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Mathematical modelling of flow and adsorption in a gas chromatograph

A. Cabrera-Codony, A. Valverde, K. Born, O. A. I. Noreldin, T. G. Myers

TL;DR

The work develops a mathematically rigorous model for gas chromatography that couples advection–diffusion in the mobile phase with adsorption/desorption kinetics. It introduces two flow regimes (constant and variable velocity) and shows that, under negligible competition between components, the multi-component system decouples into identical single-component equations, enabling a single integral solution for all components. A Laplace-transform approach yields analytic or semi-analytic expressions, validated against full numerical PDE solutions and BTEX experimental data, with two parameters per analyte fitted to data. The resulting method is significantly faster and simpler than solving all curves simultaneously, offering a practical tool for GC analysis and optimization, while highlighting avenues for future extensions to non-isothermal conditions and broader adsorbent/adsorbate systems.

Abstract

In this paper, a mathematical model is developed to describe the evolution of the concentration of compounds through a gas chromatography column. The model couples mass balances and kinetic equations for all components. Both single and multiple-component cases are considered with constant or variable velocity. Non-dimensionalisation indicates the small effect of diffusion. The system where diffusion is neglected is analysed using Laplace transforms. In the multiple-component case, it is demonstrated that the competition between the compounds is negligible and the equations may be decoupled. This reduces the problem to solving a single integral equation to determine the concentration profile for all components (since they are scaled versions of each other). For a given analyte, we then only two parameters need to be fitted to the data. To verify this approach, the full governing equations are also solved numerically using the finite difference method and a global adaptive quadrature method to integrate the Laplace transformation. Comparison with the Laplace solution verifies the high degree of accuracy of the simpler Laplace form. The Laplace solution is then verified against experimental data from BTEX chromatography. This novel method, which involves solving a single equation and fitting parameters in pairs for individual components, is highly efficient. It is significantly faster and simpler than the full numerical solution and avoids the computationally expensive methods that would normally be used to fit all curves at the same time.

Mathematical modelling of flow and adsorption in a gas chromatograph

TL;DR

The work develops a mathematically rigorous model for gas chromatography that couples advection–diffusion in the mobile phase with adsorption/desorption kinetics. It introduces two flow regimes (constant and variable velocity) and shows that, under negligible competition between components, the multi-component system decouples into identical single-component equations, enabling a single integral solution for all components. A Laplace-transform approach yields analytic or semi-analytic expressions, validated against full numerical PDE solutions and BTEX experimental data, with two parameters per analyte fitted to data. The resulting method is significantly faster and simpler than solving all curves simultaneously, offering a practical tool for GC analysis and optimization, while highlighting avenues for future extensions to non-isothermal conditions and broader adsorbent/adsorbate systems.

Abstract

In this paper, a mathematical model is developed to describe the evolution of the concentration of compounds through a gas chromatography column. The model couples mass balances and kinetic equations for all components. Both single and multiple-component cases are considered with constant or variable velocity. Non-dimensionalisation indicates the small effect of diffusion. The system where diffusion is neglected is analysed using Laplace transforms. In the multiple-component case, it is demonstrated that the competition between the compounds is negligible and the equations may be decoupled. This reduces the problem to solving a single integral equation to determine the concentration profile for all components (since they are scaled versions of each other). For a given analyte, we then only two parameters need to be fitted to the data. To verify this approach, the full governing equations are also solved numerically using the finite difference method and a global adaptive quadrature method to integrate the Laplace transformation. Comparison with the Laplace solution verifies the high degree of accuracy of the simpler Laplace form. The Laplace solution is then verified against experimental data from BTEX chromatography. This novel method, which involves solving a single equation and fitting parameters in pairs for individual components, is highly efficient. It is significantly faster and simpler than the full numerical solution and avoids the computationally expensive methods that would normally be used to fit all curves at the same time.
Paper Structure (18 sections, 77 equations, 4 figures, 3 tables)

This paper contains 18 sections, 77 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Constant velocity model
  4. Variable velocity model
  5. Solution methods
  6. Laplace transform solution
  7. Solution for the constant velocity model
  8. Solution for the variable velocity model
  9. Numerical solution
  10. Results
  11. Verification
  12. Comparison with experiment
  13. Conclusions
  14. Cross-sectional averaging for constant velocity model
  15. Derivation of variable velocity model
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Simulation using Laplace solution \ref{['Laplacesolu2']} for the concentration of each compound throughout the column at different times. From left to right and top to bottom: dimensionless times $\hat{t}=70$, $1500$, $4000$ and $6500$ (corresponding to dimensional $t=5$, 108, 290 and 470 s). Parameter values are provided in Tables \ref{['tab:Cuevasval']} and \ref{['tab:Cuevasparam']}. The outlet is located at $\hat{L}=10683$.
  • Figure 2: Comparison of the chromatograms obtained using different simulations. Top: simulation using Laplace solution \ref{['Laplacesolu2']} (solid line) against numerical solution of the full PDE system in \ref{['umultireduxDIM']} to \ref{['qmultireduxDIM']} (circles). Bottom: simulation using the Laplace solution of the variable velocity model \ref{['Laplacesolu2']} (solid line) and the constant velocity model \ref{['Laplaceconstu']} (dashed line).
  • Figure 3: Evolution of velocity (left) and pressure (right) throughout the column. Results obtained using the values in Tables \ref{['tab:Cuevasval']} and \ref{['tab:Cuevasparam']}.
  • Figure 4: Fitting of the variable velocity model (solid line) to the experimental chromatograph (striped line) reported by Nasreddine et al. Nasreddine2016. Results obtained using the values in Tables \ref{['tab:Cuevasval']} and \ref{['tab:Cuevasparam']}.