Stability of Steady States in a CSTR

These steady states are within the physically possible range of T 0 T oo) and X(0 X 1). This is in contrast to many situations in the physical sciences where equations have multiple roots but only one root is physically acceptable because the other solutions are either outside the bounds of parameters (such as negative concentrations or temperatures) or occur as imaginary or complex numbers and can therefore be ignored. 

Multiple solutions to equations occur whenever they have sufficient nonlinearity. A familiar example is equHihrium composition calculations for other than A B. The reaction composition in the reaction A i B yields a cubic polynomial that has three roots, although all but one give nonphysical concentrations because thermodynamic equilihrium (the solution for a reactor with f — co or T — co) is unique. 

However, even if there are three steady states, there is stiU the question of whether a steady state is stable even if it is in the physically significant range of parameters. We ask the question, starting the reactor at some initial situation, wiU the reactor eventually approach these solutions Or, starting the reactor near one of the steady states, wiU it remain there To answer this, we must examine the transient equations. 

For a single reaction in the CSTR, the transient mass- and energy-balance equations are 

Note that the first equation is dimensionless, while each term in the second equation has the dimensions of temperature. [We repeat these equations many times to emphasize their importance if we can understand their solutions, we will understand multiple steady states.] 

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