Startseite Bulk Polymerization of Styrene using Multifunctional Initiators in a Batch Reactor: A Comprehensive Mathematical Model
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Bulk Polymerization of Styrene using Multifunctional Initiators in a Batch Reactor: A Comprehensive Mathematical Model

  • E. Berkenwald , M. L. Laganá , P. Acuña , G. Morales EMAIL logo und D. Estenoz EMAIL logo
Veröffentlicht/Copyright: 22. Dezember 2015
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International Journal of Chemical Reactor Engineering
Aus der Zeitschrift International Journal of Chemical Reactor Engineering Band 14 Heft 1

Abstract

A detailed, comprehensive mathematical model for bulk polymerization of styrene using multifunctional initiators – both linear and cyclic – in a batch reactor was developed. The model is based on a kinetic mechanism that considers thermal initiation and chemical initiation by sequential decomposition of labile groups, propagation, transfer to monomer, termination by combination and re-initiation reactions due to undecomposed labile groups. The model predicts the evolution of global reaction variables (e.g, concentration of reagents, products, radical species and labile groups) as well as the evolution of the detailed complete polymer molecular weight distributions, with polymer species characterized by chain length and number of undecomposed labile groups. The mathematical model was adjusted and validated using experimental data for various peroxide-type multifunctional initiators: diethyl ketone triperoxide (DEKTP, cyclic trifunctional), pinacolone diperoxide (PDP, cyclic bifunctional) and 1,1-bis(tert-butylperoxy)cyclohexane (L331, linear bifunctional). The model very adequately predicts polymerization rates and complete molecular weight distributions. The model is used to theoretically evaluate the influence of initiator structure and functionality as well as reaction conditions.

Keywords: polystyrene; multifunctional initiators; kinetics; mathematical model

Funding statement: Funding: Consejo Nacional de Investigaciones Científicas y Técnicas (Grant/Award Number: “PICT 2011-1254”).

Appendix A: Basic module

Balances for the non-polymeric reagents and products

Multifunctional Initiators(ϕ=1,2,3)

(1)ddtI(ϕ)V=−ϕkd1I(ϕ)V
(2)ddtIˉ(ϕ)V=−ϕkd1Iˉ(ϕ)V
Secondary Initiator Species(ϕ>i=1,2)
(3)ddtI‾(i)V=−ikd1I‾(i)V+1−f1∑j=i+1ϕjkd1I(j)+I‾(j)V

Monomer

Assuming the “long chain approximation” (by which propagation is the only monomer-consuming reaction):

(4)ddtStV=−RpV=−kpStR⋅+2⋅R⋅V

where Rp is the global St polymerization rate, and

(5)R⋅=∑i=0∞∑n=1∞Sn⋅(i)
(6)⋅R⋅=∑i=0∞∑n=1∞⋅Sn⋅(i)

represent the total concentrations of mono- and diradicals respectively.

Radical species (i=0,1,...n=2,3,...)

Consider the mass balances of all free radical appearing in the global kinetics. Such balances provide:

(7)ddt⋅I⋅(i)V=f1(i+1)kd1I(i+1)V−2ki1St⋅I⋅(i)V
(8)ddtI⋅(i)V=∑j=i+1ϕpj(i)f1jkd1I‾(j)V−ki1StI⋅(i)V

Where pi(j) is the probability that the decomposition of the initiator of functionality j generates a monoradical with i undecomposed peroxide groups.

(9)ddt([⋅S1⋅(i)]V)=2ki1[⋅I⋅(i)][St]V−2(kp[St]+kfM[St]+ktc([R⋅]+2[⋅R⋅]))[⋅S1⋅(i)]V
(10)ddt⋅Sn⋅(i)V=2kpSt⋅Sn−1⋅(i)−⋅Sn⋅(i)V−2kfMSt+ktcR⋅+2⋅R⋅⋅Sn⋅(i)V+2ktc∑j=0i∑m=1n−1⋅Sn−m⋅(i−j)⋅Sm⋅(j)V
(11)ddtS1⋅(i)V={ki1I⋅(i)St+δi0(2ki0St3+kfMStR⋅+2⋅R⋅)}V−kpSt+kfMSt+ktcR⋅+2⋅R⋅S1⋅(i)V

Where δi0 is the Kronecker Delta (δi0= 1 if i = 0 and δi0= 0 otherwise).

(12)ddtSn⋅(i)V=kpSn−1⋅(i)−Sn⋅(i)+2kfM⋅Sn⋅(i)StV−(kfMSt+ktc(R⋅+2⋅R⋅))Sn⋅(i)V+2ktc∑j=0i∑m=1n−1⋅Sn−m⋅(i−j)Sm⋅(j)V+f2kdp∑j=i+1∞∑m=n+1∞pmj(n,i)jSm(j)V

In eq. (12), pmj(n,i) is the probability that a scission of a chain of dead polymer of length m and i peroxide groups yields a growing monoradical of chain length n with i peroxide groups.

Adding this probability over all is and ns, the following can be proved:

(13)∑i=1∞∑n=1∞∑j=i+1∞∑m=n+1∞pmj(n,i)jSm(j)=∑i=1∞∑n=1∞2iSn(i)=2PePS

where PePS is the concentration of peroxide groups in the PS chains. Note that the scission of any PS chain with undecomposed peroxide groups produces 2 monoradicals.

From eqs (9) and (10), the total concentration of diradicals may be obtained:

(14)ddt([⋅R⋅]V)=2ki1∑j=0ϕ−1[⋅I⋅(j)][St]V+2ktc[⋅R⋅]2V−2(kfM[St]+ktc([R⋅]+2[⋅R⋅]))[⋅R⋅]V

From eqs (11) and (12) and considering eq. (13), the total concentration of monoradicals may be obtained:

(15)ddt([R⋅]V)=ki1∑j=0ϕ−1[I⋅(j)][St]V+4kfM[⋅R⋅][St]V+2ki[St]3V                     −ktc([R⋅]+2[⋅R⋅])[R⋅]V+2f2kdp[PePS]V

The total radicals are calculated as R+2⋅R⋅. Using eqs (14) and (15),

(16)ddt(([R⋅]+2[⋅R⋅])V)=ki1∑j=0ϕ−1(4[⋅I⋅(j)]+[I⋅(j)])[St]V+2ki[St]3V                                        + 2f2kdp[PePS]V−ktc([R⋅]+2[⋅R⋅])2V

Peroxide groups

The total concentration of peroxide groups is

(17)Pe=∑j=1ϕjI(j)+I‾(j)+PePS

with

(18)PePS=∑i=0∞∑n=1∞iSn(i)

Peroxide groups are consumed only by decomposition reactions. Therefore, it can be written

(19)ddtPeV=−∑i=1ϕjkd1I(i)+I‾(i)V−kdpPePSV

Using this result and eq. (29), the molar concentration of undecomposed peroxide groups accumulated in the polymer can be calculated from the difference

(20)ddtPePSV=ddtPeV−∑j=1ϕjddtI(j)+I‾(j)V

Conversion and volume

Monomer conversion can be calculated from

(21)x=St0V0−StVSt0V0

where the superscript “0” indicates initial conditions.

The evolution of the reaction volume V is obtained from

(22)V=VSt01−εx

with

(23)ε=VSt0−VSfVSt0

Where VSt0 is the initial St volume, ε is the St volume contraction factor and VSf is the final volume at full conversion.

Equations (1)–(4), (7), (8), and (20) to (22) are solved simultaneously to find the evolution of speciesI(i),I‾(i), St,⋅I⋅(i), I⋅(i),R⋅+2⋅R⋅, PePS, x and V.

Appendix B: Moments module

Distribution moments equations

Define the kth moment of the distribution of diradicals (σk(i)), monoradicals (λk(i)) and polymer (μk(i)) species, characterized by their number of undecomposed peroxide groups i:

(24)σk(i)=∑n=1∞nk⋅Sn(i)⋅
(25)λk(i)=∑n=1∞nkSn(i)⋅
(26)μk(i)=∑n=1∞nkSn(i)

The evolution of the 0th, 1st and 2nd moments of the distributions of diradicals, monoradicals and polymer species are written:

(27)d(σ0(i)V)dt=2ki1[⋅I⋅(i)][St]V+2ktc∑j=0iσ0(i−j)σ0(j)V−2(kfM[St]+ktc∑i=0∞(λ0(i)+2σ0(i)))σ0(i)V
(28)dσ1(i)Vdt=2ki1⋅I⋅(i)StV+2kpStσ0(i)V+2ktc∑j=0iσ0(i−j)σ1(j)+σ1(i−j)σ0(j)V−2kfMSt+ktc∑i=0∞λ0(i)+2σ0(i)σ1(i)V
(29)dσ2(i)Vdt=2ki1⋅I⋅(i)StV+2kpSt2σ1(i)+σ0(i)V+2ktc∑j=0iσ0(i−j)σ2(j)+2σ1(i−j)σ1(j)+σ2(i−j)σ0(j)V−2kfMSt+ktc∑i=0∞λ0(i)+2σ0(i)σ2(i)V
(30)dλ0(i)Vdt=ki1I⋅(i)StV+2kfMStσ0(i)V+2ktc∑j=0iσ0(i−j)λ0(j)V+2kdp∑j=i+1∞μ0(j)V+kfMStδi0∑j=0∞λ0(j)V−kfMSt+ktc∑j=0∞λ0(j)+2σ0(j)λ0(i)V
(31)dλ1(i)Vdt=ki1I⋅(i)StV+kpStλ0(i)V+2kfMStσ1(i)V+2ktc∑j=0iλ0(i−j)σ1(j)+λ1(i−j)σ0(j)V+2kdp∑j=i+1∞μ1(j)V+kfMStδi0∑j=0∞λ0(j)+2σ0(j)V−kfMSt+ktc∑j=0∞λ0(j)+2σ0(j)λ1(i)V
(32)dλ2(i)Vdt=ki1I⋅(i)StV+kpSt2λ1(i)+λ0(i)V+2kfMStσ2(i)V+2ktc∑j=0iλ0(i−j)σ2(j)+2λ1(i−j)σ1(j)+λ2(i−j)σ0(j)V+2kdp∑j=i+1∞μ2(j)V+kfMStδi0∑j=0∞λ0(j)+2σ0(j)V−kfMSt+ktc∑i=0∞λ0(j)+2σ0(j)λ2(i)V
(33)dμ0(i)Vdt=kfMStλ0(i)V+12ktc∑j=0iλ0(i−j)λ0(j)V−ikdpμ0(i)V
(34)d(μ1(i)V)dt=kfM[St]λ1(i)V+12ktc∑j=0i(λ0(i−j)λ1(j)+λ1(i−j)λ0(j))V−ikdpμ1(i)V
(35)d(μ2(i)V)dt=kfM[St]λ2(i)V+12ktc∑j=0i(λ0(i−j)λ2(j)+2λ1(i−j)λ1(j)+λ2(i−j)λ0(j))V−ikdpμ2(i)V

The average molecular weights and polydispersity can then be calculated from

(36)Mˉn=∑i=0∞μ1(i)∑i=0∞μ0(i)
(37)Mˉw=∑i=0∞μ2(i)∑i=0∞μ1(i)
(38)D=MˉwMˉn

Appendix C: Distributions module

Radical species (i=0,1,... n=2,3,...)

Consider eqs (10) and (12). Assuming pseudosteady-state, all time derivatives may be set to zero and the following recurrence formulas can be obtained:

(39)⋅Sn⋅(i)=kpSt⋅Sn−1⋅(i)+ktc∑j=0i∑m=1n−1⋅Sn−m⋅(i−j)⋅Sm⋅(j)kpSt+kfMSt+ktcR⋅+2⋅R⋅
(40)Sn⋅(i)=kpSn−1⋅(i)+2kfM⋅Sn⋅(i)StkpSt+kfMSt+ktcR⋅+2⋅R⋅+2ktc∑j=0i∑m=1n−1⋅Sn−m⋅(i−j)Sm⋅(j)+kdp∑j=i+1∞∑m=n+1∞pmj(n,i)jSm(j)kpSt+kfMSt+ktcR⋅+2⋅R⋅

Polystyrene species (i=0,1,... n=2,3,...)

The mass balances for the PS species provide

(41)ddtSn(i)V=kfMStSn⋅(i)V+ktc2∑j=0i∑m=1n−1Sn−m⋅(i−j)Sm⋅(j)V−ikdpSn(i)V+1−f2kdp∑j=i+1∞∑m=n+1∞pmj(n,i)jSm(j)V

In order to account for the generation of monoradicals from random scission polymer chains by sequential decomposition of peroxide groups within the chains, consider a polymer chain with length n and i peroxide groups, all of which have the same thermal stability.

Let m be a uniformly distributed random variable whose value ranges 1 from 1 to n–1. The polymer chain may form 2 monoradicals, one with length m, and the other one with length n-m. These chains will have i–j and j–1 undecomposed peroxide groups respectively. If the peroxide groups are assumed to be uniformly distributed within the polymer chains in the course of polymerization, the following relation must hold:

(42)j−1n−m=i−jm

Therefore,

(43)j=i(n−m)+mn

where the brackets indicate the integer part of the expression.

The scission has then generated two monoradicals, one with length m and i-j peroxide groups, the other one with length n-m and j–1 peroxide groups.

Note that this chain scission algorithm can be modified for specific cases. For example, in the case of a linear bifunctional initiator, since all peroxide groups are located at a chain end, m=1 for every scission.

The Number Chain Length Distribution (NCLD) for the PS species is

(44)NPS(i)(n)=Sn(i)V

found by integrating eq. (41) with eqs (39) and (40) using also eqs (9) and (11) to obtain expressions for species⋅S1⋅(i) and S1⋅(i).

The concentration of the total PS species characterized by the number of undecomposed peroxide groups can be calculated with

(45)P(i)=∑n=1∞Sn(i)

The NCLD for the total polymer, characterized by chain length, can be calculated using

(46)Pn=∑i=0∞Sn(i)V

The total moles of PS are

(47)NPS=∑i=0∞∑n=1∞NPS(i)(n)

To obtain the corresponding weight Chain Length Distribution (WCLD), multiply the NCLD by sMStand replace n by s to obtain

(48)GPS(i)(s)=sMStSs(i)V

The mass of PS can then be calculated as

(49)GPS=∑i=0∞∑s=1∞GPS(i)(s)

The average molecular weights and polydispersity can then be calculated from

(50)Mˉn=GPSNPS=∑i=0∞∑s=1∞GPS(i)(s)∑i=0∞∑n=1∞Sn(i)V
(51)Mˉw=∑i=0∞∑s=1∞sGPS(i)(s)GPS=∑i=0∞∑s=1∞sGPS(i)(s)∑i=0∞∑s=1∞GPS(i)(s)
(52)D=MˉwMˉn

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Published Online: 2015-12-22
Published in Print: 2016-2-1

©2016 by De Gruyter

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https://doi.org/10.1515/ijcre-2015-0102
Schlagwörter für diesen Artikel
polystyrene; multifunctional initiators; kinetics; mathematical model

Artikel in diesem Heft

  1. Frontmatter
  2. Review
  3. Novel Membrane Reactor Concepts for Hydrogen Production from Hydrocarbons: A Review
  4. Research Articles
  5. Modeling and Simulation Study of an Industrial Radial Moving Bed Reactor for Propane Dehydrogenation Process
  6. Oxidative Degradation of Methylene Blue in Aqueous Medium Catalyzed by Lab Prepared Nickel Hydroxide
  7. CFD Modelling of a Hollow Fibre Membrane for CO2 Removal by Aqueous Amine Solutions of MEA, DEA and MDEA
  8. Performance Study of a Thermally Double Coupled Multi-Tubular Reactor by Considering the Effect of Flow Type Patterns
  9. Radiative Three-Dimensional Flow with Chemical Reaction
  10. Three-Phase Model of a Fluidized-Bed Catalytic Reactor for Polyethylene Synthesis
  11. Study on the Effect of Nickel Doping on Mo-Bi Based Catalyst for Selective Oxidation of Isobutene to Methacrolein
  12. DM Water Plant Sedimentation as a Cheap and Waste Source of Catalyst for Biodiesel Production
  13. Recovery of Biogas from Meat Industry Wastewater Using Continuously Stirred Tank Reactor (CSTR): Modeling and Optimization
  14. Comparison of CFD Simulation and Simplified Modeling of a Fluidized Bed CO2 Capture Reactor
  15. Kinetic Study for Platinum Extraction from Spent Catalyst in Cyanide Solution at High Temperatures
  16. Hydroisomerization of Biomass Derived n-Hexadecane Towards Diesel Pool: Effect of Selective Removal External Surface Sites from Pt/ZSM-22
  17. Heat Flux and Variable Thermal Conductivity Effects on Casson Flow and Heat Transfer due to an Exponentially Stretching Sheet with Viscous Dissipation and Heat Generation
  18. Optimization of Nickel Chemical Extraction from Hazardous Residue
  19. Green Approach for Biodiesel Production from Jojoba Oil Supported by Process Modeling and Simulation
  20. Differentiating Process Performance of Various Coagulants in Removal of Congo red and Orange G Dyes
  21. CFD Simulation of Hydrodynamic of a Bubble Column Reactor Operating in Churn-Turbulent Regime and Effect of Gas Inlet Distribution on System Characteristics
  22. Transformation of Phenolic Compounds by Fe(III) in the Aqueous Solution in Dark and Under Irradiation
  23. Transient Predictive Model for Dynamic Analysis, Kinetic Study, and Reactor Design of Triglycerides Transesterification to Biodiesel
  24. Screening of Alternative Carbon Sources for Recombinant Protein Production in Pichia pastoris
  25. Optimization of a Large Scale Industrial Reactor Towards Tailor Made Polymers Using Genetic Algorithm
  26. Model of Reactive Transport within a Light Photocatalytic Textile
  27. Preparation of Chitosan based Nanofibers: Optimization and Modeling
  28. Adsorption of As (III) on Iron Coated Quartz Sand: Influence of Temperature on the Equilibrium Isotherm, Thermodynamics and Isosteric Heat of Adsorption Analysis
  29. Resistance Characteristics of Coal Slime in Pipe Flow at High Pressure
  30. Activity Comparison of Acidic Resins in the Production of Valuable Glycerol Acetates
  31. Bulk Polymerization of Styrene using Multifunctional Initiators in a Batch Reactor: A Comprehensive Mathematical Model
  32. An Effective Reaction Rate Model for Gas-Solid Reactions with High Intra-Particle Diffusion Resistance
  33. Heart-Like Micro-Flow Mixer
  34. Hydrogen Production by Hydrolysis of NaBH4 with Cr-Ni-W-B Catalyst: Effects of Cold Plasma and Chromium Content
  35. Prediction of Reactivity Ratios in Free Radical Copolymerization from Monomer Resonance–Polarity (Q–e) Parameters: Genetic Programming-Based Models
  36. Steady-State Modeling of Pt-Catalyzed H2 Combustion in a Monolithic Reactor: From Micro- to Macro-kinetics
  37. Premixed Methanol–Air Combustion Characteristics in a Mini-scale Catalytic Combustor
  38. Kinetic Modeling of VOC Photocatalytic Degradation Using a Process at Different Reactor Configurations and Scales
  39. Improvement of Activated Sludge Process Using a Nonlinear PI Controller Design via Adaptive Gain
  40. Modeling and Simulation of a Pilot-Scale Bubbling Fluidized Bed Gasifier for the Gasification of High Ash Indian Coal Using Eulerian Granular Approach
  41. Performance of a Catalytic Gas–Solid Fluidized Bed Reactor in the Presence of Interparticle Forces
  42. Mathematical Modeling of Biodiesel Production under Intense Agitation
  43. Extended Aeration Activated Sludge Reactor (EAASR) for Removal of Nitrobenzene: Air Stripped or Biologically Removed?
  44. CFD Modeling of Algae Flash Pyrolysis in the Batch Fluidized Bed Reactor Including Heat Carrier Particles
  45. A Robust Extended State Observer for the Estimation of Concentration and Kinetics in a CSTR
  46. Dynamic Modeling of CATOFIN® Fixed-Bed Iso-Butane Dehydrogenation Reactor for Operational Optimization
  47. Effect of Biomass Compositions on Combustion Kinetic Parameters using Response Surface Methodology
  48. Short Communications
  49. Discussions on Particle Flux Measurement in Gas-Solids Risers
  50. Use of New Regeneration Processes for Thermolabile Amines
  51. Oxidative-Extractive Desulfurization of Liquid Fuel by Dimethyl Sulfoxide and ZnCl2 Based Ionic Liquid
  52. Corrigendum
  53. Corrigendum to: Performance Analysis of Immobilized Enzyme Semifluidised Bed Bioreactors
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© 2025 De Gruyter Brill
Heruntergeladen am 16.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ijcre-2015-0102/pdf
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