Rational approximations of the Arrhenius and general temperature integrals, expansion of the incomplete gamma function

Abstract

When analyzing materials non-isothermally using the Arrhenius equation under linear heating, a temperature integration is necessary. While the frequency factor in this equation is typically assumed to be constant, it can actually vary with temperature for certain solid-state reactions. The resulting temperature integral, known as the Arrhenius or general temperature integrals, usually have no analytical solutions. Therefore, special functions and approximation functions are often used to estimate them. In this particular study, new rational approximations for the Arrhenius and general temperature integrals were derived through the expansion of the incomplete gamma function. Two sets of these rational approximations, which exhibit excellent accuracy, are presented. One set of approximations for the Arrhenius integral matches the widely used Senum and Yang’s approximations, while the other set, which offers even greater accuracy, has not been previously reported. An obtained rational approximation has been utilized to simulate the thermal degradation of a commercially available PMMA, illustrating a practical application example.

KEYWORDS:

• Approximate integral method
• arrhenius integral
• general temperature integral
• incomplete gamma function
• non-isothermal
• thermal analysis

Disclosure statement

No potential conflict of interest was reported by the author(s).

CRediT authorship contribution statement

Alireza Aghili: Conceptualization, Methodology, Software Development, Writing – Original Draft Preparation, Writing – Review and Editing, Supervision, Project Administration.

Vahid Arabli: Data Curation, Investigation, Writing – Review and Editing.

Amir Hossein Shabani: Software Development and Visualization.

Additional information

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.