Sample Matlab Code

Description of code

This sample file computes the normalized global moments (zeroth , first , second ) of a reactive solute in Poiseuille flow between plates with surface adsorption. Three different types of surface adsorption model are implemented:

• linear kinetic model: both adsorption and desorption occur simultaneously and described using a kinetic rate;
• linear equilibrium model: both adsorption and desorption occur simultaneously and described using a equilibrium constant;
• adsorption-only model (Lungu-Moffatt model): kinetic adsorption without desorption and described only by adsorption rate.

The global moments give the fraction of solute in the fluid (), the centre of mass () and the variance of the solute distribution in the fluid (). The corresponding dimensionless transport velocity and longitudinal dispersion coefficient of the solute can be calculated by

For more details, please refer to Zhang L., Hesse M. and Wang M. Journal of Fluid Mechanics, 2017, 828: 733-752.

Note

Some of the functions require expressions from the mupad file (*.mn). Specifically,

• For method = 'num', moment_knt.mn should be open and evaluated.
• For method = 'series' and num_k >=2, Res_limit.mn should be open and evaluated.
• For method = 'num', moment_eq.mn should be open and evaluated.
• For method = 'series' and num_k >=2, Res_limit_eq.mn should be open and evaluated.

File Structure

Moment_Analysis.m: main file, calculating and plotting the global moments using different methods and models.

./functions : folder for all the basic functions used in evaluating the moments

• moment_eq.m: returns normalized global moments for equilibrium adsorption model using numerical inverse Laplace transform (method = num), or truncated series solution derived by residue theorem (method = series).
• moment_knt.m: returns normalized global moments for kinetic adsorption model using numerical inverse Laplace transform (method = num), or truncated series solution derived by residue theorem (method = series).
• moment_LM.m: returns normalized global moments for adsorption-only model using analytical solution from Lungu, E. M. & Moffatt, H. K. Journal of Engineering Mathematics, 1982, 16 (2): 121–136..
• root_tr_eq.m: returns the roots of the transcendental equation for series solution of the equilibrium adsorption model
• root_tr_knt.m: returns the roots of the transcendental equation for series solution of kinetic adsorption model
• talbot_inversion.m: returns an approximation to the inverse Laplace transform of function using Talbot's method. From Mathworks file exchange.

./mupad : folder for mupad files. These files derive analytical solutions using symbolic math, and can be used to evaluate some functions.

• moment_eq.mn: derive inverse Laplace form of moments for equilibrium adsorption model
• moment_knt.mn: derive inverse Laplace form of moments for kinetic adsorption model
• Res_limit_eq.mn: derive truncated series solution of moments using residue theorem for equilibrium adsorption model
• Res_limit_knt.mn: derive truncated series solution of moments using residue theorem for kinetic adsorption model

./figure : folder for saved figures.

Output

Comparison of global moments using different methods and models.

Reference

• Zhang L., Hesse M. and Wang M. Journal of Fluid Mechanics, 2017, 828: 733-752

• Lungu, E. M. and Moffatt, H. K. Journal of Engineering Mathematics, 1982, 16 (2): 121–136.

• Abate, J., and Whitt W. INFORMS Journal of Computing, 2006, 18(4) : 408-421.