The double-exponential form captures **both** the clarification zone (low concentrations, dominated by the r_p term) and the thickening zone (high concentrations, dominated by the r_h term). This allows simulation of the complete solids profile from effluent to underflow.
### Typical Parameter Values
| Parameter | Typical Range | Default | Unit |
|-----------|---------------|---------|------|
| v_0 | 4 – 12 | 7.5 | m/h |
| r_h | 0.3 – 0.8 | 0.576 | m³/kg |
| r_p | 2.0 – 6.0 | 2.86 | m³/kg |
| X_min | 0 – 0.1 | 0.01 | kg/m³ |
### Sensitivity
- **r_p** is the most sensitive parameter — it governs effluent suspended solids
- Takacs et al. recommend finding r_p by simulation/calibration
- v_0 and r_h primarily affect the sludge blanket position and underflow concentration
### 1D Layer Model Implementation
The settler is divided into N horizontal layers (typically 10-30). For each layer:
1. Calculate settling velocity from local concentration
2. Apply solids flux theory (gravity flux + bulk flux)
3. Update concentration via mass balance
4. Handle feed layer, overflow, and underflow boundary conditions
## Burger-Diehl Framework — PDE-Based 1D Settler
**Source**: Burger, R., Diehl, S. and various co-authors (2011-present). Multiple publications developing the framework.
### Key Characteristics
- Based on rigorous **partial differential equation** theory (hyperbolic-elliptic PDE)
- Accounts for hindered settling, compression settling, and inlet dispersion
- Every implementation detail is consistent with PDE theory (unlike ad-hoc layer models)
- More realistic prediction of underflow sludge concentration
- Essential for accurate wet-weather modelling
### Advantages Over Takacs Layer Model
- Proper handling of compression settling (important at high MLSS)
- Better sludge blanket dynamics during storm events
- Can be extended with reactive terms (ASM1 biokinetics inside settler)
### When to Use Which Model
| Scenario | Recommended Model |
|----------|-------------------|
| Steady-state design | Vesilind + flux theory |
| Dynamic simulation (standard) | Takacs 1D layer model |
| Wet-weather / high-MLSS dynamics | Burger-Diehl PDE model |
| Quick estimation | Vesilind with SVI correlation |
## Flux Theory for Clarifier Design
The solids flux approach combines the gravity settling flux with the bulk (underflow) flux:
```
J_total = J_gravity + J_bulk = v_s(X) · X + Q_u/A · X
```
Where:
- J_total = total solids flux (kg/m²/h)
- v_s(X) = settling velocity at concentration X (from Vesilind or Takacs)
- Q_u = underflow rate (m³/h)
- A = clarifier surface area (m²)
The **limiting flux** determines the maximum solids loading rate — operating above this causes blanket rise and eventual washout.
## Authoritative References
1. Vesilind, P.A. (1968). "Design of Prototype Thickeners from Batch Settling Tests." Water Sewage Works, 115, 302-307.
2. Takacs, I., Patry, G.G., Nolasco, D. (1991). "A dynamic model of the clarification-thickening process." Water Res. 25(10), 1263-1271.
3. Burger, R., Diehl, S., Nopens, I. (2011). "A consistent modelling methodology for secondary settling tanks in wastewater treatment." Water Res. 45(6), 2247-2260.
4. Torfs, E. (2015). "Different settling regimes in secondary settling tanks." PhD thesis, Ghent University.
5. Daigger, G.T. (1995). "Development of refined clarifier operating diagrams using an updated settling characteristics database." Water Environment Research, 67(1), 95-100.
