Authors’ Reply to Comments on “A New Analytical Approach to Predict Spacing Selection in Lamellar and Rod Eutectic Systems”
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The reason for the simplification proposed in the JH treatment was to make the mathematics tractable when calculating the Fourier coefficients of the general solution for liquid concentration. This was used by JH when calculating the Bo term as well as the coefficients Bn. Catalina et al. also used this simplification when calculating Bn, but, in an attempt to improving the solution accuracy, chose to use the actual CI(x) for the calculation of Bo. Unfortunately, this detail was missed by Song et al., although it is clearly stated when the JH approximation was used in Reference 2. It must be pointed out that there is still an ongoing effort aimed to refining the solution of eutectic growth and extend it from binary to multicomponent systems. Examples of such effort can be found in References 4 through 6, just to name a few.
Most of the time, the solidification processes can be described mathematically as boundary value problems, with the solid/liquid interface boundary being also the solution of the problem. Consequently, many different solutions can be obtained for the same process, depending on the assumptions used in the mathematical formulation. It is exactly the case discussed in this article. Therefore, the statement of Song et al. that a “fatal mistake” made its way in our formulation has no validity as it is the consequence of their failure to recognize the difference in the boundary conditions used in our approach compared to the JH treatment.
- 1.K. Song, H. Zhao, W. Zhao, B. Su, and Z. Wang: Metall. Mater. Trans. A, 2018, https://doi.org/10.1007/s11661-018-5026-0.
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- 4.A. V. Catalina and D.M. Stefanescu: in Proceedings of the 8th Pacific Rim Confereence on Modeling of Casting and Solidification Processes (MCSP8-2010), J.K. Choi, H.Y. Hwang and J.T. Kim, eds., April 12–15, 2010 pp. 125–32.Google Scholar