# On-Line Dynamic Data Reconciliation in Batch Suspension Polymerizations of Methyl Methacrylate

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

## 3. Simulations

#### 3.1. Sensitivity Analysis of the Mathematical Model

^{5}g/gmol for the final polymer resin, which can be easily tuned through the proper manipulation of the chain transfer constant to monomer or impurities, as the predicted weight-average molecular weight of the final polymer resin in absence of chain transfer was equal to 7.1 × 105 g/gmol.

_{w}values, given the higher reaction temperatures. However, the onset of the gel effect and the length and magnitude of the heat kick exert a dramatic effect on the final polymer properties, leading to more complex nonlinear responses. Similar words can be used to describe the trajectories of polydispersity indexes, which are initially close to 2 (as expected for free radical polymerizations controlled by disproportionation) but increase significantly during the heat kick phase. This is very interesting and indicates that it can be difficult to build simple correlations between the heat transfer coefficient and the final polymer properties. Apparently, this effect has never been described before, perhaps because most simulation and experimental studies assume that it is possible to keep the reaction temperature constant in these reaction processes.

#### 3.2. Softsensor Formulation

- Model and operation constraints:$$\mathrm{h}(\mathbf{x},\mathbf{u},\mathbf{p})=0$$$$\mathrm{g}(\mathbf{x},\mathbf{u},\mathbf{p})\le 0$$
- Search space:$${\mathbf{x}}^{\mathrm{i}}\le \mathbf{x}\le {\mathbf{x}}^{\mathrm{s}}$$$${\mathbf{p}}^{\mathrm{i}}\le \mathbf{p}\le {\mathbf{p}}^{\mathrm{s}}$$$${\mathbf{u}}^{\mathrm{i}}\le \mathbf{u}\le {\mathbf{u}}^{\mathrm{s}}$$

#### 3.3. Process

^{6}g/gmmol, and equipped with three Shodex columns and a refractometer detector (Viscotek VE 3580, Massachusetts, MA, USA). The analyses were conducted at 40 °C.

#### 3.4. Adjustment of the Reactor Pressure

#### 3.5. Process Monitoring

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Symbols, Acronyms and Abbreviations

A_{i}, B_{i}, C_{i} | Antoine constants |

Cp_{A} | Heat capacity of water [cal/g·K] |

Cp_{C} | Heat capacity of the cooling fluid [cal/g·K] |

Cp_{i} | Heat capacity of reactant i in the reacting medium [cal/g·K] |

Cp_{m} | Heat capacity of monomer [cal/g·K] |

Cp_{P} | Heat capacity of polymer [cal/g·K] |

D_{i} | Dead polymer chains of length i [mol] |

F_{c} | Volumetric flowrate of cooling fluid [g/s] |

f | Initiator efficiency |

g_{p} | Glass effect contribution |

g_{t} | Gel effect contribution |

−ΔH | Heat of reaction [kJ/mol] |

I(0) | Initial initiator mass [g] |

I | Initiator [mol] |

Inib | Inhibitor [mol] |

IP | Polydispersity |

k_{i} | Initiation rate constant [cm^{3}/mol·s] |

kd | Initiator decomposition rate constant [s^{−1}] |

kp | Propagation rate constant [cm^{3}/mol·s] |

ktd | Termination by disproportionation rate constant [cm^{3}/mol·s] |

ktm | Transfer to monomer rate constant [cm^{3}/mol·s] |

M(0) | Initial monomer mass [g] |

M | Monomer [mol] |

MMA | Molecular weight of water [g/gmol] |

MA(0) | Initial water mass [g] |

MMm | Molecular weight of monomer [g/gmol] |

MMI | Molecular weight of initiator [g/gmol] |

MMinert | Molecular weight of inert [g/gmol] |

Mn | Number-average molecular weight [g/gmol] |

M_{PVA}(0) | Initial inhibitor mass [g] |

Mw | Weight-average molecular weight [g/gmol] |

n_{inert0} | Inert at the beginning of reaction [mol] |

n_{inert} | Inert at the end of reaction [mol] |

P_{A}^{sat} | Water saturation pressure [kPa] |

P_{mma}^{sat} | Monomer saturation pressure [kPa] |

P(0) | Atmospheric pressure [kPa] |

P_{i} | Living polymer chains of length i [mol] |

P_{inert} | Inert partial pressure [kPa] |

P_{isat} | Saturation pressure of component i [kPa] |

Pp_{i} | Partial pressure of reactant i in the reacting medium [kPa] |

QP | Rate of energy exchanged with the environment [cal/s] |

QR | Rate of energy released by the polymerization reaction [cal/s] |

QT | Rate of energy exchanged with jacket [cal/s] |

R | Universal gas constant [J/mol·K] |

R^{⦁} | Radicals [mol] |

R_{p} | Propagation rate [mol/s] |

T | Reactor temperature [K] |

T_{A} | Ambient temperature [K] |

T_{C} | Jacket temperature [K] |

Te_{C} | Input cooling fluid temperature [K] |

Tg_{m} | Glass transition temperature of monomer [K] |

Tg_{p} | Glass transition temperature of polymer [K] |

t | Time [s] |

UA | Heat transfer coefficient with reactor [cal/K·s] |

UA_{A} | Heat transfer coefficient with ambient [cal/K·s] |

V_{A} | Volume of water [L] |

V_{L} | Total volume of liquid phase [L] |

V_{R} | Total reactor volume [L] |

V_{C} | Jacket volume [L] |

V_{f} | Total free volume of organic phase [mL] |

V_{f}m | Contribution of monomer to free volume [mL] |

V_{f}p | Contribution of polymer to free volume [mL] |

V_{f}pc | Critical free volume for propagation [mL] |

V_{f}tc | Critical free volume for termination [mL] |

Vm | Total volume of monomer [mL] |

Vo | Total volume of organic phase [mL] |

Vp | Total volume of polymer [mL] |

X | Monomer conversion [%] |

α_{m} | Thermal expansion coefficient for monomer |

α_{p} | Thermal expansion coefficient for polymer |

χ | Flory–Huggins interaction parameter |

λ_{K} | k^{th} moment of the chain-length distributions of living polymer chains |

μ_{K} | k^{th} moment of the chain-length distributions of dead polymer chains |

φ^{i}_{v} | Volumetric fraction of component i |

φ^{p}_{v} | Volumetric fraction of polymer |

ρ_{A} | Water density [g/cm^{3}] |

ρ_{p} | Polymer density [g/cm^{3}] |

ρ_{c} | Cooling fluid density [g/cm^{3}] |

ρ_{i} | Density of reactant i [g/cm^{3}] |

ρ_{m} | Density of monomer [g/cm^{3}] |

## Appendix A

#### Mathematical Model

## References

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**Figure 1.**Monomer conversion, as provided by the model and reported by Santos [23].

**Figure 2.**(

**a**) Temperature; (

**b**) pressure; (

**c**) weight-average molar mass (Mw); and (

**d**) polydispersity index (IP) profiles for different values of the overall heat transfer coefficient.

**Figure 3.**(

**a**) Pressure; (

**b**) Mw; (

**c**) IP; and (

**d**) conversion profiles for different initial loads of initiator.

**Figure 4.**(

**a**) Reconciled pressure profiles for reaction R1, considering (

**a**) the traditional equilibrium approach and (

**b**) the existence of fresh monomer droplets.

**Figure 5.**Reconciled reactor temperature profiles for reaction R2 considering the traditional equilibrium approach.

**Figure 6.**Reconciled (

**a**) reactor temperature; (

**b**) reactor pressure; (

**c**) jacket temperature; (

**d**) IP; (

**e**) Mw; (

**f**) conversion profiles for reaction R3, considering the formation of fresh monomer droplets.

**Figure 7.**Reconciled UA profiles for reaction R3, considering the formation of fresh monomer droplets.

**Figure 8.**Reconciled UA profiles for reaction R1, considering the formation of fresh monomer droplets.

**Figure 9.**Reconciled UA profiles for reaction R2, considering the formation of fresh monomer droplets.

Parameters | References |
---|---|

kd = 1.7 × 10^{14}∙exp(−30000/RT) s^{−1} | [19] |

kp = 7 × 10^{9}g_{p}∙exp(−6300/RT) cm^{3}/mol·s | [12] |

ktd = 1.76 × 101^{2} g_{t}∙exp(−2300/RT) cm^{3}/mol·s | [12] |

ktm = kp∙exp(−2.6 − 2888/T) cm^{3}/mol·s | [20] |

f = 0.6 | [12] |

${\mathsf{\rho}}_{\mathrm{P}}=\mathsf{\rho}\mathrm{mma}(0.754-9\cdot {10}^{-4}({\mathrm{T}-343.15))}^{-1}\mathrm{g}/\mathrm{mL}$ | [12] |

${\mathsf{\rho}}_{\mathrm{m}}=0.9654-0.00109\text{}(\mathrm{T}-273.15)-9.7\times {10}^{-7}{(\mathrm{T}-273.15)}^{2}\text{}\mathrm{g}/\mathrm{mL}$ | [12] |

${\mathsf{\rho}}_{\mathrm{A}}{=1\text{}\mathrm{g}/\mathrm{cm}}^{3}$ | [21] |

MMm = 100.12 g/gmol | [21] |

MMinert = 28.0 g/gmol | [22] |

MMA = 18.0 g/gmol | [21] |

MMI = 242.3 g/gmol | [21] |

Cp_{m} = 0.49 cal/g∙KCp_{A} = 1.0 cal/g∙K | [12,21] |

${\mathrm{Cp}}_{\mathrm{P}}=0.339+9.55\times {10}^{-4}(\mathrm{T}-298.15)\text{}\mathrm{cal}/\mathrm{g}\xb7\mathrm{K}$ ${\mathrm{P}}_{\mathrm{mma}}^{\mathrm{sat}}=\mathrm{exp}(19.8567-\frac{5441.04}{\mathrm{T}+37.32})\text{}\mathrm{mmHg}$ | [12] |

[12] | |

${\mathrm{P}}_{\mathrm{A}}^{\mathrm{sat}}=\mathrm{exp}(16.3872-\frac{3885.7}{\mathrm{T}-230.170})\text{}\mathrm{mmHg}$ | [22] |

∆H = 57.7 kJ/mol | [12] |

α_{m} = 0.001 | [12] |

Tg_{m} = 167 K | [12] |

α_{p} = 0.00048 | [12] |

Tg_{p} = 387 K | [12] |

χ = 0.5 | [12] |

VC = 10 L | Location data |

VL = 15 L | Location data |

F_{c} (g/s) | Location data |

UA_{A} (cal/K·s) | Location data |

Operational Condition | Value |
---|---|

T(0) (K) | 343.15 |

T_{c}(0) (K) | 358.15 |

Te_{c} (K) | 373.15 |

F_{c} (g/s) | 200.0 |

UA (cal/K·s) | 2 |

UA_{A} (cal/K·s) | 1.75 × 10^{−1} |

M(0) (g) | 1500 |

I(0) (g) | 15 |

M_{A}(0) (g) | 4500 |

M_{PVA}(0) (g) | 45 |

P(0) (atm) | 1 |

Npt | Niter | C_{1} | C_{2} | w | TOL |
---|---|---|---|---|---|

50 | 200 | 1 | 1 | 0.7 | 10^{−3} |

Reaction | Stirring Rate in the First 10 Min (rpm) | Stirring Rateafter 10 Min (rpm) |
---|---|---|

R1 | 800 | 600 |

R2 | 800 | 1000 |

R3 | 800 | 800 |

Initial Condition | R1 | R2 | R3 |
---|---|---|---|

T (K) | 335–350 | 335–360 | 335–350 |

T_{c} (K) | 300–360 | 320–360 | 298–330 |

UA (cal/s·K) | 10^{−1}–10^{3} | 10^{−1}–10^{3} | 10^{−1}–10^{3} |

Te_{c} (K) | 356.85 | 360.75 | 373.05 |

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**MDPI and ACS Style**

Coimbra, J.C.; Melo, P.A.; Prata, D.M.; Pinto, J.C.
On-Line Dynamic Data Reconciliation in Batch Suspension Polymerizations of Methyl Methacrylate. *Processes* **2017**, *5*, 51.
https://doi.org/10.3390/pr5030051

**AMA Style**

Coimbra JC, Melo PA, Prata DM, Pinto JC.
On-Line Dynamic Data Reconciliation in Batch Suspension Polymerizations of Methyl Methacrylate. *Processes*. 2017; 5(3):51.
https://doi.org/10.3390/pr5030051

**Chicago/Turabian Style**

Coimbra, Jamille C., Príamo A. Melo, Diego M. Prata, and José Carlos Pinto.
2017. "On-Line Dynamic Data Reconciliation in Batch Suspension Polymerizations of Methyl Methacrylate" *Processes* 5, no. 3: 51.
https://doi.org/10.3390/pr5030051