A new analytical model for the growth rate of dendrite tips

It's well known that the driving forces of dendrite growth in metal solidification typically include temperature undercooling, constitutional undercooling, and curvature undercooling, and these undercoolings are usually computed independently by their corresponding local temperature, concentration, and curvature in a model. This paper derived a new formula only depending on the temperature undercooling based on the dynamic equilibrium relationship and the Ivantsov function for computing the growth rate of dendrite tips at the equilibrium state. By taking the derivative of this formula to find the maximum growth rate, an almost constant ratio of the curvature undercooling to the temperature undercooling was found, specifically, the ratio is in the range of 0.507 to 0.524 corresponding to the temperature undercooling of 4 to 10 K. The analytical result was validated by the numerical simulations of the solidification of Inconel 718 alloy with different temperature undercoolings by using a cellular-automata mothed. The numerical results suggested a ratio of 2/3 of the curvature undercooling to the temperature undercooling for the given Inconel 718 alloy.