Eigenzeit transformation

Eigenzeit Transformation. This simple change of variable first proposed by Dostal and Mark (16) linearizes equations when a monomer appears in every term. The quantity Mdt is defined as a new variable dZ. It is widely used for stepwise polymerization without termination. It is mentioned to avoid confusion with the Z-transform. In one paper, both transformations were used in the same equation, and both were referred to in connection with the letter Z. 

P — Pseudo-steady-state assumption C = Continuous variable for chain length L = Laplace transforms Z = Z-transforms E = Eigenzeit transform M = Moments of distribution N = Numerical techniques... 

Batch Polymerization. The case of stepwise polymerization without termination was originally treated by Dostal and Mark (16) by using the Eigenzeit transformation to linearize the equations, as discussed earlier. Gee and Melville (21) extended this by the same technique to a case where the propagation rate constant varied with molecular size, contrary to the usual assumption. In the case of stepwise polymerization without termination, batch reactions can give a very narrow (Poisson) distribution. Abraham (2) and Kilkson (35) both showed that the use of the Z-transform simplified the handling of this type of mechanism. 

This change of time variables, first used in polymerization analysis by Dostal and Mark [26], is known as the eigenzeit transformation. It allows for a much easier solution of equation (3.14) since the monomer concentration is a continuously changing quantity in a batch reactor. If the temperature control is poor, the reaction rate constant would also change. 

The higher order moments can be readily generated and the set of equations integrated using the transformed initial conditions and the eigenzeit transformation. 

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