Simplified Alum–Phosphorus Module for GLM-AED
Conceptual Overview
To maintain mechanistic defensibility while keeping the implementation simple, we propose a minimal two-state kinetic model representing alum–phosphorus (Al–P) interactions in the water column.
The model captures:
- pH-dependent aluminium hydrolysis and reactivity
- Phosphate sorption onto amorphous Al(OH)₃ flocs
- Temperature effects on reaction kinetics
- Rapid settling of aluminium flocs (~10 m d⁻¹)
- Optional slow aging (loss of reactivity)
This structure is designed for scenario testing of alum dosing strategies.
State Variables
Water Column
- \(Al_r\) — Reactive aluminium floc (g Al m\(^{-3}\))
- \(P_{Al}\) — Phosphorus bound to aluminium floc (g P m\(^{-3}\))
Coupled to existing AED variable:
- \(PO_4\) — Dissolved reactive phosphorus (FRP)
Governing Equations
1. Reactive Aluminium
Alum added via inflows is assumed to rapidly hydrolyse to reactive floc.
\[ \frac{d Al_r}{dt} = L_{Al,in} - k_{age} Al_r - R_{bind} - settling \]
Where:
- \(L_{Al,in}\) = alum loading from inflow forcing
- \(k_{age}\) = first-order aging constant (loss of reactivity)
- \(R_{bind}\) = phosphate binding rate
2. Phosphate (FRP)
\[ \frac{d PO_4}{dt} = - R_{bind} \]
3. Aluminium-Bound Phosphorus
\[ \frac{d P_{Al}}{dt} = + R_{bind} - settling \]
Binding Kinetics
The core sorption/precipitation reaction is represented as:
\[ R_{bind} = k_{bind,20} \cdot Q_{10}^{\frac{T-20}{10}} \cdot f(pH) \cdot Al_r \cdot PO_4 \]
This captures:
- Second-order interaction between Al and phosphate
- Temperature-dependent reaction rates
- Strong pH control
pH Dependence
Binding efficiency is highest near neutral pH.
A Gaussian response is used:
\[ f(pH) = \exp\left( -\frac{(pH - pH_{opt})^2} {2\sigma_{pH}^2} \right) \]
Typical parameter values:
- \(pH_{opt} \approx 6.5\)
- \(\sigma_{pH} \approx 0.8\)–1.0
This ensures reduced binding efficiency at high summer pH.
Temperature Dependence
A standard Q10 formulation is applied:
\[ k_{bind}(T) = k_{bind,20} \cdot Q_{10}^{\frac{T-20}{10}} \]
Typical:
- \(Q_{10} = 1.5\)–2
Settling of Aluminium Flocs
Both \(Al_r\) and \(P_{Al}\) settle:
\[ F_{settle} = \frac{w_{Al}}{H} \cdot C \]
Where:
- \(w_{Al} \approx 10 \ \text{m d}^{-1}\)
- \(H\) = layer thickness
- \(C\) = tracer concentration
Settling can be handled directly using AED tracer mobility routines.
Optional: Sorption Capacity Limitation
To prevent unrealistic phosphorus removal at high alum doses, a capacity term may be included:
\[ R_{bind} = k_{bind} \cdot f(pH) \cdot f(T) \cdot Al_r \cdot PO_4 \left( 1 - \frac{P_{Al}}{\alpha Al_r} \right) \]
Where:
- \(\alpha\) = maximum P:Al sorption ratio
This introduces a Langmuir-type saturation constraint.
Minimal Parameter Set
The simplified model requires only:
- \(k_{bind,20}\) — Binding rate constant at 20°C
- \(Q_{10}\) — Temperature scaling factor
- \(pH_{opt}\) — Optimal pH for binding
- \(\sigma_{pH}\) — Width of pH response curve
- \(k_{age}\) — Floc aging rate (optional)
- \(w_{Al}\) — Settling velocity (~10 m d⁻¹)
Expected System Behaviour
Under constant dosing:
- Rapid FRP reduction
- Stronger binding under near-neutral pH
- Reduced effectiveness at high summer pH
Under pulsed dosing:
- Rapid FRP drawdown events
- Gradual recovery as flocs settle and age
Summary
This formulation provides:
✔ Mechanistic defensibility
✔ Strong pH control
✔ Temperature sensitivity
✔ Realistic floc settling
✔ Minimal calibration burden
While avoiding:
✘ Explicit aluminium speciation modelling
✘ Complex surface complexation chemistry
✘ Detailed redox coupling
This makes it well-suited for scenario testing of alum dosing strategies in GLM–AED.