Kinetic balances for modeling polymer MWs

It is also helpful to define moments for the bulk (ft) polymer, the total polymer in the system including live radicals  

With [D ] [P ] there is little difference in magnitude between vt and ft hut, as described below, the introduction of ft eliminates the moment closure problem created by the LCB mechanism. These moment definitions collapse the infinite set of equations for polymeric species into a manageable subset, and many of the moments have precise physical meanings. The zeroth live moment, [/cq], is the concentration of polymer radicals in the system (denoted by [Ptot] in Section 3.2.1), and the first live moment, [/ri],is the concentration of monomer units contained in all growing radicals. Similarly, [fo] is the concentration of all polymer chains in the system, and [f i ] is the concentration of monomer units bound in all polymer chains. These values are used to calculate MW averages, where Wn, is the molecular weight of the monomeric repeat unit  

Equations for [fo], [fi] and [f2] will be developed for the calculation of Afn and Afw The first step is to formulate balances (in terms of the net generation rates per unit volume, mol f s ) for live radicals, dead chains and total chains of length n, accounting for all of the consumption and generation terms from the kinetic mechanisms  

These species balances are next substituted into the moment definitions (Equations 3.57-3.59). The use of generating functions [54, 57, 58] eliminates the tedium (and possible errors) of performing the required series summations both treatments lead to the following equations describing the rates of change of the moments  

The set of moment expressions to be substituted into reactor balances consists of either the live and dead moments (Equations 3.64-3.69) or the live and bulkmoments (Equations 3.64, 3.65 and 3.70-3.72, substituting [( i] and [ 2 for [i ] and [V2] in Equation 3.65). Choosing the former, while common practice in the literature, suffers from a moment closure problem, as ( 2] and [02] depend on [V3]. The Hulburt and Katz [59] method assumes that the molecular weight distribution can be represented by a truncated series of Laguerre polynomials, approximating [V3] as 

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