Culvert formula (Water Overlay): Difference between revisions

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Flow through culverts is based on an open channel flow calculation.
Flow through [[culvert (Water Overlay)|culvert]]s is based on an open channel flow calculation.
   
   
The actual height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:
The actual datum height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:
: ''B<sub>c</sub> = max( T<sub>c</sub> , B<sub>l</sub> , B<sub>r</sub> )''
: <math>B_c = \max( T_c , B_l , B_r )</math>


The radius of the culvert:
For circular culverts, the flow height <math>h_f</math> is:
: r = D / 2.
: <math>h_f = \max(0,\min(D,\max (w_{l}, w_{r})-B_c)))</math>
For rectangular culverts, the flow depth h is:
: <math>h_f = \max(0,\min(H,\max (w_{l}, w_{r})-B_c)))</math>


The height of the water column at either end of the culvert, relative to the culvert, is calculated:
The loss coefficient for the culvert is calculated:
: ''w<sub>l</sub> = max(0, w<sub>l</sub> - B<sub>c</sub>)''
: <math>U = \sqrt{ \frac{1}{ 1 + \frac{2g \cdot n^2 \cdot  length }{  R_{h}^{\frac{4}{3}}} }}</math>
: ''w<sub>r</sub> = max(0, w<sub>r</sub> - B<sub>c</sub>)''
[[File:Hydraulic_radius.png|right]]
Flow depth d is:
: ''d = max ( 0 , min ( D , max ( w<sub>l</sub> , w<sub>r</sub> ) - B<sub>c</sub> ) ) )''


The loss coefficient for the culvert is calculated:
The hydraulic radius <math>R_h</math> is calculated as:
: U = sqrt( 1 / ( 1 + 2 * g * n<sup>2</sup> * length /  R<sub>h</sub><sup>4/3</sup> ) )
: <math>R_h = \frac{A}{P_w}</math>
 
For circular culverts, the flow area <math>A</math> and the wetted perimeter <math>P_w</math> is calculated using the formula's in the image below.
[[File:Hydraulic_radius.png|left]]{{clear}}
Where the radius of the culvert:
: <math>r = \frac{D}{2}</math>


The hydraulic radius R<sub>h</sub> is calculated using the formula's in the image on the right.
For rectangular culverts, the flow area and wetted perimeter is calculated as followed:
: <math>A = h_f \cdot D</math>
<math>
P_w =
\begin{cases}
D + 2h_f, & \text{if } h_f < H \\
2D + 2H, & \text{if } h_f \ge H
\end{cases}
</math>      


The potential flow through the culvert is then calculated:
The potential flow through the culvert is then calculated:
: ''C = U * K * sqrt( 2 * g * abs( w<sub>l</sub> - w<sub>r</sub> ) )''
: <math>Q = U \cdot K \cdot \sqrt{ 2g \cdot \Vert w_{l} - w_{r} \Vert }</math>


Finally the actual amount of water flow is calculated:
Finally the actual amount of water flow is calculated:
: ''Δf = ''Δt * C / Δx''
: <math>\Delta f = \frac{\Delta t \cdot Q} { \Delta x}</math>


Where:
Where:
* D = The [[Culvert diameter (Water Overlay)|CULVERT_DIAMETER]] attribute of the culvert.
* <math>D</math> = The [[Culvert diameter (Water Overlay)|CULVERT_DIAMETER]] attribute of the culvert, representing either the inside diameter of a circular culvert or the inside width of a rectangular culvert.
* T<sub>c</sub> = The [[Culvert threshold (Water Overlay)|CULVERT_THRESHOLD]] attribute of the culvert.
* <math>H</math> = The [[Culvert diameter (Water Overlay)|CULVERT_RECTANGULAR_HEIGHT]] attribute of the culvert, representing the inside height of a rectangular culvert.
* B<sub>c</sub> = The [[Terrain height (Water Overlay)|surface height]] of the base of the culvert.
* <math>h_f</math> = The (flow) height of the water inside the culvert.
* w<sub>left</sub> = The [[Surface water level formula (Water Overlay)|water level]] on the left side of the culvert, relative to {{datum}}.
* <math>T_c</math> = The [[Culvert threshold (Water Overlay)|CULVERT_THRESHOLD]] attribute of the culvert.
* w<sub>right</sub> = The [[Surface water level formula (Water Overlay)|water level]] on the right side of the culvert, relative to {{datum}}.
* <math>B_c</math> = The [[Terrain height (Water Overlay)|datum height]] of the base of the culvert.
* R<sub>h</sub> = The hydraulic radius in the culvert<ref name="hydradius"/>.
* <math>w_{l}</math> = The [[Surface water level formula (Water Overlay)|water level]] on the left end of the culvert, relative to {{datum}}.
* K = Flow area, based on the height of the water in the (circular) culvert.
* <math>w_{r}</math> = The [[Surface water level formula (Water Overlay)|water level]] on the right end of the culvert, relative to {{datum}}.
* g = Acceleration factor of [[Gravity (Water Overlay)|GRAVITY]], defined for the Water Overlay.
* <math>R_h</math> = The hydraulic radius in the culvert<ref name="hydradius"/>.
* L = The length of the culvert, calculated as the distance between the culvert's endpoints.
* <math>R_w</math> = The wetted perimeter.
* U = Loss coefficient for  culverts.
* <math>A</math> = The flow area.
* n = The [[Culvert n (Water Overlay)|CULVERT_N]] attribute of the culvert.
* <math>K</math> = Circular flow area, based on the height of the water in the (circular) culvert.
* C = The potential rate of water flow through the culvert.
* <math>g</math> = {{gravity}}.
* Δf = The water flow which takes place.
* <math>L</math> = The length of the culvert, calculated as the distance between the culvert's endpoints.
* Δt = Computational timestep.
* <math>U</math> = Loss coefficient for  culverts.
* Δx = Cell size.
* <math>n</math> = The [[Culvert n (Water Overlay)|CULVERT_N]] attribute of the culvert.
==See also==
* <math>Q</math> = The potential rate of water flow through the culvert in <math>m^3s^{-1}</math>.
* [[Culvert_(Water_Overlay)|Culvert]]
* <math>\Delta f</math> = The water flow which takes place.
==References==
* <math>\Delta t</math> = Computational [[Timestep formula (Water Overlay)|timestep]] in seconds.
* <math>\Delta x</math> = Cell size in meters.
 
==Related==
The following topics are related to this formula.
; Structures
: [[Culvert (Water Overlay)|Culvert]]
; Models
: [[Surface model (Water Overlay)|Surface model]]
 
{{article end
|references=
<references>
<references>
<ref name="hydradius">Hydraulic Radius Equations Formulas Calculator ∙ found at: https://www.ajdesigner.com/phphydraulicradius/hydraulic_radius_equation_pipe.php ∙ (last visited 2019-02-11)</ref>
<ref name="hydradius">Hydraulic Radius Equations Formulas Calculator ∙ found at: https://www.ajdesigner.com/phphydraulicradius/hydraulic_radius_equation_pipe.php ∙ (last visited 2019-02-11)</ref>
</references>
</references>
 
}}
{{Template:WaterOverlay_nav}}
{{WaterOverlay formula nav}}
{{Water Module buttons}}

Latest revision as of 13:20, 5 March 2024

Flow through culverts is based on an open channel flow calculation.

The actual datum height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:

For circular culverts, the flow height is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_f = \max(0,\min(D,\max (w_{l}, w_{r})-B_c)))}

For rectangular culverts, the flow depth h is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_f = \max(0,\min(H,\max (w_{l}, w_{r})-B_c)))}

The loss coefficient for the culvert is calculated:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \sqrt{ \frac{1}{ 1 + \frac{2g \cdot n^2 \cdot length }{ R_{h}^{\frac{4}{3}}} }}}

The hydraulic radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_h} is calculated as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_h = \frac{A}{P_w}}

For circular culverts, the flow area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and the wetted perimeter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_w} is calculated using the formula's in the image below.

Hydraulic radius.png

Where the radius of the culvert:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \frac{D}{2}}

For rectangular culverts, the flow area and wetted perimeter is calculated as followed:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = h_f \cdot D}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_w = \begin{cases} D + 2h_f, & \text{if } h_f < H \\ 2D + 2H, & \text{if } h_f \ge H \end{cases} }

The potential flow through the culvert is then calculated:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q = U \cdot K \cdot \sqrt{ 2g \cdot \Vert w_{l} - w_{r} \Vert }}

Finally the actual amount of water flow is calculated:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta f = \frac{\Delta t \cdot Q} { \Delta x}}

Where:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} = The CULVERT_DIAMETER attribute of the culvert, representing either the inside diameter of a circular culvert or the inside width of a rectangular culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} = The CULVERT_RECTANGULAR_HEIGHT attribute of the culvert, representing the inside height of a rectangular culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_f} = The (flow) height of the water inside the culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c} = The CULVERT_THRESHOLD attribute of the culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_c} = The datum height of the base of the culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{l}} = The water level on the left end of the culvert, relative to datum.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{r}} = The water level on the right end of the culvert, relative to datum.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_h} = The hydraulic radius in the culvert[1].
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_w} = The wetted perimeter.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} = The flow area.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} = Circular flow area, based on the height of the water in the (circular) culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} = the acceleration due to gravity, set to 9.80665.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} = The length of the culvert, calculated as the distance between the culvert's endpoints.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} = Loss coefficient for culverts.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} = The CULVERT_N attribute of the culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} = The potential rate of water flow through the culvert in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^3s^{-1}} .
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta f} = The water flow which takes place.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} = Computational timestep in seconds.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x} = Cell size in meters.

Related

The following topics are related to this formula.

Structures
Culvert
Models
Surface model

References

  1. ↑ Hydraulic Radius Equations Formulas Calculator ∙ found at: https://www.ajdesigner.com/phphydraulicradius/hydraulic_radius_equation_pipe.php ∙ (last visited 2019-02-11)
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