Frank-Kamenetskii approximation

Unfortunately, the right side of Eq. (19) cannot be integrated analytically owing to the nature of the exponential function. However, using the Frank-Kamenetskii approximation. 

The Frank-Kamenetskii approximation reduces the analysis to a one-parameter description it is useful for showing that the competition between heat production and heat removal indeed leads to the existence of criticality. However, this approximation ignores another important aspect of chemical kinetics, that of finite reaction activation energy. The effect of nonzero e in f(0) can be analyzed using a number of different approximations [4-6], By writing... 

Considering that the temperature difference 9 must be equivalent to (T - T9 in the Zeldovich-Frank-Kamenetskii-Semenov thermal flame theory, the reaction time corresponding to the reaction zone 6 in the flame can also be approximated by... 

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). 

In particular Frank-Kamenetskii. has shown (Ref 5) that the quantitative requirement for a homogeneous thermal reaction to be explosive is Ea RT. Since eq (1), being nonlinear, cannot be solved in a general manner, the solution consists of a series of approximations, such as described by Frank-Kamenetskii. (Ref 5) and later discussed critically by Gray Harper (Ref 10) (Ref 13, p 45)... 

Further in his paper, Ma ek describes on pp 45-7, the "stationary approximation , i.e., the case in which. the space distribution of temperature does not vary with.time, < T/dt = 0. Here the works of Frank-Kamenetskii. (Ref 5), Semenoff (Ref 1), Gray Harper (Ref 10) and others are applied and their, implications are discussed... 

In the "nonstationary approximation described by Ma ek on p 47 are discussed the works of Frank-Kamenetskii, Semenoff Gray Harper... 

In Table 13.2, the best approximation of the cube is obtained with a sphere of radius rsph = 1.16-r0. The Frank-Kamenetskii number then is 2.5 for a cube with a side length 2 r0. 

The initial conditions are (p = 1 and 6 = 6q (Tq — T )E/(R Ti) at t = 0. The simplest approximation to the function /(cp) is / — cp. Here a is the ratio of a transfer coefficient for fuel to that for heat, 6 is a ratio of the thermal energy at wall temperature to the activation energy (see Section B.3), y is a ratio of the thermal energy at wall temperature to the total energy released by the reaction, and eyS is the ratio of the cooling time to the characteristic time of chemical reaction at the wall temperature. Frank-Kamenetskii [28] has emphasized that in combustion, the parameters e and y are small. He also introduced the parameter 3, defined in equation (55), as occupying a role of central importance in thermal explosions. 

Later, there were improvements in the thermal theories. Probably the most significant of these is the theory proposed by Zeldovich and Frank-Kamenetskii. Because their derivation was presented in detail by Semenov [4], it is commonly called the Semenov theory. These authors included the diffusion of molecules as well as heat, but did not include the diffusion of free radicals or atoms. As a result, their approach emphasized a thermal mechanism and was widely used in correlations of experimental flame velocities. As in the Mallard-Le Chatelier theory, Semenov assumed an ignition temperature, but by approximations eliminated... 

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. 

T. Turdnyi and J. Toth, Comments to an Article of Frank-Kamenetskii on the Quasi-Steady-State Approximation, Acta Chim. Hung. 129 (1992) 903-914. 

In this regard, it may be permitted to say that the Frank-Kamenetskii s method of expanding the exponent, which has been applied in Sections 1.3 and 2.4, is, in the result, nothing else but the linear approximation of the exponential function. 

Further, the approximation e = 0 allows (4) to be solved analytically for the (infinite) slab and cylinder. This is commonly known as the Frank-Kamenetskii exponential approximation, the best known of various substitutions for an Anrhenius temperature dependence. It is a good approximation over a limited range. Other alternatives are mentioned later the most important are the quadratic and the quintic. The maximum stable temperature rise possible (ATmu) at a point corresponding to the body centre occurs at d = d r, i.e. at criticality. Tmmx is of the same order as t Semoiov value, RTm IE, or equivalently I. 

In 1939 Frank-Kamenetskii considered circumstances where Newtonian cooling was only an empirical approximation, and where the escape of heat was impeded internally by the thermal properties of the medium. (This will always be the case for a large enough system.) An internal temperature-distribution with a maximum at the middle results. For stability, this central temperature may not exceed a critical value. For a sphere with its surface at T the relationship is ... 

This approximation, valid when activation energy is high compared to ambient temperature, renders Eq.(l) analytically tractable as originally shown by Frank-Kamenetskii [1-3]. The solution can be expressed as... 

Turanyi, T., Toth, J. Comments to an article of Frank-Kamenetskii mi the quasi-steady-state approximation. Acta Chim. Hung. Models Chem. 129(6), 903-907 (1992)... 

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