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*Excerpt from Larinier, 2002[^1]*
## Hydraulic laws given by abacuses
Experiments conducted by Larinier, 2002[^1] allowed to establish abacuses that link adimensional flow \(q^*\) :
$$ q^* = \dfrac{Q/L}{\sqrt{2g}a^{1,5}} $$
to upstream head \(ha\) and the average water level in the pass \(h\) :
*Abacuses of a superactive baffles fishway for a slope of 10% (Excerpt from Larinier, 2002[^1])*
*Abacuses of a superactive baffles fishway for a slope of 15% (Excerpt from Larinier, 2002[^1])*
To run calculations for all slopes between 8% and 22%, polynomes coefficients of abacuses above are themelves adjusted in the form of slope \(S\) depending polynomes.
$$ ha/a = a_2(S) q^{*2} + a_1(S) q^* + a_0(S) $$
$$a_2(S) = - 0.354624S - 0.0153156$$
$$a_1(S) = 0.514953S + 1.25460$$
$$a_0(S) = - 2.22434S + 0.596682$$
$$ h/a = b_2(S) q^{*2} + b_1(S) q^* + b_0 $$
$$b_2(S) = - 0.559218S + 0.000504060$$
$$b_1(S) = 1.15807S + 1.07554$$
$$b_0(S) = - 2.62712S + 0.601348$$
## Calculation of \(ha\), \(h\) and \(Q\)
We can then use those coefficients to calculate \(ha\), \(h\) and \(q^*\):
$$ ha = a \left( a_2 (q^*)^2 + a_1 q^* + a_0 \right)$$
$$ h = a \left( b_2 (q^*)^2 + b_1 q^* + b_0 \right)$$
Using the positive inverse function, depending on \(ha/L\), we get:
$$ q^* = \dfrac{-a_1 + \sqrt{a_1^2 - 4 a_2 (a_0 - h_a/a)}}{2 a_2}$$
$$ Q = L q^* \sqrt{g} a^{1,5} $$
Calculation limitations of \(q^*\), \(ha/a\) and \(h/a\) are determined based on the extremities of the abacuses curves.
Flow velocity \(V\) corresponds to the minimum flow speed given the flow section \(A_w\) at the perpendicular of the baffle :
$$ V = \dfrac{Q}{A_w} $$
for superactive baffles fishways using the notation of the schema above, we have:
$$ A_w = h \times L$$
## Upstream apron elevation \(Z_{r1}\)
$$ Z_{r1} = Z_{d1} + \frac{2.6 a S - a}{\sqrt{1 + S^2}} $$
[^1]: Larinier, M. 2002. “BAFFLE FISHWAYS.” Bulletin Français de La Pêche et de La Pisciculture, no. 364: 83–101. doi:[10.1051/kmae/2002109](https://doi.org/10.1051/kmae/2002109).