Energy dissipation, empirical model

The calculation of backscatter coefficients via the approach outlined above is mathematically complex. Heidenreich 44) developed a simple empirical backscatter model which is applicable to resist exposure being based on the direct observation of chemical changes produced by backscat-tered electrons at different accelerating voltages on several substrates. The model is independent of scattering trajectory and energy dissipation calculations and is essentially a radial exponential decay of backscatter current density out to the backscatter radius determined by electron range. 

An estimate of the electron-ion recombination rate constant in high-mobility systems based on an empirical model of energy dissipation processes was provided by Warman [38]. He related the rate constant to the field dependence of the electron mobility, and proposed... 

In fact the viscosity influences both the heat balance and the mass balance. It has been shown how the heat transfer coefficient is affected by the viscosity. But the energy dissipation by the stirrer is also strongly dependent on viscosity (see Section 11.4.4). Furthermore, viscosity affects the molecular diffusion, the mass transport, the mixing time, or the residence time distribution, and therefore the reaction rate. Since the reaction rates influence the chain length and particle sizes, they have a direct effect on the polymer properties. In turn they affect the viscosity and the shear forces - there is a feedback effect. Such complex interactions cannot be described by analytical equations, so empirical models must be used. Often... 

Up to this point, the strain amplification factor can be viewed as a mere empirical approach to assign the modulus increase in CB filled compound to filler level. Equation 5.19 above essentially resulted from considerations on the hydrodynamic effects induced by the presence of solid particles ideally dispersed in a matrix with a considerably lower modulus. The empirical factor f in Equation 5.20 adds nothing in this respect and it is well known that both equations do not suit at all either highly loaded compounds, whatever is the grade of CB, or moderately loaded materials with high structure blacks. Over the last decades, several authors have developed theoretical considerations to model the likely effect of a so-called filler network structure and the associated energy dissipation process when filled compounds are submitted to increasing strain. 

Given initial conditions and Toft), these two ordinary differential equations could be solved to find e(t). But, inevitably,24 for large Reynolds numbers one finds eft.) Toft). Hence, most engineering models for the dissipation rate are largely empirical fits that attempt to model the energy flux 7di(0 instead of the individual terms on the right-hand side of (2.125). We will look at the available models in Chapter 4. 

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