The Semenov Theory

The theory of Semenov [8] was originally derived for zero-order reactions and can be applied whenever the reactant conversion is negligible and the reaction is very temperature-sensitive. It proves, however, to be too conservative in other cases. Nevertheless, it is resumed here since it gives a fundamental view of the dynamics of a thermal explosion and correct results in many practical cases. 

By renaming the two terms on the right-hand side of (4.12) as qr and qE, which represent the rate of heat production by reaction and of heat exchange with the cooling medium, respectively, the heat balance in the batch reactor can be rewritten as 

Under the assumption C = 1 at each time r, the system evolves toward steady-state conditions that can be located graphically on the Semenov diagram of Fig. 4.4 as the intersections of the curves r and q, this condition implies, indeed, that d%/dx = 0 in (4.16). For the sake of simplicity, let us first assume that %o = 7j = 1. When qE is given by line 1 with slope / i, the steady-state condition is given by point A, characterized by a low operating temperature. Point A is an attractor since its temperature is spontaneously restored after any small perturbation of the system and, consequently, in these conditions thermal explosion does not occur. 

On decreasing P, the slope of q, decreases until the critical (unstable) point B is reached where q, is tangent to /r (line 2) the corresponding critical temperature in B is denoted by Tc. The critical slope 2 is the lowest value that still leads to a bounded steady-state temperature. By further decreasing of P (line 3, 3 4 i), 

In Fig. 4.5, temperature profiles are reported in subcritical, critical, and supercritical conditions. Supercritical solutions of the simplified mathematical model proposed by Semenov are, however, purely theoretical since the assumption of negligible reactant conversion becomes very unrealistic. As an example, in the worst case where P - 0, the theory predicts an infinitely increasing temperature in the reactor. 

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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... 

The thermal expln of gaseous azomethane (Ref 3) occurs in accordance with the Semenov theory of thermal explosions as described in Ref 2. This process was studied also by Taylor Jahn (Ref 4)... 

According to the Semenov theory of chain reactions [2] the rate of oxidation depends strongly (half to first power) on the rate of production of new chain centres. However, the problem that has bedevilled combustion kinetics over many years is the chemical nature of the process. Reactions (1) and (lA) are the primary initiation reactions in hydrocarbon oxidation, to be distinguished from secondary initiation processes such as reaction (13) where radicals are produced from a stable intermediate... 

An investigation of potential thermal runaway reactions is a significant part of a thorough hazard evaluation. Important parameters of the exothermic reaction as well as of the large-scale system are discussed. Their relationship is explained through the Semenov Theory. 

For systems with a uniform temperature throughout the material, the "chemistry" and the "engineering" can be related through use of the Semenov Theory. (4) The rate of heat production was mentioned earlier as a kinetic parameter of interest and the heat transfer characteristics (in this case, rate of heat removal) as a large-scale system parameter. If the self-heating rate (rate of heat production) is determined as a function of temperature under adiabatic conditions and if there is a knowledge of the rate of heat removal as a function of temperature, information about safe operating limits for that particular system can be deduced. 

It should be pointed out that the Semenov Theory was developed for gases, is generally applied to non-viscous liquids, but does not hold for solids. Solids will not show a uniform temperature distribution because of their poor thermal conductivity. For solids, a more complex model must be used, such as the Frank-Kamenetskii Theory. (6) Discussions of this theory and others can be found in the literature. (7)... 

In conducting an investigation of thermal hazards, particularly of the thermal runaway, certain cautions must be mentioned. As discussed previously, the Semenov Theory holds for many liquids, but solids must be treated quite differently. Because the Semenov Theory is easier to apply than the theories available for solids, it is often tempting to apply the Semenov Theory to solids as well as liquids. Major errors can arise ... 

Uniting Cases. The Semenov theory has usually been considered as the limiting case of (31) as oo. However, it was pointed out by B. F. Gray that this limit, coupled with the choice of the characteristic time, results in a stretching of the dimensionless time t, such that as B -> oo one instant of real time corresponds to an infinite period of t time. Gray investigates this limit by using r = t/tit and his equations correspond to the Semenov zeroth order reaction as B- oo. 

If these dimensionless numbers are related back to the theories of selfheating, in particular the Semenov theory (Section 4.2, page 46), then it can be shown that ... 

Figures 16-18 show the comparison of the Semenov theory to the interfacial tension data of Anastasiadis et al. [20] for three different polymer systems, PDMS/ PBD 1000, PS/PBDH 3080 and PS/PMMA 10,000. The agreement is very good for the PDMS/PBD and PS/PBDH systems, whereas it is poor for the PS/PMMA blend. AcmaUy, Semenov argued that the disagreement for PS/PMMA is far beyond possible errors due to approximations of the theory and that it might indicate that the model based on the Flory-Huggins interaction term may be inadequate for the PS/PMMA system, with higher order terms being important in the excess free energy of interaction.

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