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

### 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