Weir formula (Water Overlay): Difference between revisions

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* <math>w_{l,t}</math> = The water level on the left side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>w_{l,t}</math> = The water level on the left side of the weir, relative to {{datum}}, at time <math>t</math>.
* <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_{r,t}</math> = The water level on the right side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>w_{u,t}</math> = The calculated upstream water level, relative to {{datum}}, at time <math>t</math>.
* <math>z_{th}</math> = The height of the weir at time t, according to the height adjustment mechanism.
* <math>\tau_w</math> = The [[Weir target level (Water Overlay)|WEIR_TARGET_LEVEL]] of the the weir.
* <math>T_{wm}</math> = Moment in time in seconds after which the weir can adjust its height again.
* <math>T</math> = Current simulated time in seconds.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
* <math>z^{*}_{w,t}</math> = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
* <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>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>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>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>w_{dt}</math> = The [[Weir drop threshold (Water Overlay)|weir drop threshold]] of the weir.
* <math>w_{dt}</math> = The [[Weir drop threshold (Water Overlay)|weir drop threshold]] of the weir.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir at time <math>t</math>.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir at time <math>t</math>.
* <math>\tau_w</math> = The [[Weir target level (Water Overlay)|WEIR_TARGET_LEVEL]] of the the weir.
* <math>\mu</math> = The [[Weir move step m model attribute (Water Overlay)|WEIR_MOVE_STEP_M]] of the water overlay, applicable to all weirs.
* <math>\mu</math> = The [[Weir move step m model attribute (Water Overlay)|WEIR_MOVE_STEP_M]] of the water overlay, applicable to all weirs.
* <math>\rho</math> = The [[Weir move range m model attribute (Water Overlay)|WEIR_MOVE_RANGE_M]] of the water overlay, applicable to all weirs.
* <math>\rho</math> = The [[Weir move range m model attribute (Water Overlay)|WEIR_MOVE_RANGE_M]] of the water overlay, applicable to all weirs.
* <math>T</math> = Current simulated time in seconds.
* <math>T_{wm}</math> = Moment in time in seconds after which the weir can adjust its height again.
* <math>t_{wm}</math> = The [[Weir move interval model attribute (Water Overlay)|Weir move interval]] of the water overlay, applicable to all weirs.
* <math>t_{wm}</math> = The [[Weir move interval model attribute (Water Overlay)|Weir move interval]] of the water overlay, applicable to all weirs.
* <math>z^{*}_{w,t}</math> = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
* <math>z_{th}</math> = The height of the weir at time t, according to the height adjustment mechanism.
* <math>z_{b,l}</math> = The water bottom elevation at left side of the weir the weir.
* <math>z_{b,l}</math> = The water bottom elevation at left side of the weir the weir.
* <math>z_{b,r}</math> = The water bottom elevation at the right side of the weir.
* <math>z_{b,r}</math> = The water bottom elevation at the right side of the weir.

Revision as of 10:47, 16 December 2022

Flow across weirs is calculated differently for free flow and submerged flow. Optionally, the height of the weir can variate based on a provided height values or an automatic adjustment. Therefore, we determined the height of the weir first.

First the upstream water level is calculated as followed:

Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) or it is adjusted according to the weir threshold level. When it is adjusted, a moment in time is set for that weir, during which it cannot be adjusted.

The height adjustment range values, defined by the min and max, are determined next:

Finally, the adjusted weir height is calculated and stored, as well as the moment in time at which the weir can be adjusted again.

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_{wm} = T + t_{wm}}
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} = z^{*}_{w,t}}

Now knowing the height of the weir, 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})}
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 = max(0, min( 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 r_h = \frac{h_d}{h_s}}
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 w_{u,t}} = The calculated upstream water level, 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_{th}} = The height of the weir at time t, according to the height adjustment mechanism.
  • 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 \tau_w} = The WEIR_TARGET_LEVEL of the 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 T_{wm}} = Moment in time in seconds after which the weir can adjust its height again.
  • 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} = Current simulated time 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 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 z^{*}_{w,t}} = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.


  • 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 w_{dt}} = The weir drop threshold 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 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 \mu} = The WEIR_MOVE_STEP_M of the water overlay, applicable to all weirs.
  • 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 \rho} = The WEIR_MOVE_RANGE_M of the water overlay, applicable to all weirs.
  • 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_{wm}} = The Weir move interval of the water overlay, applicable to all weirs.
  • 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_{b,l}} = The water bottom elevation at left side of the weir the weir.
  • = The water bottom elevation at the right side 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 t_{wd}} = The total amount of time a weir is dropped until the drop threshold condition is re-evaluated.
  • 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_{wd,t}} = The remaining time the weir is dropped.
  • 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 the ISO[1] standard 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}}

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.

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.

Related

The following topics are related to this formula.

Structures
Weir
Models
Surface model

References

  1. ↑ ISO FDIS 4360, 2020 Edition, March 3, 2020 - HYDROMETRY - OPEN CHANNEL FLOW MEASUREMENT USING TRIANGULAR PROFILE WEIRS ∙ Technical Committee: ISO/TC 113/SC 2 Flow measurement structures ∙ Found at: https://www.iso.org/standard/70915.html ∙ (last visited: 28-11-2022)
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