Sources of Multiplicity and Bifurcation

The last and most common nonmonotonic rate of reaction occurs under nonisothermal operation where the rate of reaction, even for a first-order reaction, is highly nonlinear, namely 

Here CA is the concentration of the reactant A. If the reaction is exothermic, i.e., if the process generates heat, then the rate equation (2.2) is nonmonotonic with respect to conversion. As the reaction proceeds, the conversion increases, i.e the concentrations of the reactants decrease (tending to cause a decrease in the rate of reaction), while the temperature increases (tending to cause an increase in the rate of reaction). These opposing effects obviously give nonmonotonic dependence upon the conversion or the concentration or the temperature. This case is investigated in more detail in Chapter 3. It is one of the natural causes of multiple steady states and bifurcation for reacting systems. 

The multiplicity of steady states and associated bifurcation phenomena are associated with open systems which allow more than one stationary nonequilibrium state for the same set of parameters. This multiplicity of a steady state may result from many causes. The most important sources of multiplicity in chemical/biological engineering processes are  

Nonmonotonic dependence of one rate process (or several) on one or more state variables. 

A recycle in a tubular reactor or axial dispersion which will be discussed in Section 7.2. 

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