Frank-Kamenetskiis Treatment

This treatment, which is due to Semenov, includes two assumptions, a uniform reactant temperature and heat loss by convection. While these may be reasonable approximations for some situations, e.g. a well-stirred liquid, they may be unsatisfactory in others. In Frank-Kamenetskii s theory, heat transfer takes place by conduction through the reacting mixture whose temperature is highest at the centre of the vessel and falls towards the walls. The mathematics of the Frank-Kamenetskii theory are considerably more complicated than those of the simple Semenov treatment, but it can be shown that the pre-explosion temperature rise at the centre of the vessel is given by an expression which differs from that already obtained by a numerical factor, the value of which depends on the geometry of the system (Table 7). 

Calcns of the critical radius for Tetryl, based on the Merzhanov Friedman treatments are compared in Table 2. Agreement is quite good. However, note that, as expected, critical radii based on the Frank-Kamenetskii-Chambre treatment (steady-state conditions) are considerably smaller than those computed via the hot spot approach. For comparison, Eq 14 (based on Ref 7) gives a r = 2.76 x 10"3cm for a Tetryl sphere at 700 K, and acr = 2.37cm at 445°K, in close agreement with Merzhanov... 

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 a very important paper, Semenov S has given a detailed analysis of these phenomena for both stationary and nonstationary (i.e., explosive) reactions. These will be discussed in greater detail in a later chapter. Frank-Kamenetskii has also given a treatment of the problem, but in a more formal and general way. 

Frank-Kamenetskii. These two approximate methods are known as the stationary and nonstationary solutions. In the stationary theory, only the temperature distribution throughout the vessel is considered and the time variation is ignored. In the nonstationary theory, the spatial temperature variation is not taken into account, a mean temperature value throughout the vessel is used, and the variation of the mean temperature with time is examined. The nonstationary problem is the same as that posed by Semenov the only difference is in the mathematical treatment. 

Comprehensive treatments of the theory and application of diffusion and chemical reaction have been given in the following classical works [D.A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, 2nd ed., Plenum Press, NY, (1969) C.N. Satterfield, Mass Transfer in Heterogeneous Catalysis, MIT Press, Cambridge, MA, (1970) R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon Press, Oxford, England, (1975)]. 

For the theoretical treatment of statistical-mechanical properties of DNA within the elastic-rod model, a Metropolis-Monte-Carlo-type approach was elaborated by Frank-Kamenetskii et al. (1985) [22]. In this approach, the DNA chain is modeled as a series of straight segments so that each Kuhn length contains k such segments. The total elastic energy is the sum of terms, each of which corresponds to a pair of adjacent straight segments and quadrat-... 

Anshelevich, V.V., Vologodskii, A.V., Lukashin, A.V., Frank-Kamenetskii, M.D., 1979. Statistical-mechanical treatment of violations of the double helix in supercoiled DNA. Biopolymers 18 2733-2744. 

Later developments included allowance for the spatial variation of radical concentration, and stationary-state treatments were put forward which were later echoed by their thermal parallels developed by Frank-Kamenetskii. 

The first attempt at analytical quantification was made by Frank-Kamenetskii. His was an iterative zero-order treatmmt. He realized that to achieve ignition when reactant was crmsumed would require an initial beat rdease rate increased above that of the equivalmt stationary-state case. For the mth-order reaction he proposed that the critical ddta should be increased. Correcting an error in his treatment leads to the empirical rdation... 

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