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Mathematical Model - ALBA (Algae-Bacteria)

Table of Contents

  1. State Variables
  2. Biological Kinetics
  3. Stoichiometry
  4. pH and Chemical Equilibria
  5. Gas-Liquid Transfer
  6. Temperature Dependence

State Variables

The model tracks 17 state variables ($C_i$), typically expressed in $g \cdot m^{-3}$ (equivalent to $mg \cdot L^{-1}$).

Symbol Description Unit
Biomass
$X_{ALG}$ Algal biomass $gCOD \cdot m^{-3}$
$X_H$ Heterotrophic bacteria $gCOD \cdot m^{-3}$
$X_{AOB}$ Ammonia Oxidizing Bacteria $gCOD \cdot m^{-3}$
$X_{NOB}$ Nitrite Oxidizing Bacteria $gCOD \cdot m^{-3}$
$X_I$ Inert particulate organic matter $gCOD \cdot m^{-3}$
Soluble Substrates
$S_S$ Readily biodegradable organic matter $gCOD \cdot m^{-3}$
$S_I$ Inert soluble organic matter $gCOD \cdot m^{-3}$
$S_{NH}$ Total Ammoniacal Nitrogen ($NH_4^+ + NH_3$) $gN \cdot m^{-3}$
$S_{NO2}$ Nitrite Nitrogen $gN \cdot m^{-3}$
$S_{NO3}$ Nitrate Nitrogen $gN \cdot m^{-3}$
$S_{PO4}$ Total Inorganic Phosphorus $gP \cdot m^{-3}$
$S_{O2}$ Dissolved Oxygen $gO_2 \cdot m^{-3}$
$S_{IC}$ Total Inorganic Carbon ($CO_2 + HCO_3^- + CO_3^{2-}$) $gC \cdot m^{-3}$
$S_{ND}$ Soluble Biodegradable Organic Nitrogen (Urea) $gN \cdot m^{-3}$
Particulate Substrates
$X_S$ Slowly biodegradable organic matter $gCOD \cdot m^{-3}$

Biological Kinetics

The model uses Liebig's Law of the Minimum for nutrient limitation, rather than multiplicative Monod terms.

General Growth Rate Equation

For a biomass $X_i$, the growth rate $\rho_{growth}$ is defined as:

$$ \rho_{growth} = \mu_{max} \cdot f(T) \cdot f(pH) \cdot f(I) \cdot \min \left( \frac{S_1}{K_1 + S_1}, \frac{S_2}{K_2 + S_2}, \dots \right) \cdot X_i $$

Where:

  • $\mu_{max}$: Maximum specific growth rate ($d^{-1}$)
  • $f(T)$: Temperature correction factor
  • $f(pH)$: pH correction factor
  • $f(I)$: Light correction factor (for algae only)
  • $S_j$: Limiting substrate concentration
  • $K_j$: Half-saturation constant

1. Algal Kinetics (Processes $\rho_1, \rho_2$)

Light Dependence ($f_I$): Uses the Haldane model modified by Bernard & Remond (2012), integrated over depth $z$:

$$ f_I(I) = \frac{I}{I + \frac{\mu_{max}}{\alpha} \left( \frac{I}{I_{opt}} - 1 \right)^2} $$

The average growth rate in the reactor is obtained by integrating $f_I(I(z))$ where $I(z)$ follows Beer-Lambert law: $$ I(z) = I_0 \cdot e^{-\epsilon \cdot TSS \cdot z} $$

Oxygen Inhibition: Photosynthesis is inhibited by high $DO$ concentrations (Hill type): $$ f_{O2, inh} = \frac{K_{O2, inh}^n}{S_{O2}^n + K_{O2, inh}^n} $$

2. Bacterial Kinetics (Processes $\rho_5 - \rho_{10}$)

Heterotrophs can grow aerobically ($\rho_5, \rho_6$) or anoxically ($\rho_8, \rho_9$).

  • Aerobic: Depends on $\frac{S_{O2}}{K_{O2} + S_{O2}}$
  • Anoxic: Depends on $\frac{K_{O2}}{K_{O2} + S_{O2}}$ (inhibition by oxygen) and presence of $NO_x$.

3. Nitrification (Processes $\rho_{14}, \rho_{17}$)

Two-step nitrification ($NH_4^+ \to NO_2^- \to NO_3^-$).

  • AOB: Growth on $S_{NH}$. Sensitive to $S_{O2}$ and $S_{IC}$.
  • NOB: Growth on $S_{NO2}$. Sensitive to $S_{O2}$ and $S_{IC}$.

Stoichiometry

The model considers 19 biological processes. The mass balance is ensured via a stoichiometric matrix $\mathbf{S}$.

$$ \frac{d\mathbf{C}}{dt} = \mathbf{S}^T \cdot \boldsymbol{\rho} + \text{Transport} + \text{Transfer} $$

Key stoichiometric coefficients ($\alpha$) are derived from the elemental composition of biomass (e.g., $C_{100}H_{183}O_{48}N_{11}P$ for algae).

Example for Algal Growth on Ammonium ($\rho_1$):

  • Consumes: $S_{NH}$, $S_{PO4}$, $S_{IC}$
  • Produces: $X_{ALG}$, $S_{O2}$

pH and Chemical Equilibria

The pH is calculated by solving the charge balance equation at each time step. The system considers the following equilibria:

  1. Ammonia: $NH_4^+ \leftrightarrow NH_3 + H^+$
  2. Carbonate: $CO_2 + H_2O \leftrightarrow HCO_3^- + H^+ \leftrightarrow CO_3^{2-} + 2H^+$
  3. Phosphate: $H_3PO_4 \leftrightarrow H_2PO_4^- \leftrightarrow HPO_4^{2-} \leftrightarrow PO_4^{3-}$
  4. Nitrite/Nitrate: $HNO_2 \leftrightarrow NO_2^-$, $HNO_3 \leftrightarrow NO_3^-$
  5. Water: $H_2O \leftrightarrow H^+ + OH^-$

Charge Balance Equation: $$ [H^+] + [NH_4^+] + \dots - [OH^-] - [HCO_3^-] - 2[CO_3^{2-}] - \dots + Z_{net} = 0 $$

Gas-Liquid Transfer

Gas transfer rates ($Q_j$) for $O_2$, $CO_2$, and $NH_3$ are modeled as:

$$ Q_j = k_L a_j \cdot (S_{j, sat} - S_j) $$

  • Oxygen: $k_L a_{O2}$ is a calibrated parameter (e.g., $34 d^{-1}$).
  • Others: Derived from diffusivity ratios: $$ k_L a_j = k_L a_{O2} \cdot \sqrt{\frac{D_j}{D_{O2}}} $$
  • Saturation ($S_{sat}$): Calculated using Henry's Law, dependent on temperature and partial pressure ($p_{gas}$).

Temperature Dependence

Growth Rates (CTMI Model)

The Cardinal Temperature Model with Inflection (Rosso et al., 1993) is used for all growth rates:

$$ f(T) = \frac{(T - T_{max})(T - T_{min})^2}{(T_{opt} - T_{min})[(T_{opt} - T_{min})(T - T_{opt}) - (T_{opt} - T_{max})(T_{opt} + T_{min} - 2T)]} $$

Defined by three parameters: $T_{min}$, $T_{opt}$, $T_{max}$.

Decay and Hydrolysis (Arrhenius)

Simple exponential dependence: $$ k(T) = k_{20} \cdot \theta^{(T - 20)} $$

Last updated: March 10, 2026 by Anibal Rojo