Frank-Kamenetskii number

Non-uniform temperature distribution in a reactor assumed model based on the Fourier heat conduction in an isotropic medium equality of temperatures of the medium and the surroundings assumed at the boundary critical values of Frank-Kamenetskii number given. 

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 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 Frank-Kamenetskii number, or parameter 8, characterizing the reaction is... 

Wood in ics CoelTicients of (44) fl 6 Half-thickness of the slab r cm Effective Ihennal diffusivity d cmVmin The Frank- Kamenetskii number Critical tempeiatiire for tbe spontaneous ignition calculated herein T, t ... 

Frank-Kamenetskii number for the degree of reaction activation parameters evaluating the ratios of the characteristic thermal and diffusion times and lengths... 

The transition of the process into a high-temperature regime is possible only if flow rate is not too large and not too small and the value of the Frank-Kamenetskii Number 6 is greater than the critical value... 

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

The more recent Thomas model [209] comprises elements of both the Semenov and Frank-Kamenetskii models in that there is a nonuniform temperature distribution in the liquid and a steep temperature gradient at the wall. Case C in Figure 3.20 shows a temperature distribution curve from self-heating for the Thomas model. The appropriate model (Semenov, Frank-Kamenetskii, or Thomas) is determined by the ratio of the heat removal from the vessel and the thermal conductivity in the vessel. This ratio is determined by the Biot number (Nm) which has been described previously as hx/X, in which h is the film heat transfer coefficient to the surroundings (air, cooling mantle, etc.), x is the distance such as the radius of the vessel, and X is the effective thermal conductivity. 

The Semenov model applies when the Biot number is close to zero, and the Frank-Kamenetskii model applies when the Biot number is large. The... 

The Frank-Kamenetskii theory has been extended to allow for reactant consumption [16—18], other geometries [19, 20] and heat transfer by convection as well as conduction, this being particularly important in gaseous systems when the Rayleigh number Ra > 600 [21]. ... 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

As appears from Fig. 14, NO/[NO] reaches a maximum of 0.65-0.68 at a given value of km [NO]r. A factor which makes km [NO]r vary a given number of times will affect the yield of nitric oxide one way or another depending on the region in which the investigation is carried out, whether it lays before or after the maximum of the characteristic curve. Thus, it could have been foreseen that an increase in the pressure or in the dimensions of the vessel would increase the quantity of nitric oxide at small km [NO]r (low explosion temperature) and decrease the yield at large fcm[NO]r. This conclusion was confirmed by Frank-Kamenetskii [7] with respect to the dimensions of the vessel and by us (see 10) and later by Sadovnikov [6] over a much wider range with respect to the influence of the pressure. 

A high Biot number means that the conductive transfer is small compared to convection and the situation is close to that considered by a Frank-Kamenetskii situation (Section 13.4.1). Inversely, a small Biot number, that is Bi < 0.2, means that the convective heat transfer dominates and the situation is close to a Semenov situation. 

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... 

In the Frank-Kamenetskii model, the surroundings temperature is set equal to the temperature of the reacting solid. Thus, there is only a small temperature gradient between this element and the wall, so only a limited heat transfer to the surroundings. This simplification establishes the above described criteria, but it is not really representative of a certain number of industrial situations. In fact, there are numerous situations where the surrounding temperature different from the initial product temperature, for example, discharging of a hot product from a dryer to a container placed at ambient temperature, and so on. Therefore, Thomas [7] developed a model that accounts for heat transfer at the wall. He added a convective term to the heat balance ... 

Sebastian (2,6) following Frank-Kamenetskii (43) arrived at the results depicted on Fig. 11.7, where the Nusselt (Nu) dimensionless number Nu = hRH/k, k is the thermal conductivity, tH is the characteristic time for removing the reaction-generated heat by conduction... 

This term specifles the ratio Si /S and has been determined explicitly by Linan and Williams [13] by the procedure they call activation energy asymptotics. Essentially, this is the technique used by Zeldovich, Frank-Kamenetskii, and Semenov [see Eq. (59)]. The analytical development of the asymptotic approach is not given here. For a discussion of the use of asymptotics, one should refer to the excellent books by Williams [ 12], Linan and Williams [13], and Zeldovich et al. [10]. Linan and Williams have called the term RT /E(Tf—To) the Zeldovich number and give this number the symbol j3 in their book. Thus... 

S is also a dimensionless number and is sometimes called the Frank-Kamenetskii s S. 

Figure 6. Typical results of Metropolis-Monte Carlo calculations on the dependence on the number of straight segments per Kuhn length, k, of a mean quantity (the mean writhing number, Wr, see Section 4.3.2.1, in the particular case) for closed polymer chain. The data are from Vologodskii and Frank-Kamenetskii (1992) [81].

Recently Wake has applied a variational treatmrait to the stationary problem, deriving critical conditions both for the class A geometries and for the cube, square rod, and equicylinder in systems where the heat transf(H is resisted by conduction in the interior and by convection at the surface. Here the condition at the boundary becomes dO/dp + N6 = 0, where N is the Biot number hLIk The limit as bf- oo corresponds to the Frank-Kamenetskii solutions. Wake uses trigonometric, rather than polynomial, expressions for this tempoature field and proceeds to derive the conditions under which solutions of the time-dependent variational equations are just possible, associating these with a critical value of 6. Results for N = oo are listed as variational (2) in the Table. For the more rorai conditions of finite Biot numbers Wake compares his results for class A geometries with the analytical forms due to Thomas. Errors are less than 0.1 % though the computational effort required is substantial. 

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... 

---