Probability Generating Functions in a Transformation Method

According to the transformation procedure, the population balance equations in terms of discrete variables (chain length, number of branch points) are transformed into a set of equations in z. The transformed set is solved and subsequently inverted to the original discrete variable domain. This process makes use of some interesting properties of certain mathematical expressions as transformed into the z-domain. Transformations and inversions are tabulated in textbooks [3). As an example we take linear AB step polymerization in a batch reactor with equal initial end group concentration Po  

The total chain concentration is Po = Pn, so by taking the zeroth moment of 

Since by definition [Eq. (104)] G(l) = /to, while the convolution property prescribes  

By putting z = 1, Eq. (110) for the total concentration ftQ is reproduced. To solve Eq. (112) we first realize that under initial conditions the pgf is given as G(z,0) = Poz integration then leads to  

This expression in the z-domain has a standard inverse form in the chain length domain, which represents a Flory distribution as the well-known solution to this problem  

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