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 :

1. $ A_i $ : drift class modified by the law
2. $ B_i $ : fix class modified by the law
3. $ B_j $ : the parameter class of the law

Parameters :

1. $ s_B $ : sensitivity treshold
2. $ \delta $ : time constant
3. $ \eta $ : adimensional exponent