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Representation of Even Integers by Cubes of Primes and Powers of 2

Acta Mathematica Hungarica
Authors:
J. Liu
and
M.-C. Liu
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On evaluate of the integral methods for the determination of the activation energy

Journal of Thermal Analysis and Calorimetry
Authors:
J. Cai
and
R. Liu

Abstract

Recently, Órfão obtained two simple equations for the estimation of the relative error in the activation energy calculated by the integral methods [2]. In this short communication, the validity of the equations has been evaluated by comparing the results calculated by the equations with the results calculated by the equation from theoretical derivation without introducing any assumption.

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Non-isothermal kinetics in solids

The precision of some integral methods for the determination of the activation energy without neglecting the temperature integral at the starting temperature

Journal of Thermal Analysis and Calorimetry
Authors:
J. Cai
and
R. Liu

Abstract

The integral methods, which are obtained from the various approximations for the temperature integral, have been extensively used in the non-isothermal kinetic analysis. In order to obtain the precision of the integral methods for the determination of the activation energy, several authors have calculated the relative errors of the activation energy obtained from the integral methods. However, in their calculations, the temperature integral at the starting temperature was neglected. In this work, we have performed a systematic analysis of the precision of the activation energy calculated by the integral methods without doing any simplifications. The results have shown that the relative error involved in the activation energy determined from the integral methods depends on two dimensionless quantities: the normalized temperature θ=T/T 0, and the dimensionless activation energy x 0=E/RT 0 (where E is the activation energy, T is the temperature, T 0 is the starting temperature, R is the gas constant).

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Errors involved in the activation energy calculated by integral methods when the frequency factor depends on the temperature ( A = A 0 T m)

Journal of Thermal Analysis and Calorimetry
Authors:
J. Cai
and
R. Liu

Abstract

The dependence of the frequency factor on the temperature (A=A 0 T m) has been examined and the errors involved in the activation energy calculated from some integral methods without considering such dependence have been estimated. Investigated integral methods are the Coats-Redfern method, the Gorbachev-Lee-Beck method, the Wanjun-Yuwen method and the Junmeng-Fusheng method. The results have shown that the error in the determination of the activation energy calculated ignoring the dependence of the frequency factor on the temperature can be rather large and it is dependent on x=E/RT and the exponent m.

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New approximation for the general temperature integral

Journal of Thermal Analysis and Calorimetry
Authors:
J. Cai
and
R. Liu

Abstract

A new approximation has been proposed for calculation of the general 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} $$\int\limits_0^T {T^m } e^{ - E/RT} dT$$ \end{document}
, which frequently occurs in the nonisothermal kinetic analysis with the dependence of the frequency factor on the temperature (A=A 0 T m). It is in the following form:
\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} $$\int\limits_0^T {T^m } e^{ - E/RT} dT = \frac{{RT^{m + 2} }} {E}e^{ - E/RT} \frac{{0.99954E + (0.044967m + 0.58058)RT}} {{E + (0.94057m + 2.5400)RT}}$$ \end{document}
The accuracy of the newly proposed approximation is tested by numerical analyses. Compared with other existed approximations for the general temperature integral, the new approximation is significantly more accurate than other approximations.
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Precision of integral methods for the determination of the kinetic parameters

Use in the kinetic analysis of solid-state reactions

Journal of Thermal Analysis and Calorimetry
Authors:
J. Cai
and
R. Liu

Abstract

In this paper, a systematic analysis of the errors involved in the determination of the kinetic parameters (including the activation energy and frequency factor) from five integral methods has been carried out. The integral methods analyzed here are Coats-Redfern, Gorbachev, Wanjun-Yuwen-Hen-Zhiyong-Cunxin, Junmeng-Fusheng-Weiming-Fang, Junmeng-Fang and Junmeng-Fang-Weiming-Fusheng method. The results have shown that the precision of the kinetic parameters calculated by the different integral methods is dependent on u (E/RT), that is, on the activation energy and the average temperature of the process.

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Parametric study of the nonisothermal n th-order distributed activation energy model involved the Weibull distribution for biomass pyrolysis

Journal of Thermal Analysis and Calorimetry
Authors:
J. Cai
and
R. Liu

Abstract

This paper describes the influences of some parameters relevant to biomass pyrolysis on the numerical solutions of the nonisothermal n th-order distributed activation energy model (DAEM) involved the Weibull distribution. Investigated parameters are the integral upper limit, the frequency factor, heating rate, the reaction order and the shape, scale and location parameters of the Weibull distribution. Those influences can be used for the determination of the kinetic parameters of the nonisothermal n th-order Weibull DAEM from thermoanalytical data of biomass pyrolysis.

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Thermal and dynamic mechanical properties of PES/PPS blends

Journal of Thermal Analysis and Calorimetry
Authors:
M. Lai
and
J. Liu

Abstract

Blends of poly(ether-sulfone) (PES) and poly(phenylene sulfide) (PPS) with various compositions were prepared using an internal mixer at 290C and 50 rpm for 10 min. The thermal and dynamic mechanical properties of PES/PPS blends have been investigated by means of DSC and DMA. The blends showed two glass transition temperatures corresponding to PPS-rich and PES-rich phases. Both of them decreased obviously for the blends with PES matrix. On the other hand, Tg of PPS and PES phase decreased a little when PPS is the continuous phase. In the blends quenched from molten state the cold crystallization temperature of PPS was detected in the blends of PES/PPS with mass ratio 50/50 and 60/40. The melting point, crystallization temperature and the crystallinity of blended PPS were nearly unaffected when the mass ratio of PES was less than 60%, however, when the amount of PES is over 60% in the blends, the crystallization of PPS chains was hindered. The thermal and the dynamic mechanical properties of the PPS/PES blends were mainly controlled by the continued phase.

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Thermal infrared mapping of the freezing phase change activity of micro liquid droplet

Journal of Thermal Analysis and Calorimetry
Authors:
F. Li
and
J. Liu

Abstract

This article is dedicated to develop an experimental approach for directly visualizing the global freezing phase change behavior of micro liquid droplets. The infrared (IR) thermograph was proposed to image the basic solidification phenomena of droplet and to acquire its temperature variations during the transient process. In particular, the volumetric recalescence event, regarded as initiation of freezing, was revealed by IR images for the first time. Preliminary results demonstrated that the involved temperature transition due to release of the latent heat can be accurately characterized by evident color break in IR images. Further, experiments were also performed simultaneously on three kinds of droplets made of pure water, dimethylsulfoxide (DMSO) and nano liquid to grasp more precise temporal and spatial temperature distribution. Types of the occurring solidification and the initial frozen volume produced from the recalescence were generally discussed. The IR monitoring method suggests a straightforward way for detecting the freezing phase change activity and its temperature evolution at micro scale.

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Thermal Analysis of Semi-Dilute Hyaluronan Solutions

Journal of Thermal Analysis and Calorimetry
Authors:
J. Liu
and
M. Cowman

Abstract

The freezing and melting of water in semi-dilute (0.5–3.0%) solutions of the polysaccharide hyaluronanhave been investigated by modulated differential scanning calorimetry.High molecular weight hyaluronan inhibited nucleation of ice and significantly depressed thefreezing temperature in a dynamic scan conducted at –3.0°C min–1. Low molecular weight hyaluronan had a weaker and more variable effect on nucleation. Theeffects on nucleation, especially by the high molecular weight hyaluronan, are attributed tothe influence of a hyaluronan network on the formation of critical ice nuclei.Both high and low molecular weight hyaluronan reduced the melting temperature of ice by 0.4–1.1°C, depending on concentration. The enthalpy change associated with this transitionwas significantly reduced. If all of the enthalpy difference is attributed to the presence of non-freezing water, approximately 3.65 g water/g hyaluronan would be non-freezing. This result appears incompatible with published studies on hyaluronan samples of low water content. An alternative hypothesis and quantitative approach to analysis of the data are suggested. The data are interpreted in terms of a small amount of non-freezing water, and amuch larger boundary layer of water surrounding hyaluronan chains, which has slightly altered thermodynamic properties relative to those of bulk water. The boundary layer water behaves similarly to water trapped in small pores in solid materials and hydrogels.

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