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Polymerization Modeling
Objective: Modeling Condensation Polymerization by using Functional
Groups
In this example, we develop a kinetic model for condensation polymerization that tracks the
concentration of functional groups instead of individual molecules. We use measurements of
molecular weights to determine the polymerization kinetics. You may download the rex file to view the
model.
Features Illustrated
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Building polymerization model using functional groups
Use of Pseudo Compounds and Derived Quantities to quantify polymer physical properties
Pressure control
Visualization of Reaction Traffic
Reaction Model
In condensation polymerization, monomer molecules join together by typically eliminating a smaller
molecule as a condensate by-product. For esterification reactions involving carboxylic acid and
alcohols, the condensate is water. The reversible reaction between a di-carboxylic acid and di-alcohol
monomer can be represented as:
HOOC-A-COOH + HO-B-OH
⇆ HOOC-A-COO-B-OH + H2O
The ester formed still has terminal alcohol and acid groups remaining, thus it can react further with
other monomer/oligomer molecules to continue growing. Terminal ends of any polymer molecules
can also react to grow the chain:
HOOC-A-COO-(/\/\/\/\/\)-COO-B-OH +
HOOC-A-COO-(/\/\/\/\/\/\/\)-COO-B-OH
⇆
HOOC-A-COO-(/\/\/\/\/\)-COO-B-COO-A-COO-(/\/\/\/\/\/\/\)-COO-B-OH + H2O
The reverse hydrolysis reaction splits the polymer molecule and can happen at any of the ester units
in the polymer chain. Given the large number of oligomerization and hydrolysis reactions possible, we
simplify the reaction model by using an approach that does not require us to track each molecule.
Instead, we only track the changes in the functional groups, thereby making an assumption that the
oligomerization and hydrolysis kinetics are independent of chain length. We track the following
functional groups:
-
T-A-COOH
T-B-OH
A-COO-B
: Terminal COOH, contained in acid monomer as well as polymer
: Terminal OH, contained only in polymer (non-monomer)
: Ester group COO
and the following compounds:
-
HO-B-OH
H2O
: Alcohol monomer
: Water
For the COOH functional group (T-A-COOH), we make no distinction between those in monomer and
those in polymer because all acids are only in the liquid phase. However, the monomer alcohol is
distributed between the liquid and gas phase, and consequently the OH functional group must be
tracked in both phases. To do this, we model the Terminal OH in monomer alcohol (HO-B-OH)
separately from the Terminal OH in the polymer molecules (T-B-OH), which are only in the liquid
phase. The experiments are run in a liquid-gas batch reactor where water and alcohol monomer are
distributed in both phases, while all other species are only in the liquid phase.
With the defined compounds, we can write the reactions as shown below:
R1 : T-A-COOH + HO-B-OH ⇆ A-COO-B + T-B-OH + H2O
R2 : T-A-COOH + T-B-OH
⇆ A-COO-B + H2O
Reaction R1 corresponds to esterification with alcohol monomer, while R2 refers to esterification with
a non-monomeric alcohol. Experimental data suggests that esterification can be catalyzed by the acid
itself as well as by a metal catalyst. So we add {R1-Cat, R2-Cat} reactions along with (R1, R2) to
represent the metal catalyzed and acid catalyzed reactions respectively. The reaction orders for the
acid catalyzed reactions are posed as:
R1-forward rate ∝ [ T-A-COOH ] [ HO-B-OH ] [H+]n
R2-forward rate ∝ [ T-A-COOH ] [ T-B-OH ] [H+]n
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R1-reverse rate ∝ [ A-COO-B ] {p1} [ H2O ] [H+]n
R2-reverse rate ∝ [ A-COO-B ] {p2} [ H2O ] [H+]n
For reverse R1 reaction, where monomer alcohol is produced, its rate is proportional to the
concentration of A-COO-B ester groups that are located next to the terminal T-B-OH alcohol groups
only. Thus we have its dependency on [A-COO-B] {p1}, where {p1} is the probability of the A-COO-B
ester group being next to T-B-OH.
It can be deduced that {p1} = [ T-B-OH ] / [ A-COO-B ]. After replacing we have:
R1-reverse rate ∝ [ T-B-OH ] [ H2O ] [H+]n
In R2-Reverse case, given that the alcohol formed does not belong to a monomer molecule, the rate
is proportional to ester groups A-COO-B that are not next to the terminal T-B-OH. The probability of
those ester groups can be calculated as:
{p2} = 1 - ( [ T-B-OH ] / [ A-COO-B ] ) = ( [ A-COO-B ] - [ T-B-OH ] ) / [ A-COO-B ]
We define the the auxiliary variable (pseudocompound) below:
Bound-A-COO-B = A-COO-B - T-B-OH
After replacing the later into R2-reverse we have the kinetics orders as
R2-reverse rate ∝ [ Bound-A-COO-B ] [ H2O ] [H+]n
As for acid catalysis factor [H+]n , we consider order n=0.5; and replace [H+]n = [ T-A-COOH ]0.5 so
the reaction orders for the acid catalysed reactions become:
R1-forward rate ∝
R2-forward rate ∝
R1-reverse rate ∝
R2-reverse rate ∝
[ T-A-COOH ]1.5 [ HO-B-OH ]
[ T-A-COOH ]1.5 [ T-B-OH ]
[ T-B-OH ] [ T-A-COOH ]0.5 [ H2O ]
[ Bound-A-COO-B ] [ T-A-COOH ]0.5 [ H2O ]
In addition, we have the reactions that are catalyzed by metallic catalyst Q, whose dependency is
proposed as:
R1-cat-forward rate ∝
R2-cat-forward rate ∝
R1-cat-reverse rate ∝
R2-cat-reverse rate ∝
[ T-A-COOH ] [ HO-B-OH ] [Q] / ( 1+k[H+]m )q
[ T-A-COOH ] [ T-B-OH ] [Q] / ( 1+k[H+]m )q
[ A-COO-B ] {p1} [ H2O ] [Q] / ( 1+k[H+]m )q
[ A-COO-B ] {p2} [ H2O ] [Q] / ( 1+k[H+]m )q
In the reaction orders above, we inhibit the effect of [Q] with a 1 / ( 1+k[H+]m )q factor. This ensures
that we can dampen the effect of catalyst when acid concentration [H+] is high. In this example, we
set site exponent q=2 and acid order m=4. Values derived earlier for {p1}, {p2} and [H+] are
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substituted into the expressions above and are entered in the rex file. You may review the kinetics in
the Kinetics→ Parameters node of the rex file.
Calculation of Polymer Average Molecular Weight
Having set up the kinetic model, we need to calculate the average molecular of the monomer-polymer
mixture:
Mn = {Mass} / {Moles}.
Each linear polymer molecule has two terminal ends, so number of moles of polymer is half the
number of terminal species. To that we also add the count of monomer alcohol to get the total moles
of the monomer-polymer mixture:
Total Moles = 0.5 ( {Moles of T-B-OH} + {Moles of T-A-COOH} ) + Moles of HO-B-OH
At the start of reaction, there is no polymer yet, so the above expression equals to the moles of
monomer species. As reaction proceeds, the number of terminal units and monomer drops reducing
the number of moles.
To obtain the mass of polymer, we need to multiply the moles of polymer fragments to their molecular
weight:
{Mass} = ∑i ( {Moles of Polymer Fragments}i * {Fragment Molecular Weight}i )
The fragment terms and their molecular weights for this illustration are below:
In the table, -A- and -B- represent the acid and alcohol monomer building blocks and F-B-OH
indicates the OH hydroxyl on the alcohol monomer molecules. They are calculated as follows:
{Moles of -A-} = ( {Moles of T-A-COOH} + {Moles of A-COO-B} ) / 2
{Moles of -B-} = ( {Moles of T-B-OH} + {Moles of A-COO-B} ) / 2 + {Moles of HO-B-OH}
{Moles of F-B-OH} = 2 {Moles of HO-B-OH}
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These formulae are entered in the Pseudocompounds node.
Setting up the REX Project
You may follow this description in the rex file provided. In the Chemistry→Compounds node, we enter
all the compounds and functional groups, plus nitrogen that is the inert present in gas phase:
The Reactions node contains the reactions described above:
In the Pseudo-Compounds node, we define the auxiliary variables to be used in the model. They are
all marked as Conserved, so their values are reported in the solution on a molar basis, as if they were
regular compounds:
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In the Kinetics node, all directions are included:
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In Kinetics→Parameters node, we set the reactions orders. In the LHHW Sites tab, we define the site
as indicated in previous section:
The Site is assigned to the reactions in Kinetics→Kinetics Sites node, and the exponent entered
there:
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Having completed the Chemistry description, we now proceed to the Estimation tree. In the
Estimation node, all the reactions and the LHHW Sites are marked for estimation. In the
Estimation→Parameters node, bounds are open for pre-exponential and activation energies. For the
Site, we only open bounds for pre-exponentials, leaving energies fixed to zero.
In Parameter relationships, we enforce the following constraints:
- We specify R1 and R1-Cat to have zero heat of reaction. Thus, Forward and Reverse
activation energies for them are same. That is enforced in the first and second parameter
relationships
- Same Activation Energy for R1 and R2, both directions. Also same for R1-Cat and R2-Cat, in
both directions. They are enforced from the third to the sixth parameter relationship
- For the reverse directions, pre-exponentials R1 and R2 are required to be equal. Similar
constraint for R1-Cat and R2-Cat reverse direction. These relationships are shown in the fifth
and sixth parameter relationships
- R1 equilibrium constant (Pre-exponential forward/reverse ratio) between 1.5 and 2.5, as
implemented in the seventh and eighth parameter relationship
- Pre-exponentials for R1/R2 forward direction ratio fixed to 2; same for R1-Cat/R2-Cat forward
ratio. Enforced in the last two parameter relationships
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In the Reactor node, we define a batch multiphase system. Pressure is set to be controlled and
outflows are enabled:
In the Reactor→Phases node, liquid and gas phases are defined, with the reactions only in the liquid
phase. In the Phase Distribution node, the compounds are assigned to the phases and Henry
parameters for H2O and monomer alcohol HO-B-OH are entered:
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In the Reactor→Flows node, an outflow is specified in Flows tab and its composition is assigned to
be same as the gas phase. As the reaction proceeds, the release of water increases the reactor
pressure and the pressure controller opens the outflow valve. To model this, this outflow is selected
to be automatically adjusted to control pressure in the Specification tab:
In the Reactor→Derived Quantities node, we specify the formula to calculate the average molecular
weight of the polymer:
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Now we can enter the experimental data, which are designed to cover the following variables:
- Temperature effect: {180, 200, 220C}
- Catalyst effect: experiments with and without catalyst Q
- Time effect: {2, 6h}
The Experimental sets are entered in the Experiments node. In Experiments→Measurements, we
select the measured items that are entered later in the Sets child node.
In the Weights node, polymer average molecular weight Mn is selected as the measurement to be
reconciled. Pressure must be weighted as it is controlled. The objective function includes both the
weighted least squares prediction error for Mn as well as a weighted term for pressure deviation from
the setpoint specified in the experimental data.
In Run Estimation→ Solution Options→ Bounds node we open the bounds for the Distillate outlet
flow, so that it can be calculated during the run to control pressure to the setpoint:
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In the Run→Solution Options node we select Kinetics Parameters as Estimate and finally we run the
project.
Model Results and Analysis
The Estimation→Results tree shows all the estimation results. The Results→Parameter node
contains the optimal kinetics parameters values.
In Model Data Comparison node, the measured and calculated profiles for key variables can be seen.
For example the chart for molecular weight along reaction time is shown below:
At low residence time, comparing the trends for sets with and without catalyst Q, we see that catalyst
has only a small effect on the molecular weight growth:
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The small impact of catalyst on the polymer formation can also be deduced from the moles traffic for
net reactions, in Reaction Traffic node. The width of arrows are proportional to the moles reacting:
Further Studies
In the Polymer Optimization example, we continue analyzing this system and turn our focus to
optimizing the alcohol-acid ratio and pressure profiles for getting the desired molecular weight.
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