An Evaluation of Solution Algorithms and Numerical Approximation Methods for Modeling an Ion Exchange Process.

Area of Science:

• Environmental Engineering
• Chemical Engineering
• Computational Modeling

Background:

• Ion exchange is crucial for drinking water treatment.
• Previous models used computationally expensive Monte Carlo simulations.
• Modeling ion exchange in particles of varying size and age presents challenges.

Purpose of the Study:

• To develop a more computationally efficient model for ion exchange processes in drinking water treatment.
• To introduce and validate a new age-averaged model (AAM).
• To compare the efficiency of different numerical schemes for a two-scale ion exchange model.

Main Methods:

• Formulated a two-scale model coupling microscale diffusion equations for resin particles with a macroscale ODE for reactor concentration.
• Introduced an age-averaged model (AAM) to simplify calculations.
• Implemented and compared two numerical schemes: Finite Element Method (FEM) with ODE solver and Krylov deferred correction (KDC) with Fast Elliptic Solver (FES).

Main Results:

• The AAM significantly reduces computational cost compared to Monte Carlo methods.
• The KDC-FES scheme demonstrates higher computational efficiency than the FEM approach, especially for increased accuracy requirements.
• Numerical results validate the accuracy and efficiency of the AAM.

Conclusions:

• The age-averaged model (AAM) provides a more efficient approach to modeling ion exchange processes.
• Higher-order numerical schemes like KDC-FES offer significant advantages in computational efficiency for this application.
• Further algorithmic improvements can enhance the efficiency of these methods for broader applications in water treatment.