Algual detachment in response to flushing with reference shear stress equal to the first time step

Louis Poirel — 2012-06-21T13:49:15Z

Definition

The rate of accidental detachment of fix algae $ b_i $ towards drift algae $ a_i $ is calculated in each section as follows :

$$ \frac{\partial B_i}{\partial t}(x,t) = -S\frac{\partial A_i}{\partial t}(x,t) = -\frac{1}{\delta}\left ( \frac{\tau_{0}(x,t) - \tau_{0}(x,0)}{\tau_{0}(x,0)} - s_B \right )^\eta B_j(x,t) $$

if $ \frac{\tau_{0}(x) - \tau_{0}(x,0)}{\tau_{0}(x,0)} > s_B $

$ \frac{\partial A_i}{\partial t}(x,t) = \frac{\partial B_i}{\partial t}(x,t) = 0 $ atherwise.

Where :

$ A_i(x,t) $ : drift algae (kg/m³)
$ B_i(x,t) $ : fix algae (kg/m)
$ B_j(x,t) $ : fix algae ($i=j$ for standard applications).
$ S(x) $ : e cross sectional area (m²)
$ \tau(x,t) $ : shear stress (N m^-2)
$ \tau_{0}(x,0) $ : shear stress at $t=0$
$ s_B $ : sensitivity treshold
$ \delta $ : time constant (s)
$ \eta $ : adimensional exponent

Specifications

Law's ID : 341
Number of acting classes : 3
Number of parameters : 3

Acting classes :

$ A_i $ : drift class modified by the law
$ B_i $ : fix class modified by the law
$ B_j $ : the parameter class of the law

Parameters :

$ s_B $ : sensitivity treshold
$ \delta $ : time constant
$ \eta $ : adimensional exponent

Algal growth (PhD Thesis, O. Fovet, 2010, p.101)

Louis Poirel — 2012-06-21T12:49:49Z

Definition

The biomass growth is calculated at time t, in each section x as follows :

$\frac{\partial B}{\partial t}(x,t) = \mu(x,t) B(x,t) F_{lim}(B'(x,t))$

where :

$F_{lim}(B'(x,t)) = \left ( 1 - \frac{B'(x,t)}{B_{Max}} \right )$
$ \mu(x,t) = \mu_{0} \theta^{T(t)-T_{0}} \frac{I(x,t)}{I_{opt}} e^{1- \frac{I(x,t)}{I_{opt}}} \textup{min} \left ( \frac{N_{i}(x,t)}{N_{i}(x,t) + K_{N_{I}}} \right )$
$ I(x,t) = I_{s}(x,t) e^{-k_{ext} h(x,t)} $
$ I_{s}(x,t) = (1-C_m(x))(1-a)R_N(t) $

Variables and parameters

$ B(x,t) $ : fix biomass modified by the law (kg m^-1)
$ B'(x,t) $ : fix biomass parameter in the law (kg m^-1)
$ B_{Max} $ : maximum value of fix biomass (kg m^-1)
$ \mu_{0} $ : reference growth rate (s ^-1)
$ \theta $ : growth coefficient
$ T(x,t) $ : water temperature (°C)
$ T_{0} $ : reference temperature (°C)
$ I(x,t) $ : light intensity at the section's bottom (W m^-2)
$ I_{s}(x,t) $ : solar light intensity (W m^-2)
$ C_{m}(x)$ : mask coefficient (cf. Temperature simulation)
$ a $ : albedo (cf. Temperature simulation)
$ R_{N}(t) $ solar radiation (W m^-2) (cf. Temperature simulation)
$ k_{ext} $ : extinction coefficient (due to turbidity)
$ h(x,t) $ : mean water level (m)
$ I_{opt}(x,t) $ : optimum light intensity (W m^-2)
$ N_{i}(x,t) $ : concentration of nutrient i (kg m^-3)
$ K_{N_{i}} $ : half-saturation constant of nutrient i (kg m^-3)

Limiting nutrient

This law can take into account up to 3 nutrients for algae growth. If less than three nutrients are used, $ K_{N_{i}} $ for unused nutrients should be set to 0.

Specifications

Law's ID : 301
Number of acting classes: 6
Number of meteo parameters : 3
Number of parameters : 9

Acting classes :

$ B(x,t) $ :the class modified by the law
$ B'(x,t) $ : the parameter class of the law
$ T(x,t) $ :Water temperature
$ N_{1}(x,t) $ : Nutrient 1
$ N_{2}(x,t) $ : Nutrient 2
$ N_{3}(x,t) $ : Nutrient 3

Meteo parameters :

$ C_{m}(x)$
$ a $
$ R_{N}(t) $

Parameters

$ B_{Max} $ : maximum value of fix biomass
$ \mu_{0} $ : reference growth rate
$ \theta $ : growth coefficient
$ T_{0} $ : reference temperature
$ k_{ext} $ : extinction coefficient
$ I_{opt} $ : optimum light intensity
$ K_{N_{1}} $ : half-saturation constant of nutrient 1
$ K_{N_{2}} $ : half-saturation constant of nutrient 2
$ K_{N_{3}} $ : half-saturation constant of nutrient 3

SIC^2: Simulation and Integration of Control for Canals

Algual detachment in response to flushing with reference shear stress equal to the first time step

Definition

Specifications

Algal growth (PhD Thesis, O. Fovet, 2010, p.101)

Definition

Variables and parameters

Limiting nutrient

Specifications