New approximation for the general temperature integral

New approximate formula for the generalized temperature integral

“…It is also shown that the errors are zero around x = 36 and 63. Compared with the published approximations proposed by Wanjun et al,11 Chen and Liu,13 and Cai et al,12, 14, 15 it can be found that the new approximation is a little more accurate in some domains of x and less accurate in other domains. However, it must be recalled that the new approximation represents a new type of approximations (i.e., the exponential approximations), and from the new approximation, some new and more reliable integral methods, Eqs.…”

“…These two types of integral methods naturally result in two types of approximations for h m ( m = 0), that is, the rational approximations and the exponential approximations. For h m with arbitrary values of m , all of the published approximated approximations are of the rational approximations 11–13. Here, we try to approximate h m in the exponential form to propose a new type of integral methods, that is to say, we suppose ln h m can be approximated by the linear combination of ln x and x as follows: …”

“…Wanjun et al11 published this kind of research first and proposed two approximate formulae for calculation of the integral I(m,T ) by using integration‐by‐parts approaches. Later, an approximation with first‐degree rational fraction was proposed by Cai and Liu,12 and a procedure to yield a series of the approximations with different complexity and accuracy is proposed by Chen and Liu 13. Recently, Cai et al have proposed two new approximations, which are more accurate 14, 15.…”

Combined kinetic analysis of solid‐state reactions: The integral method (ICKA)

“…The one recently released by Lin et al 52 can be considered the most precise as compared to previous approximations. 15,17,20 It works in the ranges 4 ≤ x ≤ 200 and −2.5 ≤ s ≤ 2.5, and has the structure of Equation (15), in which h s (x,s) is of the rational type and can be evaluated as:…”

“…where β is the linear heating rate (K s −1 ), = ∕ and ( ) = ∫ 0 ∕ ( ). In this equation, p s (x,s) is commonly known as the general temperature integral, 15 taking the form:…”