In POLLUTEv10 Example 9, the model advances beyond linear sorption by incorporating Freundlich non-linear sorption to simulate the diffusion of phenol through a clay specimen. This example reflects more realistic contaminant behavior, particularly for organic compounds that do not follow simple linear partitioning.
Problem Overview
This example simulates a laboratory diffusion test with the following conditions:
- Contaminant: Phenol
- Soil: Clay (7 cm thick)
- Transport mechanism: Diffusion only (no advection)
- Sorption model: Freundlich non-linear isotherm
- Bottom boundary: Impermeable (zero flux)
- Source type: Finite mass
Source Conditions
- Initial concentration (co): 50 mg/L
- Leachate head (Hr): 6.5 cm
Times of Interest
- 200 hr
- 400 hr
- 600 hr
- 800 hr
Conceptual Model
The system consists of:
- A finite mass source of phenol at the top
- A clay layer where diffusion and sorption occur
- An impermeable base preventing downward flux
Unlike constant concentration sources, the finite source means:
- Concentration at the top decreases over time
- Transport is influenced by both diffusion and depletion
Input Parameters
| Property | Symbol | Value | Units |
|---|---|---|---|
| Darcy Velocity | va | 0.0 | cm/hr |
| Diffusion Coefficient | D | 0.019 | cm²/hr |
| Freundlich Coefficient | Kf | 2.0 | cm³/g |
| Sorption Exponent | — | 0.628 | – |
| Soil Porosity | n | 0.46 | – |
| Dry Density | — | 1.47 | g/cm³ |
| Soil Layer Thickness | H | 7.0 | cm |
| Number of Sub-layers | — | 14 | – |
| Source Concentration | co | 50.0 | mg/L |
| Leachate Height | Hr | 6.5 | cm |
Understanding Freundlich Non-Linear Sorption
The Freundlich isotherm describes sorption as:
Where:
- S = sorbed concentration
- C = լուծ dissolved concentration
- Kf = sorption capacity
- n = non-linearity exponent
Key Implications
- Sorption is not constant (unlike linear Kd models)
- Retardation varies with concentration
- Transport becomes concentration-dependent
With n = 0.628 (< 1):
- Sorption is stronger at lower concentrations
- This causes tailing effects in concentration profiles
Key Processes Simulated
1. Diffusion
- Governed by concentration gradients
- Slower compared to Example 8 due to lower diffusion coefficient
2. Non-Linear Sorption
- Phenol interacts with clay via Freundlich behavior
- Retardation is dynamic, not constant
3. Finite Mass Source
- Source concentration decreases over time
- Results in attenuated diffusion fronts
4. Boundary Condition
- Zero flux at base → contaminant accumulates within the domain
Importance of Layer Discretization
This example highlights a critical modeling requirement:
Accuracy depends strongly on the number of sub-layers when using non-linear sorption.
Why?
- Non-linear equations require finer resolution
- Concentration-dependent retardation must be captured precisely
- Coarse discretization can lead to numerical errors
Best Practice
- Use ≥ 14 sub-layers (as in this example)
- Increase layers further for higher accuracy or steeper gradients
Graphical Output: Depth vs Concentration
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Open in new tab Concentration Profiles
- Profiles evolve over time (200 → 800 hr)
- Slower migration compared to conservative solutes
- Gradual flattening due to source depletion
Non-Linear Effects
- Stronger sorption at low concentrations causes extended tails
- Profiles are non-symmetric and more complex
Finite Source Behavior
- Peak concentrations decrease over time
- Diffusion front weakens as source mass is exhausted
Practical Applications
This example is especially relevant for:
- Organic contaminant transport (e.g., phenol, hydrocarbons)
- Landfill leachate assessments
- Clay liner performance evaluation
- Risk assessment modeling
It is particularly important when:
- Contaminants exhibit non-linear adsorption
- Long-term predictions are required
- Laboratory calibration data is available
Best Practices for POLLUTEv10 Users
- Always verify Freundlich parameters (Kf and n) from lab data
- Use fine discretization for non-linear problems
- Compare results at multiple time steps
- Be cautious when interpreting retardation (not constant!)
Conclusion
POLLUTEv10 Example 9 demonstrates how incorporating Freundlich non-linear sorption significantly enhances the realism of contaminant transport modeling.
Key takeaways:
- Non-linear sorption leads to concentration-dependent transport
- Finite sources introduce time-varying boundary conditions
- Numerical accuracy depends heavily on layer discretization
This example is essential for modeling organic contaminants in clay systems, where linear assumptions are often insufficient.