Page 104 - Numerical methods for chemical engineering

90 2 Nonlinear algebraic systems

The double-headed arrow denotes that this reaction is reversible, with an equilibrium con-
stant 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
(−NH 2 ) is balanced.
We describe the hierarchy of reactions in polycondensation using the following notation.
Let [P x ] 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:
P 1 + P 1 ⇔ P 2 + W
P 1 + P 2 ⇔ P 3 + W
P 2 + P 2 ⇔ P 4 + W (2.115)
P 1 + P 3 ⇔ P 4 + W
P 2 + P 3 ⇔ P 5 + W
The number of reactions and species in the system quickly grows very large. We can,
however, derive simple equations that describe the average chain length and breadth of the
chain length distribution with only a few state variables. We define the kth moment of the
chain length distribution, λ k ,as
∞
k
λ k = m [P m ] (2.116)
m=1
Of particular interest are the three leading moments
∞ ∞ ∞
2
λ 0 = [P m ] λ 1 = m[P m ] λ 2 = m [P m ] (2.117)
m=1 m=1 m=1
From these three values, we can calculate the number-averaged chain length, ¯ x n = λ 1 /λ 0 ,
and the weight-averaged chain length, ¯ x w = λ 2 /λ 1 . Since the weight-averaged chain length
biases more the contributions of the larger chains, ¯ x w ≥ ¯ x n , with the equality holding
only if all chains are of the same length. The ratio of these two averages, Z = ¯ x w /¯ x n , the
polydispersity, provides a simple measure of the breadth of the chain length distribution.
The larger the polydispersity, the greater the disparity in the lengths (molecular weights) of
individual chains. We now need only calculate the rates of change of these three moments
to predict, for given polymerization conditions, the characteristics of the polymer produced.
For further discussion of the use of population balances and moment equations in polymer
reaction engineering, consult Ray (1972) and Dotson et al. (1996).

Rate equations for polycondensation

We obtain rate equations for these moments by counting how each x-mer species is produced
or consumed in each individual reaction in the hierarchy above. First, we represent the
sequence of reactions (2.115) as
P m + P n ⇔ P m+n + W (2.118)