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PCGA: Polynomial collocation genetic algorithm for singular Poisson-Boltzmann equation arising in thermal explosions.

Noman Yousaf1, Rubina Nasir1, Saima Rafique1

  • 1Department of Physics, AIR University, PAF Complex, E-9, Islamabad, 44000, Pakistan.

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|April 24, 2023
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Summary

This study introduces a Polynomial Collocation Genetic Algorithm (PCGA) to solve the Frank-Kamenetzkii (FK) equation for thermal explosion analysis. The PCGA accurately predicts critical FK values for various vessel shapes, improving thermal explosion modeling.

Keywords:
Frank-kamenetzkii parameterGenetic algorithmNon-linear thermal sourcesPolynomial collocationSingular modelThermal explosion

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Area of Science:

  • Chemical Engineering
  • Reaction Kinetics
  • Mathematical Modeling

Background:

  • Thermal explosions are initiated by exothermic reactions where heat generation outpaces dissipation.
  • The Frank-Kamenetzkii (FK) parameter governs the balance between heat generation and conduction, determining explosion criticality.
  • Solving the singular nonlinear Poisson-Boltzmann equation is crucial for determining FK critical values, but analytical solutions are limited, especially for non-integer shape factors.

Purpose of the Study:

  • To develop and implement a Polynomial Collocation Genetic Algorithm (PCGA) for solving the Poisson-Boltzmann equation.
  • To accurately determine the critical Frank-Kamenetzkii (FK) parameter for various vessel geometries, including non-integer shape factors.
  • To validate the PCGA method against analytical solutions and assess its accuracy, stability, and convergence.

Main Methods:

  • The governing singular nonlinear Poisson-Boltzmann equation was transformed into coupled nonlinear algebraic equations.
  • A genetic algorithm (GA) was employed to exploit global optimization for solving these equations.
  • Polynomial collocation was utilized within the GA framework, creating the Polynomial Collocation Genetic Algorithm (PCGA).

Main Results:

  • The PCGA successfully obtained temperature distributions and critical FK values for cylindrical, parallelepiped, and arbitrary vessel shapes.
  • Solutions demonstrated good agreement with analytical results for integer shape factors, with minimal absolute errors.
  • The method exhibited stable performance across multiple runs, with high accuracy confirmed by statistical error indices.

Conclusions:

  • The PCGA is a reliable and accurate numerical method for solving the FK equation in thermal explosion analysis.
  • This approach overcomes the limitations of analytical solutions, particularly for complex geometries and non-integer shape factors.
  • The PCGA provides a stable and precise tool for predicting critical conditions in exothermic reaction systems.