Culvert formula (Water Overlay): Difference between revisions

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:

B_{c}=\max(T_{c},B_{l},B_{r})

For circular culverts, the flow height $h_{f}$ 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.

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.

References

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

Models

Surface • Ground • Rain • Evaporation • Infiltration • Elevation • Breach • Sewer • Storage • Tracer flow • Border • Order
Model formulas

Timestep • Season • Surface water level • Groundwater level
Flow formulas

Surface flow • Surface infiltration • Ground flow • Ground infiltration • Surface evaporation • Ground evaporation • Ground bottom flow
Hydraulic structure formulas

Culvert • Weir • Breach growth • Breach flow • Pump • Sewer Overflow • Inlet • Drainage
Related

Building attributes • Hydrological features • Hydraulic structures • Terrain attributes • Model attributes • Results • Simulation data

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

[1]

@@ Line 1: / Line 1: @@
 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:
 : <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:
-: <math>r = \frac{D}{2}</math>
+: <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 loss coefficient for the culvert is calculated:
+: <math>U = \sqrt{ \frac{1}{ 1 + \frac{2g \cdot n^2 \cdot  length }{  R_{h}^{\frac{4}{3}}} }}</math>
-The height of the water column at either end of the culvert, relative to the culvert, is calculated:
+The hydraulic radius <math>R_h</math> is calculated as:
-: <math>w_l = \max(0,w_l-B_c)</math>
+: <math>R_h = \frac{A}{P_w}</math>
-: <math>w_r = \max(0,w_r-B_c)</math>
-[[File:Hydraulic_radius.png|right]]
-Flow depth d is:
-: <math>d = \max(0,\min(D,\max (w_l, w_r)-B_c)))</math>
-The loss coefficient for the culvert is calculated:
+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.
-: <math>U = \sqrt{ \frac{1}{ 1 + 2 * g * n^2 * length /  R_{h}^{\frac{4}{3}} }}</math>
+[[File:Hydraulic_radius.png|left]]{{clear}}
+Where the radius of the culvert:
+: <math>r = \frac{D}{2}</math>
-The hydraulic radius R_h 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 \\
+D + 2H, & \text{if } h_f \ge H
+\end{cases}
+</math>
 The potential flow through the culvert is then calculated:
-: <math>C = U * K * \sqrt( 2 * g * abs( w_l - w_r ) )</math>
+: <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:
-: <math>Δf = ''Δt * C / Δx</math>
+: <math>\Delta f = \frac{\Delta t \cdot Q} { \Delta x}</math>
 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.
+* <math>Q</math> = The potential rate of water flow through the culvert in <math>m^3s^{-1}</math>.
+* <math>\Delta f</math> = The water flow which takes place.
+* <math>\Delta t</math> = Computational [[Timestep formula (Water Overlay)|timestep]] in seconds.
+* <math>\Delta x</math> = Cell size in meters.
 ==Related==
@@ Line 49: / Line 64: @@
 : [[Surface model (Water Overlay)|Surface model]]
-==References==
+{{article end
+|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>
 </references>
+}}
 {{WaterOverlay formula nav}}

Culvert formula (Water Overlay): Difference between revisions

Latest revision as of 13:20, 5 March 2024

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