Thermal effect models example problems

The general problem has been to extend the usefulness of the induction parameter model proposed by Oran et al. (1). This induction parameter model (IPM) is proposed as a means to enable one to estimate, relatively easily, the energy necessary to achieve ignition when using a thermal heating source Much of the calibration of this model, for example the effect of deposition volume (quench volume), can be done with one-dimensional models, and shock tube experiments. There are phenomena, however, which must be studied in two or three dimensions. Examples are turbulence and buoyancy. This paper discusses the effect of buoyancy and possible extensions to the IPM. 

The MRS closures will attract most interest for use wherever MTE methods fail. For example, in flows with rotation the Coriolis terms enter the Rij equations, but drop out in the equation for Ru — q Therefore, an MRS method probably will be essential for including rotation effects, which are of considerable importance in many practical engineering and geophysical problems. Other effects that have not yet been adequately modeled and for which MRS methods may offer some hope include additive drag reduction, ultrahigh Reynolds numbers, separation, roughness, lateral and transverse curvature, and strong thermal processes that affect the hydrodynamic motions. 

This problem was first treated in detail by Haward (1949). He considered the case of a bulk polymerization that has been compartmentalized by subdividing the reaction system into a large number of separate droplets, each of volume v. Radicals are generated exclusively within the droplets and always in pairs. An example would be the polymerizatiim of styrene in emulsified droplets dispersed in water initiated the thermal decomposition of an oil-soluble initiator which partitions almost exclusively within the monomer droplets. In the model considered by Haward, radicals are unable to exit from the droplets into the external phase. The only radical-loss process is in fact bimolecular mutual termination. It therefore follows that all the droplets must always contain an even number (including zero) of propagating radicals, and that the state of radical occupancy will change in increments of 2. The conclusion reached by Haward is that in this case the effect of compartmentalization is to reduce the overall rate of polymerization per unit volume of disperse phase. The f ysical reason for this is that, as the volume of the droplets is reduced, so are the opportunities for a radical to escape from the others—and hence to avoid mutual... 

It is evident from the foregoing discussion that the effective diffusivity cannot be predicted accurately for use under reaction conditions unless surface diffusion is negligible and a valid model for the pore structure is available. The prediction of an effective thermal conductivity is even more difficult. Hence sizable errors are frequent in predicting the global rate from the rate equation for the chemical step on the interior catalyst surface. This is not to imply that for certain special cases accuracy is not possible (see Sec. 11-10). It does mean that heavy reliance must be placed on experimental measurements for effective diffusivities and thermal conductivities. Note also from some of the examples and data mentioned later that intrapellet resistances can greatly affect the rate. Hence the problem is significant. 

In the context of the Brownian model, other models have been developed for the interpretation of the results of particular experiments. For example, for Mossbauer spectroscopy where on the one hand transitions between nuclear spins are involved and on the other hand the measurement encompasses the whole t set. These models will be briefly described in Section F.6. This is also the case for ferromagnetic resonance where models have been developed from Raikher s calculations. However the effective eigenvalue method, which allows one to derive the initial slope of the magnetization, is not valid when thermal activation process is present. New calculations have taken into account this problem. Theoretical results are summarized in Section F.7. 

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