Frank-Kamenetskii equation

The Frank-Kamenetskii equation uses a simple Arrhenius relationship for the reaction rate. The Arrhenius equation can be written as ... 

By incorporating small amounts of impurities in expls, such as LA, the thermal parameters of the Frank-Kamenetskii equation hardly change, but the induction periods notably change... 

The Frank-Kamenetskii equation has no explanation for the so-called memory effect [See Detonation (and Explosion),... 

The critical temperature for the thermal explosion of an explosive may be calculated from the Frank-Kamenetskii equation [32] ... 

TABLE 16. The critical temperatures calculated from the Frank-Kamenetskii equation for a diverse set of explosives. 

We have shown a relationship between impact and shock sensitivity and illustrated how a sensitivity index based on oxygen balance can be used to estimate sensitivity in closely related series of molecules. It is shown that the critical temperature of an explosive calculated by the Frank-Kamenetskii equation correlates fairly well with the shock sensitivity of the material. This supports the idea that the shock or impact initiation of an explosive is primarily a thermal event and not dominated by pressure driven chemistry. The concept of the "trigger linkage" in explosives is discussed and it is pointed out that insensitive explosives will require early chemistry that is thermochemically neutral or endothermic and leads to the build-up of later strongly exothermic chemistry. 

An explosive may be initiated by various methods of delivering energy to it. Bulk heating or thermal initiation can be treated satisfactorily for engineering purposes by solving the Frank-Kamenetskii equation with Arrhenius kinetics. 

The Frank-Kamenetskii" equation for a single zero-order rate process is... 

Numerical solution of the Frank-Kamenetskii equation with first order Arrhenius kinetics was first performed by Zinn and Rogers. ... 

Combustion equations with kinetics Frank-Kamenetskii... 

For the special case of a constant temperature boundary ( =1,0 = 0), Frank-Kamenetskii [3] gives the solution to Equation (5.11) as... 

The Frank-Kamenetskii model, which applies to solids and unstirred liquids, is represented by Equation (3-29) below. The heat production rate is in the numerator and the heat removal rate is in the denominator. 

As implied in the previous section, the Russian investigators Zeldovich, Frank-Kamenetskii, and Semenov derived an expression for the laminar flame speed by an important extension of the very simplified Mallard-Le Chatelier approach. Their basic equation included diffusion of species as well as heat. Since their initial insight was that flame propagation was fundamentally a thermal mechanism, they were not concerned with the diffusion of radicals and its effect on the reaction rate. They were concerned with the energy transported by the diffusion of species. 

Frank-Kamenetskii first considered the nonsteady heat conduction equation. However, since the gaseous mixture, liquid, or solid energetic fuel can undergo exothermic transformations, a chemical reaction rate term is included. This term specifies a continuously distributed source of heat throughout the containing vessel boundaries. The heat conduction equation for the vessel is then... 

The solution of the Poisson-Boltzmann equation with. the application to thermal explosions) 5) D.A. Frank-Kamenetskii, "Diffusion and Heat Exchange in Chemical Kinetics, pp 202-66, Princeton Uni v-Press, Princeton, NJ (1955) (Quoted from MaSek s paper) 6) L.N. Khitrin, "Fizika Goreniya i Yzryva (Physics of Combustion and Explosion), IzdMGU, Moscow (1957)... 

In deriving a kinetic equation for mechanism (379), Frank-Kamenetskii assumed the surface to be uniform. The kinetic equation answering to this assumption is obtained immediately from (61) ... 

The parameter 8 is called the form factor or the Frank-Kamenetskii number. When a solution of Equation 13.26 exists, a stationary temperature profile can be established and the situation is stable. When there is no solution, no steady state can be established and the solid enters a runaway situation. The existence, or not, of a solution to the differential Equation 13.26, depends on the value of parameter 8, which therefore is a discriminator. The differential equation can be solved for simple shapes of the solid body, for which the Laplacian can be defined ... 

In this equation, h is the convective heat transfer coefficient at the external side of the wall and Ts the temperature of the surroundings. In this equation, the heat capacity of the wall is ignored, which is justifiable in most industrial situations. The border conditions are the same as for the Frank-Kamenetskii model, that is, the problem is considered symmetrical ... 

In 1940 Frank-Kamenetskii made an attempt to formulate mathematical conditions for the applicability of this approach [37]. A strict formulation for the problem of a mathematical status for the principle of quasi-stationarity was suggested by Sayasov and Vasilieva [38] in terms of the theory of singularly perturbed differential equations. 

The depletion of CA at the interface has to be compensated by diffusion from the bulk solution. The usual algebraic equation for this process is given in terms of an appropriate driving force (Frank-Kamenetskii, 1979 Froment and Bischoff, 1979) as... 

The behavior of systems described by Equation (6) is well-known. It was first analyzed by Frank-Kamenetskii (JO in the 1940 s in a study of the theory of thermal explosions. He was interested in the transient behavior of a fixed volume of reactant, which is also describable by Equation (6), with z... 

Effects of Reaction Parameters An unusual feature of Equation (15) is that it does not appear to involve the heat of reaction and heat capacity. This is because they were algebraically eliminated by combining the Frank-Kamenetskii relation,... 

A thermal explosion theory as developed by D. A. Frank-Kamenetskii (Ref 5) led to the equation AT = -(Q/A) -B/BTe eCE/R1 >(T-T° > The ignition temp expln limits so ealed agree with those experimentally found for azomethane by Rice (Ref 1), MeNOg by Appin, N20 by Zel dovich and Yakovlev (Ref 4) and for H2S by Yakovlev and Shantarovich (Ref 3). The expl limit ealed for N20 was subsequently found experimentally. With Et azide, the ealed... 

Both Frank-Kamenetskii and Rice have examined this equation to see under what conditions no stationary solutions are possible, and they find that for spherical vessels there is a limit (which then defines the explosion limit) given by... 

Equations such as (XIV.6.3) with additional linear terms are well-known equations, and solutions are available for vessels of simple geometry such as long cylinders, flat vessels with linear face dimensions large compared to their separation, and spheres or spherically symmetrical vessels. " Such solutions were discussed originally by Bursian and Sorokin, and additional cases have been presented by Lewis and Von Elbe," Semenoff, and Frank-Kamenetskii. ... 

A. J. Lotka, J. Am. Chem. Sor.j 42, 1596 (1920), was the first to describe a set of kinetic equations for a periodic chemical process. These equations seem to have inspired the suggestion of Frank-Kamenetskii. 

The formula of Zel dovich and Frank-Kamenetskii [7], [34] may be obtained from the simplest result of von Karman by performing an asymptotic expansion of the integral 1 for large values of jS and retaining only the first term in the expansion. In view of equation (43), by transforming from the variable t to z = P ( T)/(l/a + t) in equation (64), we find that... 

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. 

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