**Author details**

wavevectors simulated are *n* ¼ 0, 1, and 2. The experimental data are close to the

**Figure 3** shows the experimental scatter for Vanadium from the absolute zero to the melting point, Thermo-Calc and Ferreira et al. model's calculations [15, 16]. The predictions for *n* ¼ 1 agrees, for the whole temperature range, with the experimen-

**Figure 4** shows the molar specific heat for Titanium, the experimental data from Chase [24] and found in Desai [23]. Chase experimental data, in green, follow *n* ¼ 2 for the whole temperature range. In this case, Chase's experiment's thermodynamic conditions allow concluding that no phase transition at *T* ¼ 1156*K* takes place, which configures a non-fundamental state specific heat. On the other hand, after the transition temperature, Desai's [23] experimental data and Thermo-Calc agree with the theoretical model for *n* ¼ 0 from 1156 to 1941 K, configuring a fundamen-

The model previously proposed by Ferreira et al. [15, 16] based on the critical radius of phase nucleation to determine the total numbers of modes, and consequently, the Density of State successfully predicted the molar specific heat capacity of transitional elements. In Cr and V, the experimental data follow the theoretical prediction curves with *n* ¼ 2 and *n* ¼ 1, respectively. Furthermore, the model's calculation for Nb agrees with the experimental data except for the set found in Kirillin et al. [18]. The thermophysical properties of Niobium at high temperatures and experimental difficulties might be the reasons responsible for the slight deviation observed between the predictions and experimental data at high temperatures. For Titanium, non-fundamental states and fundamental state molar heat capacity were predicted experimentally and theoretically, as Chase's experiments follow the

The authors acknowledge the financial support provided by FAPERJ (The Scientific Research Foundation of the State of Rio de Janeiro), CAPES and CNPq

(National Council for Scientific and Technological Development).

theoretical calculations for *n* ¼ 0.

*Recent Advances in Numerical Simulations*

tal data, and Thermo-Calc.

tal state specific heat.

model's theoretical predictions for *n* ¼ 2.

**Acknowledgements**

**274**

**4. Conclusions**

Ivaldo Leão Ferreira<sup>1</sup> \*, José Adilson de Castro<sup>2</sup> \* and Amauri Garcia<sup>3</sup> \*

1 Faculty of Mechanical Engineering, Federal University of Pará, UFPA, Belém, PA, Brazil

2 Graduate Program in Metallurgical Engineering, Fluminense Federal University, Volta Redonda, RJ, Brazil

3 Department of Manufacturing and Materials Engineering, University of Campinas – UNICAMP, Campinas, SP, Brazil

\*Address all correspondence to: ileao@ufpa.br, joseadilsoncastro@id.uff.br and amaurig@fem.unicamp.br

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
