The Prout-Tompkins Model

The Prout-Tompkins model is the oldest described in the literature [8] and it is also the simplest, since it is based on only one reaction and one rate equation  

This expression describes the characteristic S-shaped conversion curve as a function of time. Under isothermal conditions, the maximum reaction rate and consequently the maximum heat release rate is obtained for a conversion of 0.5  

The kinetic constant can be calculated from the maximum heat release rate measured under isothermal conditions  

This model gives a symmetrical peak with its maximum at half conversion. Hence the model is unable to describe non-symmetrical peaks as they are often observed in practice. Moreover, in order to obtain a reaction rate other than zero, some product B must be present in the reaction mass. Therefore, the initial concentration of B (CBo) or the initial conversion (X0) is a required parameter for describing the behavior of the reaction mass. This also means that the behavior of the reacting system depends on its thermal history, that is, on the time of exposure to a given temperature. This simple model requires three parameters the frequency factor, the activation energy, and the initial conversion that must be fitted to the measurement in order to predict the behavior of such a reaction under adiabatic conditions. 

---

Figure 12.3 Heat release rate and conversion under isothermal condition for the Prout-Tompkins model.

Show that for the Prout-Tompkins model the activation energy calculated from the heat release rate and from the isothermal induction time is the same. 

To test the Bawn model, the early stages of decomposition should give a linear plot of In a against t and the later stages a linear plot of ln(l - a) against t. The same holds for the Prout-Tompkins model (Table 3.3.), so the two models cannot readily be distinguished on kinetic grounds. 

A number of formal kinetic models exist to describe autocatalytic reactions (cf. [1]). In what follows only the simplest model, the Prout-Tompkins model, is described following [1]. It is based on the reaction equation... 

Here we describe three of them the Prout-Tompkins [8], the Benito-Perez [6] models, and a model stemming from the Berlin school [1, 10, 11]. These models describe the phenomenon in a simple way and are the most used in practice, especially with respect to process safety. 

Branching nucleation forms the starting point for a derivation of the Prout-Tompkins reaction model [13] (see below). The main rate equations for nucleation are summarized in Table 3.1. and illustrated in Figure 3.1. 

It should be pointed out that sigmoidal rate plots are sometimes observed for reactions of solids. One of the rate laws used to model such reactions is the Prout-Tompkins equation, the left-hand side of which contains the function ln(a/ (1 — a)) where a is the fraction of the sample reacted (see Section 7.4). The left-hand sides of Eqs. (6.72) and (7.68) have the same form, and both result in sigmoidal rate plots. These cases illustrate once again how gready different types of chemical processes can give rise to similar rate expressions. 

A more detailed analysis( ) shows that the special conditions which lead to this equation are (1) a branching coefficient independent of a (which is reasonable if this simply represents the number of individual nuclei in each microcluster, but surely less plausible in terms of a model in which branching is due to strain or cracking) (2) a termination coefficient which is equal to the branching coefficient times a/a (this should result in the same rate constant for acceleratory and decay periods, whereas they usually differ and sometimes even have different temperature coefficients) and (3) a symmetrical decomposition curve with a,- = 0.5 (potassium permanganate is the best example of this). Equation (34) presupposes the existence of some initial decomposition (nucleus formation) during the induction period integration between limits (ao, to) and (a, t) yields the Prout-Tompkins equation( )... 

Model fitting used the extended Prout-Tompkins model,... 

Thanks to its versatility, this model has proved to describe a great number of autocatalytic reaction systems [5]. Systems with a slow initiation reaction are called strong autocatalytic. Because the rate of the initiation reaction is low, product is formed slowly, leading to a long induction time under isothermal conditions. For such systems, the initial heat release rate is low or practically zero. Consequently, the reaction may remain undetected for a relatively long period of time (Figure 12.4). When the reaction accelerates, such an acceleration appears suddenly and may lead to runaway. A strong autocatalytic reaction is formally equivalent to a Prout-Tompkins mechanism. 

The sigmoidal decomposition curves can be interpreted using the Prout and Tompkins model. This model assumes that the decomposition is governed by the formation and growth of active nuclei which occur on the surface as well as inside the crystals. The formation of product molecules sets up further strains in the crystal since the surface array of product molecules has a different unit cell from the original substance. The strains are relieved by the formation of cracks. Reaction takes place at the mouth of these cracks owing to lattice imperfections and spreads down into the crevices. Decomposition on these surfaces produces further cracking and so the chain reaction spreads. 

---