Frank-Kamenetskii

Many authors contributed to the field of diffusion and chemical reaction. Crank (1975) dealt with the mathematics of diffusion, as did Frank-Kamenetskii (1961), and Aris (1975). The book of Sherwood and Satterfield (1963) and later Satterfield (1970) discussed the theme in detail. Most of the published papers deal with a single reaction case, but this has limited practical significance. In the 1960s, when the subject was in vogue, hundreds of papers were presented on this subject. A fraction of the presented papers dealt with the selectivity problem as influenced by diffitsion. This field was reviewed by Carberry (1976). Mears (1971) developed criteria for important practical cases. Most books on reaction engineering give a good summary of the literature and the important aspects of the interaction of diffusion and reaction. 

Frank-Kamenetskii, D.A., 1961, Diffusion and Heat Transfer in Chemical Kinetics, Edition, Plenum Press. 

At the Installation, Sakharov worked with many colleagues, in particular Yakov Zcldnvich and David Frank-Kamenetskii. Sakharov made key contributions to the Soviets first full-fledged H-bomb, tested in 1955. He also made many contributions to basic physics, perhaps the most important being his thesis that the universe is composed of matter (rather than all matter having been annihilated against antimatter) is likely to be related to charge-parity (CP) noninvariance. 

Using the Frank-Kamenetskii (1969) transformation we present the relation in Eq. (3.27) in the following form (Zel dovich et al. 1985) ... 

Frank-Kamenetskii DA (1969) Diffusion and heat transfer in chemical kinetics, 2nd edn. Plenum, New York... 

Lyamichev, V., Frank-Kamenetskii, M., Soyfer, V. Protection against UV-induced pyrimidine dimerization in DNA by triplex formation. Nature, Vol.344, No. 6266, (1990), pp. 568-570, ISSN 1476-4687... 

Cherny D.Y., Belotserkovskii B.P., Frank-Kamenetskii M.D., Egholm M., Buchardt O., Berg R. H., Nielsen, P.E. DNA unwinding upon strand-displacement binding of a thymine-substituted polyamide to double-stranded DNA. Proc. Natl Acad. Sci. USA 1993 90 1667-1670... 

Ya.B. Zel dovich and D.A Frank-Kamenetskii. A theory of thermal flame propagation. Acta Physicochimica URSS, IX 341-350, 1938. 

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. 

Mirkin SM, Frank-Kamenetskii MD (1994) Annu Rev Biophys Biomol Struct 23 541... 

Mirkin SM, Lyamichev VI, Drushlyak KN, Dobrynin VN, Filippov SA, Frank-Kamenetskii MD (1987) Nature 330 495... 

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. 

Combustion equations with kinetics Frank-Kamenetskii... 

We follow the analysis of Frank-Kamenetskii [3] of a slab of half-thickness, rG, heated by convection with a constant convective heat transfer coefficient, h, from an ambient of Too. The initial temperature is 7j < 7 ,XJ however, we consider no solution over time. We only examine the steady state solution, and look for conditions where it is not valid. If we return to the analysis for autoignition, under a uniform temperature state (see the Semenov model in Section 4.3) we saw that a critical state exists that was just on the fringe of valid steady solutions. Physically, this means that as the self-heating proceeds, there is a state of relatively low temperature where a steady condition is sustained. This is like the warm bag of mulch where the interior is a slightly higher temperature than the ambient. The exothermiscity is exactly balanced by the heat conducted away from the interior. However, under some critical condition of size (rG) or ambient heating (h and Too), we might leave the content world of steady state and a dynamic condition will... 

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

Calculate the radius of a spherical pile of cotton gauze saturated with cottonseed oil to cause ignition in an environment with an air temperature ( /],) of 35 °C and 100 °C. Assume perfect heat transfer between the gauze surface and the air. The gauze was found to follow the Frank-Kamenetskii ignition model, i.e. 

Critical heat production rates (i.e., heat production rates that still do not lead to a runaway), are often determined by small scale experiments. However, the effect of scale-up on these rates, as discussed in [161], must be taken into account. An indication of the effect of scaling in an unstirred system is shown in Figure 3.2. In this figure, the heat production rate (logarithmic scale) is shown as a function of the reciprocal temperature. Point A in the figure represents critical conditions (equivalent heat generation and heat removal) obtained in a 200 cm3 Dewar vessel set-up. It can be calculated from the Frank-Kamenetskii theory on heat accumulation [157, 162] that the critical conditions are lowered by a factor of about 12 for a 200 liter insulated drum. These conditions are represented by... 

FIGURE 3.3. Comparison of Critical Temperatures for Frank-Kamenetskii and Semenov Models (Right Cylinder Configuration). 

When the reactor is scaled up to 60 cm radius, however, the operating point is between the two curves. This means that the reaction can be safely run at 50°C in a well-agitated process vessel of 60 cm radius with the heat transfer coefficient as stated above becauseerating point is below the Semenov curve. In case the agitation is lost, however, the Frank-Kamenetskii curve becomes the better predictor of runaway temperatures, and because the operating point is above this curve, the estimate is that the reaction will run away. The calculation of the Frank-Kamentskii method is available in ASTME-1231 [166]. 

The second model, proposed by Frank-Kamenetskii [162], applies to cases of solids and unstirred liquids. This model is often used for liquids in storage. Here, it is assumed that heat is lost by conduction through the material to tire walls (at ambient temperature) where the heat loss is infinite compared to the rate of heat conduction through the material. The thermal conductivity of the material is an important factor for calculations using this model. Shape is also important in this model and different factors are used for slabs, spheres, and cylinders. Case B in Figure 3.20 indicates a typical temperature distribution by the Frank-Kamenetskii model, showing a temperature maximum in the center of the material. 

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... 

FIGURE 3.20. Typical Temperature Distributions during Self-Heating in a Vessel. A = The Semenov Model B = The Frank-Kamenetskii Model C = The Thomas Model... 

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. 

Examples of the use of the Semenov and Frank-Kamenetskii models are presented by Fisher and Goetz [210]. 

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... 

In the next section, the flame speed development of Zeldovich, Frank-Kamenetskii, and Semenov will be discussed. They essentially evaluate this term to eliminate the unknown ignition temperature 7] by following what is now the standard procedure of narrow reaction zone asymptotics, which assumes that the reaction rate decreases very rapidly with a decrease in temperature. Thus, in the course of the integration of the rate term lv in the reaction zone, they extend the limits over the entire flame temperature range T0 to T. This approach is, of course, especially valid for large activation energy chemical processes, which are usually the norm in flame studies. Anticipating this development, one sees that the temperature term essentially becomes... 

This term specifies the ratio SJS 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. (4.59)]. 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... 

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