Critical Heat Losses to Avoid Self-Heating in Coal Piles

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Abstract
The Frank-Kamenetskii theory of the self-heating of a reactive slab is extended to include volumetric heat losses. The method of non-dimensionalising the heat loss highlights the important role of the Frank-Kamenetskii parameter in the heat dissipation term. Two types of heat loss are considered: firstly that without any dependence on temperature and secondly that with a global (and linear) temperature dependence. It is shown that in either case there is a critical non-dimensional heaf-loss rate beyond which thermal runaway will not occur. This value corresponds to when the actual heat loss is a certain small fraction of the volumetric heat generation. The fraction depends, through the Frank-Kamenetskii parameter, on the constants governing the chemical reaction— in particular the activation energy and heat of reaction. In that such volumetric heat losses are to be expected with the burning of low rank coals with high moisture content, the theory gives some indication of the critical water content necessary to deter the self-heating of coal piles. The complete problem (involving species concentration equations ) must be addressed to get more accurate estimations of this quantity. The advantage of this theory is its simplicity whilst still encapsulating the major effect of heat loss on temperature profiles. It should serve as a useful “ldquo; benchmark” case for comparison to more detailed work' at a later stage.