SIC^2: Simulation and Integration of Control for Canals

Exchange laws

Louis Poirel — 2012-06-21T14:35:03Z

An exchange law modifies one or more quality classes, based on parameters specific to the law, hydraulic variables, and possibly concentrations of some quality classes.

Thus, the exchange terms relative to law $i$ are calculated as follows:

$$\left(E_{k_1}^i, \dots, E_{k_n}^i\right)=L^i\left(C_{k_1}, \dots, C_{k_n}, p_1, \dots, V, h, \dots\right)$$

The exchange term of a class $k$ is the sum of its exchange term in every law where this class is present :

$$ E_k=E_{k}^{i_1}+E_{k}^{i_2}+\cdots+E_{k}^{i_n}$$

Engelund-Hansen (1967)

Louis Poirel — 2012-06-21T14:30:37Z

Engelund-Hansen formula

Transport capacity is calculated as follows :

$$C_{eq}=0.05\rho_S\frac{LU^2}{Q}\frac{(JR)^{\frac{3}{2}}}{\sqrt{g} (\rho_S/\rho-1)^2d}$$

where :

$ \rho_S$ is the sediment's density (kg/m³)
$ L $ is the stream's width (m)
$ U $ is the mean velocity (m/s)
$ Q $ is the water discharge (m³/s)
$ J $ is the slope (m/m)
$ R $ is the hydraulic radius (m)
$ g $ is the gravity (m/s²)
$ \rho $ is the density of water (kg/m³)
$ d $ is the sediment diameter

Specifications

Law's ID : 561
Number of acting classes : 4
Number of parameters : 6

Acting classes :

$ C_i $ : transported class modified by the law
$ C_j $ : fix class modified by the law
$ C_k $ : transported parameter class ($i=k$ for standard use)
$ T $ : water temperature

Parameters :

$ d $ : sediment diameter
$ \rho_S$ : sediment density
$ p $ : sediment porosity
$ \alpha$
$ i_{ech}$ : exchange formula : 1 = Han, 2 = Hazen
$\beta$

Sediment transport

Louis Poirel — 2012-06-21T14:08:26Z

Generalities

The calculation of sediment exchanges involves three steps:

Calculating an equilibrium concentration $C_{eq}$ according to a proposed law;
Calculating an adaptation time following of Han's formula$t_A=\beta\frac{Ru*}{VW}$ or Hazen's formula: $t_A=\beta\frac{R}{W}$ where $W$ is the fall velocity;
Calculation of the exchange term$E=\frac {\alpha C_{eq}-C }{t_A}$ ;
where $\alpha$ and $\beta$ are dimensionless parameters.

The fall rate is calculated according to Zanke's formula:

$$W=1.1*10\frac{\nu}{d}\left(\sqrt{1+\frac{0.01 g (\rho_S/\rho-1)d^3}{\nu^2}}-1\right)$$

This law is equal to Stokes' law for small diameters, and Newton's law for larger particles.

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

Exponential growth with fixed coefficient

Louis Poirel — 2012-06-21T06:19:57Z

Definition

This law changes the concentration of a class depending on an other class concentration (or eventually itself), and two fix coefficients $ k $ and $\alpha_k$ as shown here:

$ \frac{dC_i}{dt}=k C_j^{\alpha_k} $

Classic use

Numerous solutes have a first-order kinetics with an equation similar to :

$ \frac{dC_{Ni}}{dt}= - k_{Ni} C_{Ni} $

where $ k_{Ni} $ is the reaction constant (which is the inverse of a time). For instance, bacteriological oxygen demand (DBO_5 $ $), has a constant around 0.3 days ^-1. This degradation models the process of water auto-purification in streams.

Specifications

Law's ID : 201
Number of acting classes : 2
Number of parameters : : 2

Acting classes :

$ C_i $ : the class modified by the law
$ C_j $ : the parameter class of the law

Parameters :

$ k $ : reaction constant
$ \alpha_k $ : reaction order

Evolution of the temperature

Louis Poirel — 2012-06-21T06:19:52Z

Definition

This law changes water temperature depending on meteorology and water temperature as explained in «Temperature simulation».

Specifications

Law's ID : 101
Number of acting classes : 1
Number of parameters : 0

Acting classes :

Water temperature