Analytically Solved Solid Fraction Model for the Newtonian Thermal Analysis of Casting
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THERMAL analysis is a practical tool widely used to examine the solidification characteristics of metals.[1] In the Newtonian thermal analysis of casting, a single thermocouple is placed at the center of the cup and the instantaneous temperature of the melt is measured.[2] The aims of thermal analysis include the computation of the latent heat and determination of the solid fraction. First, the Newtonian baseline, a fictitious cooling rate curve that assumes no phase transformation, is defined.[3] Then the latent heat and solid fraction are determined by computing the area between the experimental cooling curve and the baseline.[4] The mathematical model of the Newtonian method assumes a linear relationship between the heat flux and the temperature of the cooling sample. It simply defines the cooling rate of the sample in the single-phase interval as dT hA ¼ ðT Ta Þ; dt mcp
½1
where h is the convective heat transfer coefficient, A is the total surface area, m is the mass, cp is the specific heat, T is the temperature, t is the time, and Ta is the ambient temperature.[5] The solution of the cooling rate as a function of time gives the Newtonian baseline (zero curve) in terms of the given parameters (Eq. [2]): dT hA t ¼ ae mcp ; ½2 dt zc where a is the integration constant from the initial condition and the subscript zc denotes the zero curve.[6] Solidification, or any phase transformation that produces heat, can be represented as a heat source in the cooling equation. Therefore, the contribution of solidification to the cooling rate can be defined using the
KADIR CAN ERBAS ¸ , formerly Part-time Lecturer with the Basic Sciences Unit, TED University, Ziya Go¨kalp Caddesi, Ankara, Turkey. Contact e-mail: [email protected] Manuscript submitted November 18, 2014. Article published online April 8, 2016 3026—VOLUME 47A, JUNE 2016
solid fraction (fs), specific heat (cp), and latent heat of fusion (Lf) as in Eq. [3]: dT hA Lf dfs ¼ : ðT T a Þ þ dt mcp cp dt
½3
Since Eq. [3] is valid in the solidification interval, the integration of the equation between the start and end times of solidification gives the latent heat (Lf/cp).[7] The required parameters are found by fitting the single-phase region of the cooling curve to the function given in Eq. [2].[8] The Newtonian method is widely used for metallurgical applications; however, some of the many defects discussed in the literature are summarized below. C¸etin and Kalkanlı report that the Newtonian method is unreliable for the estimation of latent heat with potential errors being seen in the variations in pouring temperature and the reaction between the sample and the sand cup.[9] Another critique of the melt-cup interaction is reported by Emadi et al., who suggest using small-thermal capacity crucible to achieve a good estimation of latent heat; otherwise, the results of the analysis reflect the thermal properties of metal-cup system rather than the metal itself.[10] To avoid the complications due to the mold, Erbas¸ proposes a new baseline, two-capacitive system baseline
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