Startseite Prediction of Reactivity Ratios in Free Radical Copolymerization from Monomer Resonance–Polarity (Q–e) Parameters: Genetic Programming-Based Models
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Prediction of Reactivity Ratios in Free Radical Copolymerization from Monomer Resonance–Polarity (Q–e) Parameters: Genetic Programming-Based Models

  • K. Shrinivas , Rahul P. Kulkarni , Saif Shaikh , Ravindra V. Ghorpade , Renu Vyas , Sanjeev S. Tambe EMAIL logo , S. Ponrathnam und Bhaskar D. Kulkarni
Veröffentlicht/Copyright: 3. April 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

The principal deficiency of the widely utilized Alfrey–Price (AP) scheme for computing reactivity ratios in the widely used free radical copolymerization is that it ignores important factors, such as the steric effects. This often leads to inaccurate reactivity ratio predictions by AP model. Accordingly, in this study, exclusively data-driven, Q–e parameter-based new models have been developed for the reactivity ratio prediction in free radical copolymerization. In the model development, a novel artificial intelligence formalism known as “genetic programming (GP)” that performs symbolic regression has been employed. The GP-based models possess a different functional form than AP model. Further, parameters of GP-based models were fine-tuned using Levenberg–Marquardt (LM) nonlinear regression method. A comparison of AP, GP and GP-LM as well as artificial neural network (ANN)-based models indicates that GP and GP-LM models exhibit superior reactivity ratio prediction accuracy and generalization performance (with correlation coefficient magnitudes close to or greater than 0.9) when compared with AP and ANN models. The GP-based reactivity ratio prediction models developed here due to their higher accuracy and generalization capability have the potential of replacing the widely used AP models.

Keywords: reactivity ratio; free radical copolymerization; Alfrey–Price scheme; genetic programming; symbolic regression

Acknowledgment

RV thanks the Department of Science and Technology (DST), New Delhi, for the award of “Women Scientist Fellowship.”

Appendix A

The GP procedure begins by creating a random population of symbolically coded candidate solutions (expressions) to a given data-fitting problem; each candidate solution is described in the form of a “tree” structure representing a different mathematical data-fitting expression. A tree typically consists of multiple symbolically coded building blocks (branches) comprising function and terminal nodes. These blocks when combined together form a complete mathematical expression. Within a tree each node may be connected to “sub-branch” nodes. However, connections to nodes other than in the adjoining layer are forbidden. In mathematical terms, a tree represents a directed graph originating from a single “root” vertex or node where each node is connected to a number of sub-branch nodes. An illustrative tree structure defining an expression is shown in Figure 3 (panel a) wherein two branches emanate from the root node. The function node shown in the figure defines a mathematical operator; a set of possible operators is given below:

  1. Arithmetic operators: addition, subtraction, multiplication, division

  2. Trigonometric and other mathematical operators: sine, cosine, tan, cot, square root, logarithm, exponentiation, etc.

  3. Conditional operators: IF-THEN-ELSE

  4. Boolean operators: AND, OR, etc.

Figure 3: Illustration of a tree structure and selection, crossover and mutation operations in a generic genetic programming implementation. (a) Schematic of a tree structure encoding an expression “(v-x) * [sqrt (v ÷ z)]”; (b) selection of tree branches for crossover; (c) crossover operation, and (d) mutation operation.
Figure 3:

Illustration of a tree structure and selection, crossover and mutation operations in a generic genetic programming implementation. (a) Schematic of a tree structure encoding an expression “(v-x) * [sqrt (v ÷ z)]”; (b) selection of tree branches for crossover; (c) crossover operation, and (d) mutation operation.

The operator set used in a GP-implementation typically depends upon the nature of the data-driven modeling problem being addressed. For instance, in a trivial data-fitting problem, four basic arithmetic operators (+, –, ×, ÷) may be adequate, whereas to model data from, say, an exothermic chemical reactor, an exponentiation operator may be additionally needed to account for the temperature effects.

A terminal node (“leaf” of a branch) defines an “operand” (entity on which an operator acts) the examples of which are variables, constants (elements of the parameter vector, α), and zero-arity functions, i.e. functions with no arguments, such as rand (random number). Accordingly, the candidate solution defined in the tree structure of Figure 3 (panel a) is “(v-x) * [sqrt (v ÷ z)].”

GP implementation

To solve the symbolic regression problem defined in eq. (4), the GP implementation uses four steps:

  1. Creation of initial population: Randomly generate an initial population of candidate solutions to the symbolic regression problem (eq. (4)) wherein each solution is coded using a tree structure.

  2. Fitness evaluation: Compute the fitness score of each candidate solution in the current population using the available example input-output data. Fitness score is an indicator of how well a given candidate solution fits the example input-output data. It is used in selecting highly ‘fit’ candidates for ‘mating’ to produce offspring candidate solutions. In essence, fitness score helps in eliminating ‘inferior’ solutions and allows the best candidate solutions in the current population to pass on their genetic material across generations. There exists a number of ways to determine the fitness score of a candidate solution depending upon the specific problem being solved [3]. For instance, the fitness of a candidate solution can be evaluated using a sum-squared-error (SSE) dependent fitness function of the following form:

    (10)Rj2=11+Δj2;j=1,2,…,Np

    where Δj2 refers to the SSE, computed as

    (11)Δj2=∑i=1Ntrnyi−yi,jmdl2

    where yi refers to the desired (target) value of the function output from the training dataset, i denotes the index of the training data; yi,jmdl represents the output predicted by the jth candidate solution when ith training input vector is used for prediction.

  3. Generate a new population of candidate solutions by:

    • Selection: Here, individual candidate solution trees are chosen from the current population to form a mating pool of parent solutions for undergoing crossover (recombination) operation. The basic idea of selection operation is that only fitter candidate solutions (assessed on the basis of high fitness scores) enter the mating pool. There exist several methods such as Roulette-wheel selection, tournament selection, elitist mating [47], etc. for conducting selection of parents.

    • Crossover: Randomly form pairs of parent trees from the mating pool to undergo crossover, and create a pair of new offspring solutions from each parent pair by exchanging randomly chosen parts of the parent trees. A number of crossover strategies exists (see e.g. Banzhaf and Langdon [48], however their principal aim is to introduce variation in the population by breeding offspring candidate solutions inheriting parts of each parent. An illustration of one of the strategies, namely, two parent–two offspring crossover is shown in Figure 3 wherein the branches of the parent trees I and II chosen to undergo crossover are shown in panel b while the pair of offspring formed following the crossover is portrayed in panel c.

    • Mutation: Modify (mutate) contents of randomly chosen function and/or terminal node(s) of offspring candidate solutions with a small probability. Like crossover, mutation can be carried out in various ways such as “branch” mutation and “node” mutation. For instance, in the node mutation [Figure 3 (panel d)] an operator (operand) of a randomly chosen function (terminal) node is replaced by another operator (operand), whereas in the branch mutation a randomly chosen branch is replaced by a randomly generated another branch. The population of candidate solutions resulting upon mutated offspring forms a new generation of candidate solutions.

Repeat steps 2 and 3 over a number of generations till convergence is achieved. The convergence criterion could be that the fitness of the best solution shows no improvement over a large number of successive generations, or GP has evolved over a pre-specified number of generations.

It may be noted that GP-based models while are good at interpolation, these are not so good at extrapolation. This weakness, which is shared by all nonlinear data-driven models, can be overcome by gathering more data in regions where extrapolation is desired.

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Published Online: 2015-4-3
Published in Print: 2016-2-1

©2016 by De Gruyter

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
https://doi.org/10.1515/ijcre-2014-0039
Schlagwörter für diesen Artikel
reactivity ratio; free radical copolymerization; Alfrey–Price scheme; genetic programming; symbolic regression

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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