The 3-D Image of The Temperature Integral And Its Self-Similarity


 How to derive the accurate value of temperature integral is a vital problem for the non-isothermal kinetic analysis. In the past six decades, researchers provided various methods to solve above problem, but the error usually becomes divergent when the value of x (x=Ea/RT) is too small or too large, no matter whether it is a numerical method or an approximation method. In this paper, we present a new series method and elaborately design a computer program to calculate the value of temperature integral. Finally, we reveal the mysterious relationship between the integral, the temperature and the activation energy, and we find an extremely interesting phenomenon that the 3-D image of the temperature integral is of self-similarity according to the fractal theory.


Introduction
The non-isothermal chemical kinetics is extensively introduced in chemical engineering [1,2], mineral technology [3], metallurgical technology [4], materials science [5], biomass energy [6,7], and other relevant disciplines. The reaction rate of a solid-involved reaction is now generally accepted as where α denotes the conversion ratio of the concerned substance at time t, f(α) is the mechanism function, and k is the rate constant expressed by Arrhenius equation: where A, T, Ea is pre-exponential factor, temperature, and activation energy of the reaction, respectively.
For a linear heating process, the heating rate β is usually a function of time, Combining all above equations, re-arranging and integrating Equation (1), we obtain where T0 is the initial reaction temperature, and the initial reaction extent α0 usually equals zero.
Then the integral of Boltzmann factor Ψ(T) is derived from Equation (4), The integral of Boltzmann factor is a transcendental function and any attempts to give the analytical solution failed finally [8]. It was encountered by scientists in the early thermogravimetric analyses [9][10][11][12][13]. Reich and Levi [14] made mistakes since they didn't recognize it is a transcendental function. Although different efforts have been made for solving this puzzle during the past six decades [15][16][17][18][19][20][21][22][23][24][25][26][27][28], troubles still exist occasionally since 'there is no true value for the integral' [27]. The temperature integral has been playing an 'enigmatic' role in the kinetic analysis. That is, it has appeared to be a necessary evil to be dealt with [25].
According to Flynn's categorization, these solutions of the temperature integral are classified into three categories: series solutions, complex approximations [29,30], and simple approximations. The series solutions listed by Flynn all have some drawbacks [25] and are inapplicable because of unbearable errors under some particular occasions.
The numerical calculation based on quadrature method was occasionally mentioned in previous papers [24,28]. However, the residual error usually increases with the rise of upper limit temperature of the integration, or increases with the decrease of x (denotes Ea / (RT)). As a disquieting situation, even the published paper [24] misled readers and gave the wrong quadrature results as standard integral values. All in all, the accurate value of the temperature integral is always attractive in the kinetic analysis.

Theory
All approximations to the temperature integral makes a detour, errors and troubles cannot be eradicated. The only way to get the exact numerical solution of the integral is to develop an appropriate series solution which is easy to reach convergence.
Assuming T is not much larger than T0, here we provide a new series solution based on Taylor's expansion, Where ΔT is the difference between T and T0, T = T0 + ΔT, and n denotes the order of Taylor expansion.
Then the exact derivative value at temperature T0 could be calculated using Equations (7) -(9).
(n = 1) The n-th-order derivative expression shown as Equation (9) is established according to Leibniz formula [32]. With the help of the general form of n-th-order derivative equation, we designed a computer program to calculate the integral of Boltzmann factor with Equation (6). The numerical value can converge within a sufficiently small residual error when the derivative order n is large enough and ΔT is small enough.
Computing results show that the value usually converges in a reasonably little error when n is more than 4 and ΔT is no more than 50 K. Consequently, a multi-step integral method is applied to compute the integral value when the difference between upper and lower limit temperature is larger than 50 K. The corresponding error analysis of present calculation method is also provided in the supplemental material.

Results
The dependence of integral value on Ea and T has always kept as mysterious relationships. Here we show the true value of the integral in Fig.1 a. This image is drawn based on 200×200 calculation nodes, and each node is computed with the Taylor order of 20 to ensure the numerical value is accurate enough. It can be seen that the integral value declines in order of magnitude with the multiple growth of the activation energy, which proves the integral value is extremely sensitive to the activation energy.
Therefore, the approximation method often provides an inaccurate value and fails to predict the kinetic process. The integral value tends to be equivalent to the value of ΔT when Ea is nearby 0 kJ/mol, while the integral value declines in order of magnitude and tends to be 0 when Ea is larger than 75 kJ/mol.  Competing Interests: There are no conflicts of interest to declare.