Frank criterion

In connection with the expansion of the exponent in a series studied by Frank-Kamenetskii [3], a single criterion is sufficient to describe the dependence of the reaction rate on the temperature in the only temperature region of interest to us, that in which the reaction rate is high ... 

For the treatment of practical cases, it is often necessary to assess other shapes other than a slab, infinite cylinder, or sphere. In such a case, it is possible to calculate the Frank-Kamenetskii criterion for some commonly used shapes. For a cylinder of radius r and height h, the critical value of the Frank-Kamenetskii criterion is given by [7]... 

In the specific case of chemical drums with a height equal to three times the radius, the Frank-Kamenetskii criterion is 8cri, = 2.37 [1]. A cube with a side length 2ro, can be converted to its thermally equivalent sphere. The Semenov number then becomes... 

Step 5 The heat exchange at the wall with the overall heat transfer coefficient of 50 Wm 2 K can also be considered, following the Thomas model. Thus, the Frank-Kamenetskii criterion (8) is to be compared with the Thomas criteria... 

As a fifth attempt, an increase of the heat transfer at the wall in the Thomas model is not practicable and would not be efficient, since the major part of the resistance to heat transfer is the conductivity in the product itself, as shown by the high value of the Biot criterion, 300, which is closer to Frank-Kamenetskii conditions than to Semenov conditions. 

It should be emphasized that Oh and Cavendish assumed that the reactions only occur on the surface of the channel wall. This assumption is less realistic for a layer of washcoat (typically y-alumina) dispersed with catalyst applied on to the wall surface. Ramanathan, Balakotaiah, and West showed that the diffusion in the washcoat has a profound influence on the light-off behavior of a monolith converter. They derived an analytical light-off criterion based on a onedimensional two-phase model with position-dependent heat and mass transfer coefficients. The derivation of this criterion is based on the two key assumptions a positive exponential approximation (i.e., the Frank-Kameneskii approximation) and negligible reactant consumption in the fluid phase. The light-off is defined as the occurrence of multiple steady states with the attainment of the ignited steady state. Here, we discuss only the results of their analysis, without going into the details of their derivation. 

Figure 11 Effect of reactant consumption on the stability ofa first-order exothermic reaction, heat loss (a) against reciprocal adiabatic flame temperature (B ). Curve (a). Adler-Enig temperature-concentration inflection criterion curve (b) Bowes-Thomas temperature-time inflection criterion curve (c) Gray-Sherrington quadratic Uapunov function curve (d). Frank-Kamenetskii s empirical criterion (the curves show a )...

When considered in this framework, the resulting critical thickness condition is independent of any physical mechanism by which the interface misfit dislocation is actually formed in the system. This approach to critical thickness is an outgrowth of the pioneering work of Frank and Van der Merwe (1949) and the criterion was formulated in the way described here by Van der Merwe and van der Berg (1972). The most commonly observed mechanism for formation of misfit dislocations is through glide of a threading dislocation. In this section, the notion of critical thickness is re-examined on the basis of this mechanism. 

Letter from Frank R. McCoy to Paul Rice (WSRC), "Single Failure Criterion," February 11, 1991,... 

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