# Search Results

## Abstract

The general method of evaluating the temperature integral for temperature dependent frequency factors have been considered. The values of the temperature integral as evaluated by the present method are in excellent agreement with those obtained numerically.

# Approximations for the temperature integral

## Their underlying relationship

## Abstract

The temperature integral cannot be analytically integrated and many simple closed-form expressions have been proposed to use
in the integral methods. This paper first reviews two types of simple approximation expressions for temperature integral in
literature, i.e. the rational approximations and exponential approximations. Then the relationship of the two types of approximations
is revealed by the aid of a new equation concerning the 1^{st} derivative of the temperature integral. It is found that the exponential approximations are essentially one kind of rational
approximations with the form of *h*(*x*)=[*x*/(*Ax*+*k*)]. That is, they share the same assumptions that the temperature integral *h*(*x*) can be approximated by *x*/*Ax*+*k*). It is also found that only two of the three parameters in the general formula of exponential approximations are needed
to be determined and the other one is a constant in theory. Though both types of the approximations have close relationship,
the integral methods derived from the exponential approximations are recommended in kinetic analysis.

## Abstract

Three rational fraction approximations for the temperature integral have been proposed using the pattern search method. The
validity of the new approximations has been tested by some numerical analyses. Compared with several published approximating
formulas, the new approximations is more accurate than all approximations except the approximations proposed by Senum and
Yang in the range of 5≤*E/RT*≤100. For low values of *E/RT*, the new approximations are superior to Senum-Yang approximations as solutions of the temperature integral.

## Abstract

*E*is the activation energy,

*R*the universal gas constant and

*T*the absolute temperature. The exponent

*m*arises from the temperature dependence of the pre-exponential factor. This paper has proposed two new approximate formulae for the generalized temperature integral, which are in the following forms:

*h*

_{m}(

*x*) is the equivalent form of the generalized temperature integral. For commonly used values of

*m*in kinetic analysis, the deviations of the new approximations from the numerical values of the integral are within 0.2 and 0.03%, respectively. In contrast to other approximations, both the present approaches are simple, accurate and can be used easily in kinetic analysis.

## Abstract

A new approximate formula for temperature integral is proposed. The linear dependence of the new fomula on *x* has been established. Combining this linear dependence and integration-by-parts, new equation for the evaluation of kinetic
parameters has been obtained from the above dependence. The validity of this equation has been tested with data from numerical
calculating. And its deviation from the values calculated by Simpson's numerical integrating was discussed. Compared with
several published approximate formulae, this new one is much superior to all other approximations and is the most suitable
solution for the evaluation of kinetic parameters from TG experiments.

## Abstract

*A*=

*A*

_{0}

*T*

^{m}). It is in the following form:

## Abstract

*m*in kinetic analysis, the deviation of most approximations from the numerical values of the integral is within 0.7%, except the first approximation (within 4.0%). Since they are simple in calculation and hold high accuracy, the approximations are recommended to use in the evaluation of kinetic parameters from non-isothermal kinetic analysis.

## Abstract

The accuracy and scope of application of previously reported approximations of the temperature integral were evaluated. The
exact solution was obtained independently by solving the temperature integral numerically be Simpson's rule, the trapezoidal
rule and the Gaussian rule.
Two new approximations have been proposed:

*X=E/RT*. The first equation gives higher accuracy, with a deviation of less than 1% and 0.1% from the exact solution for

*X*≥7 and

*X*≥10, respectively. The second equation has a wider scope of application, with a deviation of less than 1% for

*X*≥4 and of less than 0.1% for

*X*≥35.