Page 112 - Biodegradable Polyesters

90 4 Synthesis, Properties, and Mathematical Modeling of Biodegradable Aliphatic Polyesters

k 5
tG + tG ⇌ bG + G (4.10)
k ′
5
k 6
tG + tG −−→ bDG + W (4.11)
Reactions (4.6)–(4.9) represent the typical esterification reactions, while reac-
tion (4.10) is the polycondensation reaction, occurring mainly in the second step
of polyester formation. Finally, reaction (4.11) is a side reaction resulting in digly-
col repeating units, with ether linkages in the oligomeric chain. k (i = 1, 6) and
i
′
k (i = 1, 5) representing the kinetic rate constants of the six elementary reactions
i
−1
(l mol −1 min ).
Development of the Mathematical Model In order to develop a mathematical model
for the esterification reaction the following assumptions are made:
• All kinetic rate constants are independent of polymer chain length (discussed
in Section 4.4.1).
• All the water produced during the reaction is instantaneously vaporized and
removed.
• All glycol vaporized is totally returned to the reactor. This assumption is correct
according to the specially designed experimental device used [43].

On the basis of the reaction mechanism Equations 4.6–4.11, the reaction rates
can be expressed in terms of the different functional groups present in the reactor
and the corresponding rate constants [38]. The terms in parentheses denote mole
numbers of each component.

R ={4k (SA)(G)−(k ∕K )(tSA)(W)}∕V 2 (4.12)
1 1 1 1
R ={2k (tSA)(G)− 2(k ∕K )(bSA)(W)}∕V 2 (4.13)
2 2 2 2

R ={2k (SA)(tG)−(k ∕K )(tSA)(W)}∕V 2 (4.14)
3 3 3 3

R ={k (tSA)(tG)− 2(k ∕K )(bSA)(W)}∕V 2 (4.15)
4
4
4
4
R ={k (tG)(tG)− 4(k ∕K )(bG)(G)}∕V 2 (4.16)
5
5
5
5
R ={k (tG)(tG)}∕V 2 (4.17)
6
6
′
where, K = k ∕k (i = 1, 5) denote the equilibrium rate constants. The volume of
i
i
i
thereactionmixture canbeexpressed as
(SA)MW SA (G)MW G W OLIG (W)MW W
V = + + − (4.18)

SA G OLIG W