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Complete analytic solutions for convection-diffusion-reaction-source equations without using an inverse Laplace

Albert S Kim1

  • 1Civil and Environmental Engineering, University of Hawaii at Manoa, 2540 Dole Street Holmes 383, Honolulu, Hawai'i, 96822, USA. albertsk@hawaii.edu.

Scientific Reports
|May 17, 2020
PubMed
Summary

Researchers developed a general 1D analytic solution for the convection-diffusion-reaction-source (CDRS) equation, crucial for modeling transient mass-transfer phenomena. This breakthrough addresses the analytical intractability of complex systems with adsorption and desorption processes.

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Area of Science:

  • Environmental science and engineering
  • Chemical engineering
  • Applied mathematics

Background:

  • Transient mass-transfer phenomena involve convection, diffusion, and reaction.
  • The unsteady convection-diffusion-reaction (CDR) equation models these coupled processes.
  • Existing analytical solutions for CDR are limited, and the CDRS equation with sources/sinks is generally intractable.

Purpose of the Study:

  • To derive a general 1D analytic solution for the unsteady convection-diffusion-reaction-source (CDRS) equation.
  • To establish a general formalism for deriving solutions under specific boundary conditions.
  • To address the analytical intractability of mass-transfer phenomena with adsorption/desorption.

Main Methods:

  • Application of a one-sided Laplace transform technique.
  • Assumption of constant diffusivity, velocity, and reactivity.
  • Development of a general formalism for Dirichlet/Dirichlet and Dirichlet/Neumann boundary conditions.

Main Results:

  • A general 1D analytic solution for the CDRS equation was obtained.
  • The derivation was facilitated by specific conditions on the source function and initial concentration.
  • A general formalism for deriving solutions under common boundary conditions was provided.

Conclusions:

  • The study successfully provides a general analytic solution for the previously intractable CDRS equation.
  • The methodology offers a pathway to analytically solve complex mass-transfer problems in various natural and engineered systems.
  • This work advances the mathematical modeling of transient phenomena involving adsorption and desorption.