regression errors
$\mu ,\nu$	Residual sum of squares, total sum of squares
$∆T$	Total undercooling [°C]
$∆T_{max}$	Mean nucleation undercooling [°C]
$∆T_{\sigma }$	Standard deviation of nucleation undercooling [°C]
$∆T_c$	The undercooling contributions of solute diffusion [°C]
$∆T_t$	The undercooling contributions of thermal diffusion [°C]
$∆T_r$	The undercooling contributions of solid/liquid interface curvature [°C]
$∆T_k$	The undercooling contributions of attachment kinetics [°C]
$∆{\overline{T}}_S,∆{\overline{T}}_V$	Surface, bulk nuclear undercooling [K]
$∆T_{\sigma ,S},∆T_{\sigma ,V}$	Standard deviation of surface, bulk nuclear [K]
$n_{S,max},n_{V,max}$	Surface, bulk grain density [m−2]
$\mathsf{\Gamma }$	Gibbs–Thomson coefficient ($\mathsf{\Gamma }=2\times {10}^{-7}$ in this study)
$\gamma$	The plastic multiplier
${\beta }_T,{\beta }_S$	The thermal expansion coefficient, the solute expansion coefficient for species s