The Heat Explosion Theory

Before discussing the safety assessment of chemical processes under normal as well as under upset conditions in detail, the classical heat explosion theory shall be treated. The first scientists to investigate the so-called runaway of an exothermic chemical reaction were Semenov and Frank-Kamenetzidi [18,19]. They were the pioneers in investigating and describing the self-heating process of reacting systems up to an explosion-like temperature rise in its dependence on different heat loss conditions to the environment. The criteria they derived are still valid today and form the basis of any safety assessment. 

Frank-Kamenetzkii, on the other hand, described the other limiting case of a totally unstirred exothermic reacting system. This is also of practical relevance, as this is a good model for particle beds as such exist in drying and storage operations. The main heat transfer resistance prevails within the reaction mass itself The wall is taken as an infinitely large isothermal reservoir in this case, neglecting all boundary layer effects. 

Schematic presentation of the temperature profiles according to Semenov and Frank-Kamenefcddi 

It shall be assumed that a simple conversion of A to B shall be performed in a batch reactor. It is further assumed that A is available in a close to infinitive amount, this way allowing the neglection of any consumption of A with progressing reaction time. Such a process may formally be described by a zero-order reaction rate law. The teniperature dependence of the reaction rate shall follow the Arrhenius relationship. With these prerequisites the heat production rate of this reaction can be expressed  

If the rate constant is referred to a reference temperature Tq, which in the case of batch processes is equivalent to the cooling temperature. 

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Two proximations allow the solution of this balance. First, the dependency of the reaction rate on temperature shall follow the approximation already used in the discussion of the heat explosion theory. The second step neglects the development of radial concentration profiles. 

It is an interesting fact that a formally equivalent equation was obtained when discussing the heat explosion theory. At that time a zero-order reaction in a cooled BR was being examined. For the sensitivity the following expression is deduced ... 

This equation, also shows a singularity. A further examination of this singularity yields a condition which is already known from the heat explosion theory as a rule of thumb... 

As stated in Preface, the basic concept of the thermal explosion theory is that whether the thermal explosion or the spontaneous ignition of a chemical of the TD type, including every gas-permeable oxidatively-heating substance, having an arbitrary shape and an arbitrary size, placed in the atmosphere under isothermal conditions, occurs or not is decided, based on the balance between the rate of heat generation in the chemical and the rate of heat transfer from the chemical to the atmosphere at the critical state for the thermal explosion which exists at the end of the early stages of the self-heating process. 

Incidentally, the effect of the concentration of a chemical of the TD type on the rate of the exothermic decomposition reaction, in the early stages of the selfheating process, of the chemical is assumed to be of the zeroth order in the thermal explosion theory [1], In other words, it is assumed in the thermal explosion theory that the concentration of the chemical remains virtually constant while the self-heating process is in the early stages, because the consumption of the chemical caused by the reaction can be neglected while the self-heating process is in the early stages. 

Now, when a solid chemical of the TD type is placed in the atmosphere maintained at a temperature below T, which is shown in Fig. 2 in Section 1.2, a spatially gradient distribution of temperature is effected in the solid chemical. However, the temperature as a whole does not vary with time in other words, the spatially gradient distribution of temperature effected in the self-heating solid chemical placed in the atmosphere maintained at a temperature below T is stationary. The stationary equation of the thermal explosion theory, Eq. (22), is thus obtained by considering the value of the derivative with regard to the time to be zero in Eq. (20). This approach is called the stationary theory of the thermal explosion [7]. ... 

The differential equations, in which one of the unknown variables, e.g., temperature, enters in a nonlinear manner, i.e., as an exponential function, while its derivative enters in a linear form, are often dealt with in the thermal explosion theory. Such an equation is called quasilinear in mathematics [21]. For instance, we have assumed in Eq. (50) presented in Section 2.5 that the value of dTIdt remains virtually constant while the self-heating process of 2 cm of a chemical of the TD type, subjected to the adiabatic self-heating test, is in the early stages. 

Correlation among the pattern of the TG-DTA curve of a self-heating powdery chemical, the two types of self-heating behaviors, Le., the TD type and the quasi-AC type, and the two equations of the thermal explosion theory... 

Group in. Powdery chemicals to each of which neither of the two equations of the thermal explosion theory can be applied to calculate the Tc. When confined in the closed cell and subjected to the adiabatic self-heating test started from a 7 2 cm of a powdery chemical of this group warms slowly up to the 7j, but the temperature remains near the F, until the chemical has finished melting. Once, however, the chemical finishes melting in the course of time, an apparently sudden quasi-autocatalytic reaction of the resultant liquefied chemical starts [25]. 

Like most rules of thumb, this one, too, may lead to dangerous misjudgments with respect to the thermal process safety, if not applied correctly. The following shall demonstrate this. One of the fundamental theories used for the safety assessment of exothermic chemical reactions is explained in detail in Chapter 4.2 the thermal or heat explosion theory by Semenov [18], The central statement of this theory is that an explosion-like runaway of an exothermic chemical reaction will always occur... 

In the search for a better approach, investigators realized that the ignition of a combustible material requires the initiation of exothermic chemical reactions such that the rate of heat generation exceeds the rate of energy loss from the ignition reaction zone. Once this condition is achieved, the reaction rates will continue to accelerate because of the exponential dependence of reaction rate on temperature. The basic problem is then one of critical reaction rates which are determined by local reactant concentrations and local temperatures. This approach is essentially an outgrowth of the bulk thermal-explosion theory reported by Fra nk-Kamenetskii (F2). 

Extension of the hydrodynamic theory to explain the variation of detonation velocity with cartridge diameter takes place in two stages. First, the structure of the reaction zone is studied to allow for the fact that the chemical reaction takes place in a finite time secondly, the effect of lateral losses on these reactions is studied. A simplified case neglecting the effects of heat conduction or diffusion and of viscosity is shown in Fig. 2.5. The Rankine-Hugoniot curves for the unreacted explosive and for the detonation products are shown, together with the Raleigh line. In the reaction zone the explosive is suddenly compressed from its initial state at... 

The self-heating and ignition of baled or loose wool in bulk storage is discussed and analysed, and steady state thermal explosion theory is applied to the prediction of critical masses and induction periods for storage and transportation situations in relation to ambient temperature. Results obtained were consistent with current safety practices. 

One experiment which does not seem to fit into the network of the salt-gradient theory was that of Wright and Humberstone (1966), who impacted water on molten aluminum and obtained explosions. These results are at variance with those of Anderson and Armstrong, but the latter worked at 1 bar whereas the former used a vacuum environment. It might be possible that, under vacuum, it is much easier to achieve intimate contact between the aluminum and water and, under these conditions, there may be sufficient reaction between the aluminum and water to allow soluble aluminum salts to form. This salt layer could then form the superheated liquid which is heated to the homogeneous nucleation temperature and explodes. 

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). 

Such reactions have been used to explain the three limits found in some oxidation reactions, such as those of hydrogen or of carbon monoxide with oxygen, with an "explosion peninsula between the lower and the second limit. However, the phenomenon of the explosion limit itself is not a criterion for a choice between the critical reaction rate of the thermal theory and the critical chain-branching coefficient of the isothermal-chain-reaction theory (See Ref). For exothermic reactions, the temperature rise of the reacting system due to the heat evolved accelerates the reaction rate. In view of the subsequent modification of the Arrhenius factor during the development of the reaction, the evolution of the system is quite similar to that of the branched-chain reactions, even if the system obeys a simple kinetic law. It is necessary in each individual case to determine the reaction mechanism from the whole... 

Cook (Ref 17, p 36) designates the available energy as A, and states that this property, as well as the heat of explosion Q, and the ratio A/Q are the important quantities determining the total blast or "avaiable work potential or "available energy . The theory is presented in Chapter 11 of Ref 17, pp 265ff and is considered more reliable than experimental procedures, at least for CHNO expls. The experimental procedures referred to by Cook for determination of (A) include Trauzl Block Test and Ballistic Mortar Test. New methods have been proposed, such as determination of peak pressure or/and total energy ... 

Another Commission was appointed in France to inquire into coal-mine explosions. Influenced by Berthelot s work and theories, its attention was directed to the question of the heat evolved by an explosive and the resulting temperature of the products (Ref 11, p 45) Note I As a result of the work of the Commission, a max temp of 1500°C for expls used in coal layers (explosifs grisou-couche) was established and 1900°C for expls used for blasting rock in gaseous coal mines (explosifs grisou-roche) (Vol 3 of Encycl, p C369-L)... 

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