Form features Multiple conditioning

We now consider the dependence of the stationary-state solution on the parameter d. To represent a given stationary-state solution we can take the dimensionless temperature excess at the middle of the slab, 0ss(p = 0) or 60,ss-With the above boundary conditions, two different qualitative forms for the stationary-state locus 0O,SS — <5 are possible. If y and a are sufficiently small (generally both significantly less than i), multiplicity is a feature of the system, with ignition on increasing <5 and extinction at low <5. For larger values of a or y, corresponding to weakly exothermic processes or those with low temperature sensitivity, the hysteresis loop becomes unfolded to provide... 

Summarizing the statements of these three most commonly used models, it appears that the so-called mass action and phase-separation models simulate a third condition which must be fulfilled with respect to the formation of micelles a size limiting process. The latter is independent of the cooperativity and has to be interpreted by a molecular model. The limitation of the aggregate size in the mass action model is determined by the aggregation number. This is, essentially, the reason that this model has been preferred in the description of micelle forming systems. The multiple equilibrium model as comprised by the Eqs. (10—13) contains no such size limiting features. An improvement in this respect requires a functional relationship between the equilibrium constants and the association number n, i.e.,... 

Reversible exothermic reactions have an ultimate decrease of rate with temperature, and so the heat generation curve turns down (as in Example 10.4.a-l) however, the qualitative features remain the same. The heat generation curve for complex reactions can have more than one hump, and thus more than three steady states are possible for a given operating condition. The humps also tend to be smaller, leading to more readily obtained transitions between steady states, and so on—Westerterp [30]. Also, other types of multiple steady states and instabilities can occur. For example, with certain forms of rate expressions highly nonlinear in concentration, just the mass balance Eq. 10.4.a-i may have more than one solution. This is summarized in Perlmutter [31] (as well as many other techniques). 

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