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A mathematical model for differential thermal analysis

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  • A mathematical model of a differential thermal analysis (DTA) system was formulated so that influence of the various physical parameters on the DTA peak could be determined. The specific DTA apparatus simulated had cylindrical sample holes drilled into a nickel block considered to have a negligible thermal resistance, and the specific reaction was the α to β quartz crystal transformation with zero-order kinetics. For this specific DTA system the thermal resistance of the sample was the controlling factor causing the differential temperature; consequently, the model was sublimated to a heat transfer problem involving a moving phase boundary within a cylinder being heated. The ordinary explicit finite difference method was adapted to describe the temperature profile in an infinite-cylindrical sample, and special equations were derived to consider the moving phase boundary. A digital computer solution of these equations produced graphical DTA peaks whose shape was largely dependent upon the values of the governing physical parameters for the apparatus and the samples. The results compared well with previous theoretical investigations of a differential thermal analyzer, and it is felt that the results of this study are more accurate than those obtained by other investigators. In addition, good qualitative agreement was found between the results of the present model and the experimental peaks of the two previous investigations of the α-β phase transformation in quartz. Theoretical variations in the heating rate generated the same general trends in the maximum peak temperature and the peak area as indicated by previous experimental results. Finally, the effects of the heat of transformation and thermal diffusivity on the shape of the DTA peak were determined. Recommendations for the application of this model to a two-dimensional case are made for a cylinder. Specifically, a procedure for treating the movement of a phase boundary of variable shape is suggested.
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