Weir formula (Water Overlay): Difference between revisions

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The height of the water at each end of the weir, relative to the weir, is calculated:
The height of the water at each end of the weir, relative to the weir, is calculated:
: <math>h_s = max(0, max( w_l, w_r ) - z_{w,t})</math>
: <math>h_s = max(0, max( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
: <math>h_d = max(0, min( w_l, w_r ) - z_{w,t})</math>
: <math>h_d = max(0, min( w_{l,t}, w_{r,t} ) - z_{w,t})</math>


Optionally, when a Weir drop threshold is configured, the weir height <math>z_w,t</math> at time <math>t</math> can update depending on the water level and the remaining weir drop time:
Based on the relative water heights, the weir is judged to have either a submerged flow or a free flow, based on the following ratio:
:<math>z_{w,t} =
: <math>r_h = \frac{h_d}{h_s}</math>
\begin{cases}
:<math>
z_b, & \mbox{if }t_{wd,t} > 0 \\
Q =
z_w, & \mbox{otherwise }  
\end{cases}
</math>
:<math>t_{wd,t} =  
\begin{cases}
\begin{cases}
t_{wd}, & \mbox{if } max( w_l, w_r) > w_{dt} \mbox{ and } t_{wd,t} <= 0 \\
min ( Q_s , Q_f), & \text{if } r_h > 0.5 \\
t_{wd, t-1} - \Delta t, & \mbox{otherwise}
Q_f, & \text{otherwise}
\end{cases}
</math>
 
 
Based on the relative water heights, the weir is judged to have either a submerged flow or a free flow, based on the following ratio:
: <math>h_{ratio} = \frac{h_d}{h_s}</math>
: <math>Q =
\begin{cases}
min ( Q_{submerged}, Q_{free}) & \mbox{if } h_{ratio} > 0.5 \\
Q_{free}, & \mbox{otherwise}
\end{cases}
\end{cases}
</math>
</math>


For free flow, capacity is calculated directly:
For free flow, it is calculated directly:
: <math>Q_{free} = f_{w,d} \cdot c_w \cdot w_w \cdot ( h_s - h_d )^{3/2}</math>
: <math>Q_f = f_{w} \cdot c_w \cdot b \cdot ( h_s - h_d )^{3/2}</math>


For submerged flow, the following calculation is used:
For submerged flow, the following calculation is used:
: <math>Q_{submerged} = U_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }</math>
: <math>Q_s = U_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }</math>


Finally the actual amount of water flow is calculated:
Finally the actual amount of water flow is calculated:
: <math>\Delta w = \Delta t \cdot \frac{Q}{\Delta x}</math>
: <math>\Delta w = \frac{\Delta t \cdot Q}{\Delta x}</math>


Where:
Where:
* <math>h_s</math> = The height of the water column relative to the top of the weir, on the side with the highest [[Surface water level formula (Water Overlay)|water level]].
* <math>w_{l,t}</math> = The water level on the left side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>h_d</math> = The height of the water column relative to the top of the weir, on the side with the lowest [[Surface water level formula (Water Overlay)|water level]].
* <math>w_{r,t}</math> = The water level on the right side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>w_l</math> = The water level on the left side of the weir, relative to {{datum}}.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
* <math>w_r</math> = The water level on the right side of the weir, relative to {{datum}}.
* <math>h_s</math> = The height of the water column relative to the top of the weir, on the side with the highest [[Surface water level formula (Water Overlay)|water level]], at time <math>t</math>.
* <math>w_{dt}</math> = The [[Weir drop threshold (Water Overlay)|weir drop threshold]] of the weir.
* <math>h_d</math> = The height of the water column relative to the top of the weir, on the side with the lowest [[Surface water level formula (Water Overlay)|water level]], at time <math>t</math>.
* <math>z_{w,t}</math> = The height of the weir at time t, depending on drop mechanism.
* <math>f_{w}</math> = Dutch weir factor, set to 1.7.  
* <math>z_w</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir.
* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
* <math>z_b</math> = The water bottom elevation at the weir.
* <math>C_d</math> = Corresponding Coefficient of discharge. Part of the Dutch weir factor.
* <math>t_{wd}</math> = The [[Weir drop time s (Water Overlay)|total amount of time]] a weir is dropped until the drop threshold condition is re-evaluated.
* <math>b</math> = The breadth of weir crest, adjustable using the [[Weir width (Water Overlay)|WEIR_WIDTH]].
* <math>t_{wd,t}</math> = The remaining time the weir is dropped.
* <math>f_{w,d}</math> = Dutch weir factor, set to 1.7.
* <math>c_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
* <math>w_w</math> = The [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
* <math>Q</math> = The potential rate of water flow across the weir.
* <math>Q</math> = The potential rate of water flow across the weir.
* <math>h_{ratio}</math> = The ratio of water heights on either side of the culvert.
* <math>r_h</math> = The ratio of water heights on either side of the culvert.
* <math>Q_{free}</math> = The potential rate of water flow across the weir, based on a free flow calculation.
* <math>Q_{free}</math> = The potential rate of water flow across the weir, based on a free flow calculation.
* <math>Q_{submerged}</math> = The potential rate of water flow across the weir, based on a submerged calculation.
* <math>Q_{submerged}</math> = The potential rate of water flow across the weir, based on a submerged calculation.
* <math>U_{loss}</math> = Loss coefficient for submerged weirs, set to 0.9.
* <math>U_{loss}</math> = Loss coefficient for submerged weirs, set to 0.9.
* <math>A</math> = Flow area, based on the highest water column height relative to the top of the weir, and [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
* <math>A</math> = Flow area, based on the highest water column height relative to the top of the weir, and [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
* <math>g</math> = Acceleration factor of gravity, set to 9.80665.
* <math>g</math> = acceleration due to gravity.
* <math>\Delta w</math> = The water flow which takes place.
* <math>\Delta w</math> = The water flow which takes place.
* <math>\Delta t</math> = Computational timestep.
* <math>\Delta t</math> = Computational timestep.
* <math>\Delta x</math> = Cell size.
* <math>\Delta x</math> = Cell size.
===Free flow coefficients===
To clarify the free flow formula in comparison with other definitions:
When <math>h \approx 0.01</math> meters, the following holds.
:<math>f_{w} \cdot c_w \approx C_d \cdot \sqrt{g}</math>
:<math>C_d=0,633 \cdot (1-\frac{0.0003}{h})^\frac{3}{2}</math>
For <math>h</math> generally larger than that, the flow can be calculated up to 5.5% smaller than expected when using the second pair of coefficients. However, this can be corrected with the weir coefficient if necessary.
where:
* <math>f_{w}</math> = Dutch weir factor, set to 1.7.
* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir, set to a default of 1.1
* <math>C_d</math> = Calculated Coefficient of discharge.
* <math>g</math> = acceleration due to gravity.
* <math>h</math> = Weir head; the height of the water column relative to the top of the weir.


==Related==
==Related==

Revision as of 15:32, 15 December 2022

Flow across weirs is calculated differently for free flow and submerged flow.

The height of the water at each end of the weir, relative to the weir, 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 h_s = max(0, max( w_{l,t}, w_{r,t} ) - z_{w,t})}

Based on the relative water heights, the weir is judged to have either a submerged flow or a free flow, based on the following ratio:

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 = \begin{cases} min ( Q_s , Q_f), & \text{if } r_h > 0.5 \\ Q_f, & \text{otherwise} \end{cases} }

For free flow, it is calculated directly:

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_f = f_{w} \cdot c_w \cdot b \cdot ( h_s - h_d )^{3/2}}

For submerged flow, the following calculation is used:

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_s = U_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }}

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 w = \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 w_{l,t}} = The water level on the left side of the weir, relative to datum, at time 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} .
  • 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,t}} = The water level on the right side of the weir, relative to datum, at time 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} .
  • 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 z_{w,t}} = The WEIR_HEIGHT of the weir, at time 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} .
  • 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_s} = The height of the water column relative to the top of the weir, on the side with the highest water level, at time 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} .
  • 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_d} = The height of the water column relative to the top of the weir, on the side with the lowest water level, at time 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} .
  • 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 f_{w}} = Dutch weir factor, set to 1.7.
  • 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 C_w} = The WEIR_COEFFICIENT of the weir.
  • 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 C_d} = Corresponding Coefficient of discharge. Part of the Dutch weir factor.
  • 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} = The breadth of weir crest, adjustable using the WEIR_WIDTH.
  • 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 across the weir.
  • 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 ratio of water heights on either side 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_{free}} = The potential rate of water flow across the weir, based on a free flow calculation.
  • 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_{submerged}} = The potential rate of water flow across the weir, based on a submerged calculation.
  • 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}} = Loss coefficient for submerged weirs, set to 0.9.
  • 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} = Flow area, based on the highest water column height relative to the top of the weir, and WEIR_WIDTH of the weir.
  • 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} = acceleration due to gravity.
  • 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 w} = 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.
  • 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.

Free flow coefficients

To clarify the free flow formula in comparison with other definitions: When 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 \approx 0.01} meters, the following holds.

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 f_{w} \cdot c_w \approx C_d \cdot \sqrt{g}}
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 C_d=0,633 \cdot (1-\frac{0.0003}{h})^\frac{3}{2}}

For 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} generally larger than that, the flow can be calculated up to 5.5% smaller than expected when using the second pair of coefficients. However, this can be corrected with the weir coefficient if necessary.

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 f_{w}} = Dutch weir factor, set to 1.7.
  • 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 C_w} = The WEIR_COEFFICIENT of the weir, set to a default of 1.1
  • 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 C_d} = Calculated Coefficient of discharge.
  • 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} = acceleration due to gravity.
  • 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} = Weir head; the height of the water column relative to the top of the weir.

Related

The following topics are related to this formula.

Structures
Weir
Models
Surface model
Water Module
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Water Module
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Tutorials
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Overview
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Theory
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Benchmarks
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
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