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/src/assets/docs-*
/compodoc-fr
/docs-fr_pdf
/docs-en_pdf
/release
# dependencies
/node_modules
/docs-fr/javascripts/mathjax
/docs-en/javascripts/mathjax
/src/date_revision.ts
# IDEs and editors
......
../CHANGELOG.md
\ No newline at end of file
# Calculation of the head loss on a water intake grid
<div style="position: relative"><a id="grille-conventionnelle" style="position: absolute; top: -60px;"></a></div>
## Conventional grid
Conventional grid planes: perpendicular to the flow and slightly inclined to the horizontal
### Formula
Use of the F1 formula of Raynal et al (2012) to calculate the head losses.
$$\xi = K_F * K_O * K_\beta = a * \left ( \frac{O}{1-O} \right )^{1.6} * \left ( 1 - \cos{\beta} \right )^{0.39}$$
<div style="position: relative"><a id="grille-orientee" style="position: absolute; top: -60px;"></a></div>
## Oriented grid
Flow-oriented and near-vertical grid planes.
![Oriented grid](grille-orientee.jpg)
*Courret, D. et Larinier, M. Guide pour la conception de prise d’eau ichtyocompatibles pour les petites centrales hydroélectriques, 2008. <https://doi.org/10.13140/RG.2.1.2359.1449>.*
### Formula
Use of the F2 formula of Raynal et al (2012) to calculate the head losses.
$$\xi = K_F * K_O * K_\alpha = a * \left ( \frac{O}{1-O} \right )^{1.6} * \left ( 1 + c * \left ( \frac{90 - \alpha}{90} \right )^{2.35} * \left ( \frac{1 - O}{O} \right )^{3} \right )$$
<div style="position: relative"><a id="grille-inclinee" style="position: absolute; top: -60px;"></a></div>
## Inclined grid
Grid planes perpendicular to the flow, and inclined with respect to the horizontal
![Inclined grid](grille-inclinee.jpg)
![Inclined grid](grille-inclinee-b.jpg)
*Courret, D. et Larinier, M. Guide pour la conception de prise d’eau ichtyocompatibles pour les petites centrales hydroélectriques, 2008. <https://doi.org/10.13140/RG.2.1.2359.1449>.*
### Formula
Use of the F3 formula of Raynal et al (2012) to calculate the head losses.
$$\xi = K_{F, b} * K_b * K_\beta + K_{Fent} * K_{entH} = a * \left ( \frac{O_b}{1-O_b} \right )^{1.65} * \left ( \sin \beta \right )^{2} + c * \left ( \frac{O_{entH}}{1-O_{entH}} \right )^{0.77}$$
## Parameters
<div style="position: relative"><a id="cote-du-sommet-immerge-du-plan-de-grille" style="position: absolute; top: -60px;"></a></div>
### Elevation of the immersed vertex of the grid plane
May be different from the water level if the top of the grid plane is drowned.
<div style="position: relative"><a id="largeur-de-la-section" style="position: absolute; top: -60px;"></a></div>
### Section width
#### Conventional or inclined grid
Must also correspond to the width of the grid plane.
<div style="position: relative"><a id="vitesse-dapproche-moyenne-pour-le-debit-maximum-turbine-en-soustrayant-la-partie-superieure-eventuellement-obturee" style="position: absolute; top: -60px;"></a></div>
### Average approach speed for the maximum turbinated flow, subtracting the upper part, if any, blocked
"Maximum" value of the approach speed taken into account in the calculation of the head loss in a safety approach.
<div style="position: relative"><a id="inclinaison-par-rapport-a-lhorizontale" style="position: absolute; top: -60px;"></a></div>
### Inclination with respect to the horizontal
#### Conventional grid
Scope of the formula: 45 ≤ β ≤ 90°
#### Oriented grid
Vertical grid planes (β = 90°).
The slight inclination of the grid planes (β≈ 75/80°), often set up for screening purposes, can be neglected.
#### inclined grid
Scope of the formula: 15° ≤ β ≤ 90°
Recommended for fish guidance: β ≤ 26°
<div style="position: relative"><a id="orientation-par-rapport-a-la-direction-de-lecoulement" style="position: absolute; top: -60px;"></a></div>
### Orientation with respect to the direction of flow
#### Conventional grid
Grid planes perpendicular to the flow (α = 90°)
#### Oriented grid
Scope of the formula: 30° ≤ α ≤ 90°
Recommended for fish guidance: α ≤ 45°
#### inclined grid
Grid planes perpendicular to the flow (α = 90°)
<div style="position: relative"><a id="vitesse-normale-moyenne-pour-le-debit-maximum-turbine" style="position: absolute; top: -60px;"></a></div>
### Average normal speed for maximum turbinated flow rate
#### Conventional grid
Recommended to avoid plating fish on the grid plane (physical barrier) or their premature passage through (behavioural barrier): VN ≤ 0.5 m/s.
#### Oriented or inclined grid
Recommended to avoid plating fish on the grid plane (physical barrier) or their premature passage through (behavioural barrier): VN ≤ 0.5 m/s.
Above the average value calculated here, it is essential to refer to the recommendations derived from the experimental characterization of the actual velocity values.
<div style="position: relative"><a id="rapport-de-forme-des-barreaux" style="position: absolute; top: -60px;"></a></div>
### Bar shape ratio
#### Oriented grid
Validity range of the formula: ratio b / p close to 0.125
<div style="position: relative"><a id="rapport-espacementepaisseur-des-barreaux" style="position: absolute; top: -60px;"></a></div>
### Ratio of spacing / bar thickness
#### Oriented grid
Scope of validity of the formula: 1 ≤ e / b ≤ 3
<div style="position: relative"><a id="obstruction-globale-du-plan-de-grille-barreaux-entretoises-elements-de-supports-longitudinaux-et-transversaux-retenue" style="position: absolute; top: -60px;"></a></div>
### Overall obstruction of the grid plane (bars + spacers + longitudinal and transverse support elements) retained
To be determined from the grid plans.
#### Conventional grid
Scope of validity of the formula: 0.2 ≤ O ≤ 0.60
#### Oriented grid
Scope of the formula: 0.35 ≤ O ≤ 0.60
#### inclined grid
Obstruction due to the bars and longitudinal support elements retained \(O_b\). To be determined from the grid plans.
Scope of validity of the formula: 0.28 ≤ Ob ≤ 0.53
<div style="position: relative"><a id="profil-des-barreaux" style="position: absolute; top: -60px;"></a></div>
### Bar profile
![Bar profile](profil-barreaux.png)
*Raynal, Sylvain. « Étude expérimentale et numérique des grilles ichtyocompatibles ». Sciences et ingénierie en matériaux, mécanique, énergétique et aéronautique - SIMMEA, 2013.*
#### Conventional grid
The shape coefficient of the bars \(a\) is 2.89 for the rectangular profile (PR) and 1.70 for the hydrodynamic profile (PH).
#### Oriented grid
The shape coefficient of the bars is 2.89 for the rectangular profile (PR) and 1.70 for the hydrodynamic profile (PH).
The shape coefficient of the bars \(c\) is 1.69 for the rectangular profile (PR) and 2.78 for the hydrodynamic profile (PH).
#### inclined grid
The shape coefficient of the bars \(a\) is 3.85 for the rectangular profile (PR) and 2.10 for the hydrodynamic profile (PH).
<div style="position: relative"><a id="obstruction-effective-due-aux-entretoises-et-elements-de-support-transversaux-rapportee-a-la-section-decoulement" style="position: absolute; top: -60px;"></a></div>
### Effective obstruction due to spacers and transverse support elements in relation to the flow cross-section
#### inclined grid
To be determined from the grid plans.
Scope of the formula: OentH ≤ 0.28
<div style="position: relative"><a id="coefficient-de-forme-moyen-des-entretoises-et-elements-transversaux-ponderes-selon-leurs-parts-respectives" style="position: absolute; top: -60px;"></a></div>
### Average shape coefficient of spacers and transverse elements, weighted according to their respective shares
#### inclined grid
To be determined from the grid plans.
For example, 1.79 for cylindrical spacers, 2.42 for rectangular spacers, and around 4 for square beams and IPNs.
# Jet impact
The downstream fish evacuation outlet ends with a device that empties into the plant's tailrace. This module calculates the position and velocity at the point of impact of the free fall or water vein on the surface of the tailrace water taking into account the initial angle and velocity of the jet and the drop height.
Excerpt from Courret, Dominique, and Michel Larinier. Guide for the design of ichthyocompatible water intakes for small hydroelectric power plants, 2008. https://doi.org/10.13140/RG.2.1.2359.1449, p.24:
> Speeds in the structure and at the point of impact in the tailrace must remain below about 10 m/s, with some organizations even recommending that they not exceed 7-8 m/s (ASCE 1995). (...) The head between the outlet and the water body must not exceed a dozen metres to avoid any risk of injury to fish on impact, whatever their size and mode of fall (free fall or fall in the water vein) (Larinier and Travade 2002). The discharge must also be made in an area of sufficient depth to avoid any risk of injury from mechanical shock. Odeh and Orvis (1998) recommend a minimum depth of about a quarter of the fall, with a minimum of about 1 m.
## Formula
With \(g\): gravity acceleration = 9.81 m.s-2
### Fall height
$$H = 0.5 * g * \frac{D^{2}}{\cos \alpha^{2} * V_0^{2}} - \tan \alpha * D$$
### Impact abscissa (horizontal distance covered)
$$D = \frac{V_0}{g * \cos \alpha} \left ( V_0 * \sin \alpha + \sqrt{ \left ( V_0 * \sin \alpha \right )^{2} + 2 * g * H } \right )$$
### Flight time
$$t = \frac{D}{V_0 \cos \alpha} $$
### Horizontal speed at impact
$$V_x = V_0 \cos \alpha$$
### Vertical speed at impact
$$V_z = V_0 \sin \alpha - t * g$$
### Speed at impact
$$V_t = \sqrt{ \V_x^{2} + V_z^{2} }$$
# Backwater curve
The calculation of the backwater curve involves the following differential equation:
$$\frac{dy}{dx}=\frac{I_f - J(h)}{1-F^2(h)}$$
where \(I_f\) is the slope of a canal, \(J\) the formula giving us the local pressure drop (depending on the water level), \(y\) here refers to the height of water.
Thus, for a rectangular channel of width \(b\) and a Strickler coefficient \(K\):
$$J=\frac{Q^2 (b+2y)^{4/3}}{K^2 b^{10/3}y^{10/3}}$$
and
$$F^2=\frac{Q^2}{gb^2y^3}$$
The integration of the equation can be done by one of the following methods: [Runge-Kutta 4](../../methodes_numeriques/rk4.md), [Explicit Euler](../../methodes_numeriques/euler_explicite.md), [trapezes integration](../../methodes_numeriques/integration_trapezes.md).
Depending on the flow regime, the calculation can be carried out:
* from downstream to upstream for the river regime with definition of a downstream boundary condition.
* from upstream to downstream for torrential regime with definition of an upstream boundary condition
If we take the example of a rectangular channel, [the proposed scilab code example for solving an ordinary differential equation](../../methodes_numeriques/euler_explicite.md) is amended as follows:
```scilab
b=0.3;
K=50;
If=0.005;
Q=0.01;
function z=DQ(y);z=Q-K*(b*y)^(5/3)/(b+2*y)^(2/3)*sqrt(If); endfunction
yn=fsolve(0.5,DQ);
tmax=0;
t0=10;
dt=-0.5;
function z=f(y,t);z=(If-Q^2*(b+2*y)^(4/3)/(K^2*(b*y)^(10/3)))/(1-Q^2/(9.81*b^2*y^3)); endfunction
y0=0.12;
```
which gives us the normal depth, and the water line. Depending on the numerical method used, we can have large errors in the case of an F2 backwater curve (downstream condition below normal height), because the waterline slopes are much steeper, and therefore much more prone to errors related to linear interpolation. We can therefore deduce that on the one hand the choice of the resolution method is important, and on the other hand it is essential to take a critical look at the solutions (with an interpretation of the processes we are trying to model).
# Slope
This tools allows to calculate the missing value of the four quantities:
- upstream elevation (\(Z_1\)) in m;
- downstream elevation (\(Z_2\)) in m;
- length (\(L\)) in m;
- slope (\(I\)) in m/m;
## Formula
$$I = \frac{(Z_1 - Z_2)}{L}$$
# Uniform flow
The uniform flow is characterized by a water height called the normal height. The normal height is reached when the water line is parallel to the bottom, the load is then itself parallel to the water line and thus the head loss is equal to the slope of the bottom:
\(I_f = J\)
With:
- \(I_f\): bottom slope in m/m
- \(J\): head loss in m/m
The head loss {J} is calculated here using Manning-Strickler's formula:
$$J=\frac{U^2}{K^{2}R^{4/3}}=\frac{Q^2}{S^2K^{2}R^{4/3}}$$
With:
- \(K\): Strickler coefficient in m<sup>1/3</sup>/s
In uniform flow, we obtain the formula:
$$Q=KR^{2/3}S\sqrt{I_f}$$
Based on the which, flow \(Q\), slope \(I_f\) and Strickler calculation \(K\) can be calculated analytically.
To calculate normal height \(h_n\) , one can solve \(f(h_n)=Q-KR^{2/3}S\sqrt{I_f}=0\)
using Newton's method:
$$h_{k+1} = h_k - \frac{f(h_k)}{f'(h_k)}$$
with:
- \(f(h_k) = Q-KR^{2/3}S\sqrt{I_f}\)
- \(f'(h_k) = -K \sqrt{I_f}(\frac{2}{3}R'R^{-1/3}S+R^{2/3}S')\)
To calculate the geometrical parameters of the section, the calculation module uses the flow calculation equation and solves the problem by dichotomy.
# Parametric section
This module calculates the hydraulic quantities associated to:
- a section with a defined geometrical shape ([See section types managed by Cassiopée](types_sections.md))
- a draft \(y\) in m
- a flow \(Q\) in m<sup>3</sup>/s
- a bottom slope \(I_f\) in m/m
- a roughness expressed with the Strickler's coefficient \(K\) in m<sup>1/3</sup>/s
The calculated hydraulic quantities are:
- Width at mirror (m)
- Wet perimeter (m)
- Hydraulic surface (m<sup>2</sup>)
- Hydraulic radius (m)
- Average speed (m/s)
- Specific head (m)
- Head loss (m)
- Linear variation of specific energy (m/m)
- Normal depth (m)
- Froude number
- Critical depth (m)
- Critical head (m)
- Corresponding depth (m)
- Impulsion (kg⋅m⋅s<sup>-1</sup>)
- Conjugate depth
- Tractive force (Pa)
## Width at mirror, wet perimeter and surface
[See the dedicated page for the parameters specific to each type of section](types_sections.md)
### Rectangular section
- Width at mirror : \(B=L\)
- Surface : \(S=L.y\)
- Perimeter : \(P=L+2y\)
### Trapezoidal section
- Width at mirror : \(B=L+2..m.y\)
- Surface : \(S=(L+m.y)y\)
- Perimeter : \(P=L+2y\sqrt{1+m^2}\)
### Circular section
- Width at mirror : \(B=D\sin\theta\)
- Surface : \(S=\frac{D^2}{4} \left(\theta - \sin\theta.\cos\theta \right)\)
- Perimeter : \(P=D.\theta\)
### Parabolic section
- Width at mirror : \(B=\frac{B_b}{y_b^k}y^k\)
- Surface : \(S=\frac{B_b}{y_b^k}\frac{y^{k+1}}{k+1}\)
- Perimeter : \(P=2\sum _{i=1}^{n}\sqrt{\frac{1}{n^2}+\frac{1}{4}\left( B\left(\frac{i.y}{n}\right)-B\left(\frac{(i-1).y}{n}\right) \right)^2}\) for \(n\) large enough
## Hydraulic radius (m)
$$R = S / P$$
## Average speed (m/s)
$$U = Q /S$$
## Specific head (m)
$$H(y) = y + \frac{U^2}{2g}$$
## Head loss (m/m)
Cassiopée uses Manning Strickler formula:
$$J=\frac{U^2}{K^{2}R^{4/3}}=\frac{Q^2}{S^2K^{2}R^{4/3}}$$
## Linear variation of specific energy (m/m)
$$\Delta E_s = I_f - J$$
## Normal depth (m)
[See the uniform flow calculation.](regime_uniforme.md)
## Froude number
The Froude number expresses the ratio between the mean fluid velocity and the surface wave velocity. \(c\).
$$ c = \sqrt{\frac{gS}{B}}$$
$$ Fr = \frac{U}{c} = \sqrt{\frac{Q^2B}{gS^3}}$$
## Critical head (m)
The critical height is reached when the average velocity of the fluid is equal to the velocity of the waves on the water surface.
The critical height is therefore reached when the Froude number \(Fr=1\).
For any section, the critical height is calculated as follows \(y_c\) by solving \(f(y_c)=Fr^2-1=0\)
We use Newton's method by posing \(y_{k+1} = y_k - \frac{f(y_k)}{f'(y_k)}\) with :
- \(f(y_k) = \frac{Q^2 B}{g S^3} - 1\)
- \(f'(y_k) = \frac{Q^2}{g} \frac{B'.S - 3 B S'}{S^4}\)
## Critical head (m)
This is the head calculated for a draft equal to the critical depth. \(H_c = H(y_c)\).
## Corresponing depth (m)
For a fluvial (respectively torrential draft) \(y\), corresponding depth is the torrential (respectively fluvial) draft for the which \(H(y) = H(y_{cor})\).
## Hydraulic impulsion (kg⋅m⋅s<sup>-1</sup>)
The impulsion \(I\) is the sum of the amount of movement and the resultant of the pressure force in a section:
$$I=\rho Q U + \rho g S y_g$$
With :
- \(\rho\) : the density of water (kg/m<sup>3</sup>)
- \(y_g\) : the distance from the centre of gravity of the section to the free surface (m)
The distance from the centre of gravity of the section to the free surface \(y_g\) can be found from the formula :
$$S.y_g = \int_{0}^{y} (y-z)B(z)dz$$
With \(y\) the depth and \(B(z)\) the width at mirror for a draft \(z\)
Formulas of \(S.y_g\) for the different section shapes are :
- rectangular section: \(S.y_g = \frac{L.y^2}{2}\)
- trapezoidal section: \(S.y_g = \left (\frac{L}{2} + \frac{m.y}{3} \right )y^2\)
- circular section: \(S.y_g = \frac{D^3}{8}\left (\sin\theta - \frac{\sin^3\theta}{3} - \theta \cos\theta \right )\)
- parabolic section: \(S.y_g=\frac{B_b.y^{k+2}}{y_b^k(k+1)(k+2)}\)
## Tractive force (Pa)
$$ \tau_0 = \rho g R J $$
# Manning-Strickler's formula
## Definition
Manning-Strickler formula is written as follows:
$$V = K_s R_h^{2/3} i^{1/2}$$
with:
- \(V\) la vitesse moyenne de la section transversale en m/s
- \(K_s\) Strickler's coefficient
- \(R_h\) Hydraulic radius in m
- \(i\) slope en m/m
The Strickler coefficient \(K_s\) varies from 20 (rough stone and rough surface) to 80 (smooth concrete and cast iron).
## Chow's table (1959)
<table>
<thead>
<tr>
<th></th>
<th colspan="3">Strickler coefficient KS</th>
<th colspan="3">Manning coefficient n</th>
</tr>
</thead>
<tbody>
<tr>
<td>Type of channel and description</td>
<td>Minimum</td>
<td>Normal</td>
<td>Maximum</td>
<td>Minimum</td>
<td>Normal</td>
<td>Maximum</td>
</tr>
<tr>
<td colspan="7">Natural streams - minor streams (top width at floodstage < 30 m / 100 ft)
</td>
</tr>
<tr>
<td>1. Main channels</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>  a. clean, straight, full stage, no rifts or deep pools
</td>
<td>40</td>
<td>33</td>
<td>30</td>
<td>0,025</td>
<td>0,03</td>
<td>0,033</td>
</tr>
<tr>
<td>  b. same as above, but more stones and weeds
</td>
<td>33</td>
<td>29</td>
<td>25</td>
<td>0,03</td>
<td>0,035</td>
<td>0,04</td>
</tr>
<tr>
<td>  c. clean, winding, some pools and shoals
</td>
<td>30</td>
<td>25</td>
<td>22</td>
<td>0,033</td>
<td>0,04</td>
<td>0,045</td>
</tr>
<tr>
<td>  d. same as above, but some weeds and stones
</td>
<td>29</td>
<td>22</td>
<td>20</td>
<td>0,035</td>
<td>0,045</td>
<td>0,05</td>
</tr>
<tr>
<td>  e. same as above, lower stages, more ineffective slopes and sections
</td>
<td>25</td>
<td>21</td>
<td>18</td>
<td>0,04</td>
<td>0,048</td>
<td>0,055</td>
</tr>
<tr>
<td>  f. same as "d" with more stones
</td>
<td>22</td>
<td>20</td>
<td>17</td>
<td>0,045</td>
<td>0,05</td>
<td>0,06</td>
</tr>
<tr>
<td>  g. sluggish reaches, weedy, deep pools
</td>
<td>20</td>
<td>14</td>
<td>13</td>
<td>0,05</td>
<td>0,07</td>
<td>0,08</td>
</tr>
<tr>
<td>  h. very weedy reaches, deep pools, or floodways  with heavy stand of timber and underbrush</td>
<td>13</td>
<td>10</td>
<td>7</td>
<td>0,075</td>
<td>0,1</td>