Example. Steady-state modeling of a condensation polymerization reactor

Example. Steady-state modeling of a condensation polymerization reactor 

The most common form of nylon, nylon-6,6, is made by polycondensation of the two monomers hexamethylene diamine (HMD) and adipic acid (ADA). The first step in the reaction sequence is the condensation of two monomers to form a dimer with an amide linkage, —CONH—, and a water molecule (the condensate). 

The double-headed arrow denotes that this reaction is reversible, with an equilibrium constant on the order of 100. The dimer still has functional groups on each end, and so may continue to react to produce even larger molecules. If the water is removed by evaporation to drive the equilibrium to the right, polymer chains with molecular weights of 20-30 kg/mol may be produced. To achieve very high molecular weights, we use equimolar amounts of each monomer so that the stoichiometry between the acid ends (—COOH) and the base ends (-NH2) is balanced. 

We describe the hierarchy of reactions in poly condensation using the following notation. Let [P c] be the total concentration of chains in the system that contain x monomer units, i.e., are x-mers. We then have the following hierarchy of reactions  

The number of reactions and species in the system quickly grows very large. We can, however, derive simple equations tiiat describe the average chain length and breadth of the chain length distribution with only a few state variables. We define the Ath moment of the chain length distribution, X], as 

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