Pasa Analytics | Filtration Simulation - Contaminant Adsorption and Migration

Fundamental Transport Equation with Adsorption

Contaminant transport in porous media (filter) is governed by:

where the source term γ represents adsorption/desorption on the filter substrate:

with:

• θ: filter porosity
• ρ_s: filter density
• K_d: partition coefficient (adsorption)

Scaling and Retardation Factor

The equation can be rewritten as a classical advection-diffusion equation with effective coefficients:

The retardation factor R > 1 represents the slowing down of the concentration front due to adsorption: higher R means more effective contaminant retention.

Complete Adsorption-Diffusion Model

The full system consists of the fluid and filter domains considered as a coupled system in which concentrations in fluid (c_ℓ) and filter (c_f) are considered. The reduction of the fluid concentration corresponds to the loading of the filter (or simply to the adsorption of compounds by the filter, whose concentration increases.

This model integrates:

• α: contaminant adsorption coefficient on filter
• κ: filter-to-fluid diffusion coefficient (desorption)
• ε: saturation factor (adsorption dependence on filter loading)

Special Case: Variable Adsorption Coefficient

As the filter loads, adsorption efficiency decreases. The effective adsorption coefficient α(t) decreases with increasing filter loading c_f.

This nonlinearity leads to progressive filter saturation and contaminant breakthrough.

This section presents simulation of substance migration from polymer tube wall to flowing fluid.

Geometric Configuration

• Inner diameter: 4 mm
• Wall thickness: 0.6 mm
• Length: 1 m
• Flow rate: 100 mL/h
• Inner volume: 12.57 mL

Substance Properties

Two substances with contrasting diffusivities are simulated:

Pure Diffusion Regime

Without flow, migration is governed by Fick's second law:

After 1 hour of diffusion, fluid concentrations reach:

• Substance 1: 0.0408 mg in inner volume (12.57 mL)
• Substance 2: 0.2420 mg

Advection-Diffusion Regime

With flow, the transport equation becomes:

where velocity profile v is solution of Navier-Stokes equations:

For a flow rate of 100 mL/h (bulk velocity ≈ 2.21 mm/s), Reynolds number is very low (Re ≪ 1), flow is laminar with parabolic profile (Poiseuille).

Results for Continuous Flow

For a total flowed volume of 400 mL (duration: 4 hours), cumulative contaminant masses are:

Flow Rate Profile Impact

Three scenarios compared for same total volume (400 mL):

Interpretation: Interruptions and slowdowns significantly increase migration due to prolonged contact time between fluid and tube wall. Continuous processing minimizes contamination.

Steady State and Beer-Lambert Law

In steady state, neglecting diffusion from filter (κ ≈ 0), the system reduces to:

whose solution is an exponential decay:

This relationship, analogous to Beer-Lambert law in optical absorption, characterizes contaminant attenuation in the filter.

Breakthrough Curve

The key parameter for filter sizing is the breakthrough curve: outlet concentration evolution as function of time or treated volume.

Optimal Filtration Parameters

Filtration system optimization relies on trade-off between:

• Efficiency: contaminant retention capacity (high α, high R)
• Lifetime: volume treated before breakthrough (filter capacity)
• Pressure drop: impact on flow rate (Darcy's law, Navier-Stokes in porous media)

with K the porous medium permeability (m²), function of porosity and pore size.

The complete model integrates:

1. Migration: diffusion from tube wall + advection by flow
2. Filtration: contaminant adsorption on filter media
3. Coupling: filter inlet concentration = tube outlet concentration

Case Study

Consider a system where:

• Fluid flows through 1 m tubing (wall migration)
• Then through 10 cm thick filter (adsorption)
• Flow rate: 100 mL/h
• Substance 1 (previous parameters)

Sensitivity Analysis

Coupled system efficiency depends on several parameters: