Advection–Diffusion Transport Equation

In addition to algal proliferation, the ALGEQUEAU module has evolved to take into account other quality parameters including sediment transport simulation.

The module aims at simulating the evolution and transport of temperature and several classes of nutrients, algae, or sediment in the fixed and transported parts of an hydraulic network.

Quality classes

Quality in the network is given at any point by a vector :

$$ C(x,t)=\left(\begin{array}{c} C_1(x,t)\\ \vdots\\ C_n(x,t)\\ \end{array} \right)\ $$

where $C_k$ may be :

  • a temperature (°C) ;
  • a nutrient concentration (mg/L) ;
  • a concentration of drift algae (kg/m3) ;
  • a linear density of fixed algae (kg/m)
  • a sediment concentration in water (g/m3) ;
  • a linear density of sediment in the bed (m3/m).

Advection-Diffusion equation

The advection diffusion transport equation is solved for transported classes :

$$ \frac{\partial SC_k}{\partial t}+ \frac{\partial QC_k}{\partial x}= \frac{\partial}{\partial x}\left(DS\frac{\partial C_k}{\partial x}\right) + SE_k $$

where :

  • $S$ is the wetted area (m2) ;
  • $Q$ is the discharge (m3/s) ;
  • $D$ is the diffusion coefficient (m2/s);
  • $E_k$ is an exchange rate ([$C_k$]/s with [$C_k$] the unit of class $C_k$).

For fixed classes, there is no advection-diffusion term in the equation :

$$ \frac{dC_k}{dt}(x,t) = E_k $$

Calculation of the exchange rate

The vector $E$ of exchange rates for each class is computed from the list of exchange formulas given by the user, depending on the classes vector $C$ and some parameters $p$

$$ E_k = \mathcal{F}_k(C,p) $$

where $F_k$ sums the contributions of all the exchange formulas affecting $C_k$.

Numerical resolution of the equation

The solution to the equation is calculated using a conjugate gradient method.