Stabilization at high flow velocities

For homogeneous (completely flowing) open systems a steady-state point becomes unique and stable at a very high constant velocity of the flow [35]. In this case the concentrations of gas-phase components rapidly become almost constant and their ratios are close to those for the input mixture. This fact is independent of a concrete type of the w(c) function. To confirm this postulate, let us consider eqns. (125) for a balance polyhedron Da. Since vm is very high, the inequality (127) is fulfilled automatically and we can write 

As Z)0 is a convex restricted coinvariant set, it contains at least one steady-state point of eqn. (139). Note that, if starting from some vin, for any two different solutions of eqn. (139) lying in T)0, cl(t) and c2(t), the function c1 ( ) - c2(t) is monotonically reducing to zero, the steady state is unique, and any solution lying in Z)0 tends to this steady state at t - oo. It is the distance to this point that will be the global Lyapunov function for eqn. (139) in t)0. Let us investigate at which values of vin the function c ( ) — c2(t) decreases monotonically. 

Since the inequality (140) must be fulfilled for arbitrarily close c1 and c2 values, we obtain 

Due to the convexity of D0 (here it is merely a simplex), the local condition (141) is sufficient to claim that eqn. (140) is valid. Inequality (141) is fulfilled if the maximum eigenvalue /max of the matrix V2 (F -I- F T) is lower than v JV at any c from D0. 

An accurate formula for the upper limit of these Amax in Da cannot be given. Hence it is recommended that individual vin values are found for every kinetic model. The stability of the matrix 

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