A Membrane-based Process Modeling Simulation Using Emperical Numerical Methods

• General info
• Key incorporated theoretical aspects
• Mathematical methodology
• Installation
• Simulation results
• Use cases

A chemistry tool for modeling solute molecules transport through a membrane. It uses data from a Visual MINTEQ simulation of a solution where equilibrium has transpired.

• Advections
  • Solute addvection
  • Percipitate addvection
• Convection
• Percipitate suspension
• Indirect equilibrium Incorporation (From MINTEQ)
• pH specification (From MINTEQ)
• Osmotic pressure
• Solute Accumulation
• Fouling
• Conservation of mass investigation

The concept is an emperical numerical method based on the generalized advection-diffusion equation.

The movement equation of the emperical model is solved numerically as a finite difference explicit upwind scheme in accordance to the forward Euler method as:

$$c_{i,j}=c_{i,j-1} D \frac{(c_{i 1,j-1}-2c_{i,j-1} c_{i-1,j-1})}{\Delta x^2}\Delta t - J_{tot,v} \frac{(c_{i,j-1}-c_{i-1,j-1})}{\Delta x} \Delta t$$

The boundary conditions is defined as follows:

$$ c_{L_x,j}-c_{L_x,j-1}=D(\frac{c_{L_x \Delta x,j-1}-c_{L_x,j-1}}{\Delta x^2} \frac{c_{L_x,j-1} c_{L_x-\Delta x,j-1}}{\Delta x^2})\Delta t - J_{tot,v}\frac{(c_{L_x,j-1}\cdot (1-\sigma_i)- c_{L_x-\Delta x,j-1})}{\Delta x}\Delta t $$

The diffusive and advective stabilities is defined in isolated enviorments by the following conditions:

$$D\frac{\Delta t}{\Delta x^2} \leq \frac{1}{2} ~~~ And ~~~ J_{tot,v}\frac{\Delta t}{\Delta x} <1$$ And $$J_{tot,v}\frac{\Delta t}{\Delta x} <1$$

While in a system where both of these stabilities are relevant a new term must be upheld:

$$1-J_{tot,v}\frac{\Delta t}{\Delta x} - 2D\frac{\Delta t}{\Delta x^2} \geq 0$$

OS X & Windows:

git clone https://github.com/Andemanden/Computational-Solute-Transport-Across-Membranes.git

Linux:

git clone https://github.com/Andemanden/Computational-Solute-Transport-Across-Membranes.git --depth 1 --branch=master ~/dir-name

Current MATLAB version:
MathWorks Page

Visual MINTEQ:
The MINTEQ Page

Pictures to be added

This code has already been used in this article: ¹

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