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Clear All Modify Search  # New approximate formulae for the generalized temperature integral

Journal of Thermal Analysis and Calorimetry
Authors: H. Chen and N. Liu

## Abstract

The generalized temperature integral

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {T^m } \exp ( - E/RT)dT$$ \end{document}
frequently occurs in non-isothermal kinetic analysis. Here 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:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} h_m (x) = \frac{x} {{(1.00141 + 0.00060m)x + (1.89376 + 0.95276m)}} \hfill \\ h_m (x) = \frac{{x + (0.74981 - 0.06396m)}} {{(1.00017 + 0.00013m)x + (2.73166 + 0.92246m)}} \hfill \\ \end{gathered}$$ \end{document}
where 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.

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# Approximations for the temperature integral

## Their underlying relationship

Journal of Thermal Analysis and Calorimetry
Authors: H. Chen and N. Liu

## 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 1st 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.

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# Study on pyrolysis kinetics of walnut shell

Journal of Thermal Analysis and Calorimetry
Authors: H. Yuan and R. Liu

## Abstract

Slow pyrolysis of walnut shell which is a cheap and abundantly available solid waste was carried out using thermogravimetric analysis. The effects of raw material heating rate on the pyrolysis properties and kinetic parameters were investigated. A two-step consecutive reaction model were used to simulate the pyrolysis process. The kinetic parameters were established by using the pattern search method. Comparison between experimental data and the model prediction indicated that the two-step consecutive reaction model can better describe the slow pyrolysis of walnut shell as the formation of an intermediate during the pyrolysis process was taken into account.

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# A procedure to approximate the generalized temperature integral

Journal of Thermal Analysis and Calorimetry
Authors: H. Chen and N. Liu

## Abstract

In the paper a new procedure to approximate the generalized temperature integral

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {T^m } \exp ( - E/RT)dT$$ \end{document}
, which frequently occurs in non-isothermal thermal analysis, are presented. A series of the approximations for the temperature integral with different complexity and accuracy are proposed from the procedure. For commonly used values of 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.

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# HPLC analysis of magnolol and honokiol in Magnoliae Cortex after solvent sublation

Acta Chromatographica
Authors: H. Dong Dong and X. Liu Liu

## Summary

A novel method for separating and concentrating magnolol and honokiol from Magnoliae Cortex by solvent sublation and analysis of the compounds by high-performance liquid chromatography (HPLC) has been established. The optimum conditions for solvent sublation were use of n-butanol as sublation solvent, sample solution at pH 2, nitrogen flow 50 mL min–1, and sublation time 50 min. The floating product obtained under the optimum conditions was determined by HPLC analysis on a C18 reversed-phase column, with 22:78 (%, v/v) water-methanol as isocratic mobile phase at a flow rate of 1.00 mL min–1. When the method was used for quantification of magnolol and honokiol in Magnoliae Cortex recovery ranged from 98.1 to 106.1%, RSD was from 3.07 to 4.80%, and LOD for honokiol and magnolol were 0.94 and 1.14 ng mL–1, respectively.

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# Determination of the total selenium in different materials by pixe

Journal of Radioanalytical and Nuclear Chemistry
Authors: Q. Liu, H. Shao, and G. Moschini

## Abstract

The determination of the total selenium in different materials is now a routine task for many laboratories. A few problems, however, still remain concerning the choice of an efficient digestion technique and an accurate and precise detection method. For this purpose, we investigated the action of various reagents used for the wet digestion of different materials. Efficient digestion combined with preconcentration were successfully applied to biological samples. Using PIXE, selenium can be detected at 5 ppb level in a short time. The overall performance of wet digestion and PIXE methods were tested with some standard reference materials.

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# The existence problem for colour critical linear hypergraphs

Acta Mathematica Hungarica
Authors: H. L. Abbott and A. C. Liu
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# Deduction of an improved formula for coincidence counting and its experimental verification

Journal of Radioanalytical and Nuclear Chemistry
Authors: Y. Liu, H. Du, and L. Yao

## Abstract

An improved accurate coincidence correction formula has been deduced on the basis of Cox's theory considering the complex situations of differences in pulse shaping width as well as a relative delay existing between the two channels. The correctness has been examined by experiments.

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# Preferential solvation of lysozyme by dimethyl sulfoxide in binary solutions of water and dimethyl sulfoxide

Journal of Thermal Analysis and Calorimetry
Authors: T. Kamiyama, H. Liu, and T. Kimura

## Abstract

To reveal the denaturation mechanism of lysozyme by dimethyl sulfoxide (DMSO), thermal stability of lysozyme and its preferential solvation by DMSO in binary solutions of water and DMSO was studied by differential scanning calorimetry (DSC) and using densities of ternary solutions of water (1), DMSO (2) and lysozyme (3) at 298.15 K. A significant endothermic peak was observed in binary solutions of water and DMSO except for a solution with a mole fraction of DMSO (x 2) of 0.4. As x 2 was increased, the thermal denaturation temperature T m decreased, but significant increases in changes in enthalpy and heat capacity for denaturation, ΔH cal and ΔC p, were observed at low x 2 before decreasing. The obtained amount of preferential solvation of lysozyme by DMSO (∂g 2/∂g 3) was about 0.09 g g−1 at low x 2, indicating that DMSO molecules preferentially solvate lysozyme at low x 2. In solutions with high x 2, the amount of preferential solvation (∂g 2/∂g 3) decreased to negative values when lysozyme was denatured. These results indicated that DMSO molecules do not interact directly with lysozyme as denaturants such as guanidine hydrochloride and urea do. The DMSO molecules interact indirectly with lysozyme leading to denaturation, probably due to a strong interaction between water and DMSO molecules.

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# Determination of dispersity of crushed granite

Journal of Radioanalytical and Nuclear Chemistry
Authors: D. J. Liu and X. H. Fan

## Summary

Experimental crushed granite column breakthrough curves, using 99Tc as spike tracer and 3H as invariant tracer, were analyzed by different linear regression techniques. Dispersity of crushed granite and the retardation factor of 99TcO4 - on the crushed granite were determined simultaneously by one linear regression. Dispersity of crushed granite was also obtained with 3H as invariant tracer by the other linear regression. The dispersities found by spike source and invariant source methods are compared. Experimental results show that the dispersity found by the spike source method is close to that found by the invariant source method. This indicates that dispersity is only a characteristic of the dispersion medium.

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