Published in:
27-11-2018 | Discussion/Reply
Authors’ Reply to Comments on “A New Analytical Approach to Predict Spacing Selection in Lamellar and Rod Eutectic Systems”
Authors:
Adrian V. Catalina, Subhayu Sen, Doru M. Stefanescu
Published in:
Metallurgical and Materials Transactions A
|
Issue 2/2019
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Excerpt
The authors, Song
et al., of the article[
1] claim that a “fatal mistake” was found in our paper[
2] that discusses the spacing selection in binary eutectic systems. The issue that Song
et al. bring into question is the correctness of Eq. [9a] of Reference
2, which is also presented in an explicit form as Eq. [4] in Reference
1 and reproduced here for clarity from our original paper:
$$ \begin{aligned} B_{\text{o}} & = \left[ {\left( {f_{\alpha } \nu_{\alpha } + f_{\beta } \nu_{\beta } } \right) - \left( {f_{\alpha } \nu_{\alpha } k_{\alpha } + f_{\beta } \nu_{\beta } k_{\beta } } \right)} \right] \cdot \left( {C_{\text{E}} + B_{\text{o}} } \right) \\ & \quad + \left[ {\nu_{\alpha } \left( {1 - k_{\alpha } } \right) - \nu_{\beta } \left( {1 - k_{\beta } } \right)} \right] \cdot \sum\limits_{n = 1}^{\infty } {B_{n} \frac{{\sin (n\pi f_{\alpha } )}}{n\pi }} . \\ \end{aligned} $$
(9a)
At this point, we would like to invite the authors of Reference
1 to carefully consider the boundary condition given by Eq. [3] of their paper, which is identical to Eq. [8] of our paper, and integrate it within the appropriate limits, also by accounting that
CI(
x) is the liquid concentration at the solid/liquid interface as given by Eq. [1] of Reference
1 (
i.e., for
z = 0). We trust that they will obtain the result shown in Eq. [4] of Reference
1. Certainly, this is different from Eq. [10] of Song
et al. The reason for these different results resides in the fact that Song
et al. substituted
CI(
x) for the eutectic concentration,
CE, in the boundary condition (see Eqs. [7-1] and [7-2] of Reference
1). This substitution is a simplification proposed in the original treatment of Jackson and Hunt (JH)[
3] which, as explained by JH in their note, can be used “with quite good accuracy for most cases.” Therefore, when setting
να = 1 and
νβ = 1, the result obtained by Song
et al. for the
Bo term (Eq. [10] of Reference
1) becomes identical to that of JH, except for being expressed in a different form. Consequently, the results of Song
et al. are also correct, as they are exactly the JH solution. …