Weir formula (Water Overlay): Difference between revisions

Latest revision as of 13:21, 5 March 2024

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:

w_{u,t}=max(w_{l,t},w_{r,t})

Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) $z_{w,t}$ 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.

z_{w,t}^{*}={\begin{cases}z_{th},&{\text{if}}&\|{w_{u,t}-\tau _{w}}\|>\mu {\text{ and }}\tau _{w}>-10000\\z_{w,t-1}^{*},&{\text{if}}&T_{wm}>T\\z_{w,t},&{\text{otherwise}}\end{cases}}

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

z_{min,w}=max(z_{w,t}-\rho ,min(z_{b,l},z_{b,r}))

z_{max,w}={\begin{cases}z_{w,t},&{\text{if}}&w_{u,t}<z_{w,t-1}^{*}\\z_{w,t}+\rho ,&{\text{otherwise}}\end{cases}}

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

z_{th}={\begin{cases}min(z_{max,w},max(z_{min,w},z_{w,t-1}^{*}+\mu )),&{\text{if}}&w_{u,t}<\tau _{w}\\min(z_{max,w},max(z_{min,w},z_{w,t-1}^{*}-\mu )),&{\text{otherwise}}\end{cases}}

T_{wm}=T+t_{wm}

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:

h_{s}=max(0,max(w_{l,t},w_{r,t})-z_{w,t})

h_{d}=max(0,min(w_{l,t},w_{r,t})-z_{w,t})

Based on the relative water heights, the weir uses either a submerged flow $Q_{s}$ or a free flow $Q_{f}$ formula, based on the following ratio:

r_{h}={\frac {h_{d}}{h_{s}}}

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:

Q_{f}=f_{dw}\cdot C_{w}\cdot b\cdot (h_{s}-h_{d})^{3/2}

For submerged flow, the following calculation is used:

Q_{s}=f_{loss}\cdot A\cdot {\sqrt {2\cdot g\cdot (h_{s}-h_{d})}}

with:

A=b\cdot (h_{s}-h_{d})

Finally the actual amount of water level change is calculated:

\Delta w={\frac {\Delta t\cdot Q}{\Delta x\cdot \Delta x}}

Where:

$w_{l,t}$ = The water level on the left side of the weir, relative to datum, at time $t$ .
$w_{r,t}$ = The water level on the right side of the weir, relative to datum, at time $t$ .
$w_{u,t}$ = The calculated upstream water level, relative to datum, at time $t$ .

$z_{th}$ = The height of the weir at time t, according to the height adjustment mechanism.
$\tau _{w}$ = The WEIR_TARGET_LEVEL of the the weir.
$T_{wm}$ = Moment in time in seconds after which the weir can adjust its height again.
$z_{w,t}$ = The WEIR_HEIGHT of the weir, at time $t$ .
$z_{w,t}^{*}$ = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
$z_{b,l}$ = The water bottom elevation at left side of the weir the weir.
$z_{b,r}$ = The water bottom elevation at the right side of the weir.
$z_{min,w}$ = The minimum allowed height for weir w.
$z_{max,w}$ = The maximum allowed height for weir w.
$\mu$ = The WEIR_MOVE_STEP_M of the water overlay, applicable to all weirs.
$\rho$ = The WEIR_MOVE_RANGE_M of the water overlay, applicable to all weirs.
$T$ = Current simulated time in seconds.
$t_{wm}$ = The WEIR_MOVE_INTERVAL_S of the water overlay, applicable to all weirs.

$h_{s}$ = The height of the water column relative to the top of the weir, on the side with the highest water level, at time $t$ .
$h_{d}$ = The height of the water column relative to the top of the weir, on the side with the lowest water level, at time $t$ .
$r_{h}$ = The ratio of water heights on either side of the culvert.
$Q$ = The potential rate of water flow across the weir.
$Q_{f}$ = The potential rate of water flow across the weir, based on a free flow calculation.
$Q_{s}$ = The potential rate of water flow across the weir, based on a submerged calculation.
$f_{dw}$ = Dutch weir factor, set to 1.7.
$C_{w}$ = The WEIR_COEFFICIENT of the weir.
$b$ = The breadth of weir crest, adjustable using the WEIR_WIDTH.
$f_{loss}$ = Loss coefficient for submerged weirs, set to 0.9.
$A$ = Flow area, based on the highest water column height relative to the top of the weir, and WEIR_WIDTH of the weir.
$g$ = the acceleration due to gravity, set to 9.80665.
$\Delta w$ = The water level change in meters, which takes place.
$\Delta t$ = Computational timestep in seconds.
$\Delta x$ = Cell size in meters.

Free flow coefficients

To clarify the free flow formula in comparison with the ISO^[1] standard definitions: When $h\approx 0.01$ meters, the following holds.

f_{dw}\cdot C_{w}\approx C_{d}\cdot {\sqrt {g}}

C_{d}=0,633\cdot (1-{\frac {0.0003}{h}})^{\frac {3}{2}}

where:

$f_{dw}$ = Dutch weir factor, set to 1.7.
$C_{w}$ = The WEIR_COEFFICIENT of the weir, set to a default of 1.1
$C_{d}$ = Calculated Coefficient of discharge.
$g$ = acceleration due to gravity.
$h$ = Weir head; the height of the water column relative to the top of the weir.

For $h$ 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.

References

↑ 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)

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

[ref-ISO-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)

[1]

@@ Line 1: / Line 1: @@
-Flow across [[Weir (Water Overlay)|weir]]s is calculated differently for free flow and submerged flow.
+Flow across [[Weir (Water Overlay)|weir]]s 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.
-The height of the water at each end of the weir, relative to the weir, is calculated:
+First the upstream water level is calculated as followed:
-: <math>h_s = max(0, max( w_l, w_r ) - z_{w,t})</math>
-: <math>h_d = max(0, min( w_l, w_r ) - 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:
+:<math>w_{u,t} =  max( w_{l,t}, w_{r,t})</math>
-:<math>z_{w,t} =
+Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) <math>z_{w,t}</math> 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.
+:<math>z^{*}_{w,t} =
 \begin{cases}
-z_b, & \mbox{if }t_{wd,t} > 0 \\
+z_{th}, & \text{if} & \|{w_{u,t} - \tau_w}\| > \mu \text{ and } \tau_w > -10000 \\
-z_w, & \mbox{otherwise }
+z^{*}_{w,t-1}, & \text{if} & T_{wm} > T \\
+z_{w,t}, & \text{otherwise}
 \end{cases}
 </math>
-:<math>t_{wd,t} =
+The height adjustment range values, defined by the min and max, are determined next:
+:<math>z_{min,w} = max ( z_{w,t} - \rho, min (z_{b,l}, z_{b,r} ) )</math>
+:<math>z_{max,w} =
 \begin{cases}
-t_{wd}, \mbox{if } max( w_l, w_r) > w_{dt} \mbox{ and } t_{wd,t} <= 0 \\
+z_{w,t}, & \text{if} &  w_{u,t} < z^{*}_{w,t-1} \\
-t_{wd, t-1} - \Delta t, \mbox{otherwise}
+z_{w,t} + \rho, & \text{otherwise}
 \end{cases}
 </math>
+Finally, the adjusted weir height is calculated and stored, as well as the moment in time at which the weir can be adjusted again.
-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_{th} =
-: <math>h_{ratio} = \frac{h_d}{h_s}</math>
-: <math>Q =
 \begin{cases}
-min ( Q_{submerged}, Q_{free}) & \mbox{if } h_{ratio} > 0.5 \\
+min(z_{max,w}, max( z_{min,w}, z^{*}_{w,t-1} + \mu )), & \text{if} & w_{u,t} < \tau_w \\
-Q_{free}, & \mbox{otherwise}
+min(z_{max,w}, max( z_{min,w}, z^{*}_{w,t-1} - \mu )), & \text{otherwise}
 \end{cases}
 </math>
+:<math> T_{wm} = T + t_{wm}</math>
+:<math>z_{w,t} = z^{*}_{w,t}</math>
+Now knowing the height of the weir, 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,t}, w_{r,t} ) - z_{w,t})</math>
+: <math>h_d = max(0, min( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
+Based on the relative water heights, the weir uses either a submerged flow <math>Q_s</math> or a free flow <math>Q_f</math> formula, based on the following ratio:
+: <math>r_h = \frac{h_d}{h_s}</math>
+:<math>
+Q =
+\begin{cases}
+min ( Q_s , Q_f), & \text{if } r_h > 0.5 \\
+ Q_f, & \text{otherwise}
+\end{cases}
+</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_{dw} \cdot C_w \cdot b \cdot ( h_s - h_d )^{3/2}</math>
 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 = f_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }</math>
-Finally the actual amount of water flow is calculated:
+with:
-: <math>\Delta w = \Delta t \cdot \frac{Q}{\Delta x}</math>
+: <math>A = b \cdot (h_s-h_d)</math>
+Finally the actual amount of water level change is calculated:
+: <math>\Delta w = \frac{\Delta t \cdot Q}{\Delta x \cdot \Delta x}</math>
 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>w_{u,t}</math> = The calculated upstream water level, relative to {{datum}}, at time <math>t</math>.
-* <math>w_r</math> = The water level on the right side of the weir, relative to {{datum}}.
-* <math>w_dt</math> = The [[Weir drop threshold (Water Overlay)|weir drop threshold]] of the weir.
+* <math>z_{th}</math> = The height of the weir at time t, according to the height adjustment mechanism.
-* <math>z_{w,t}</math> = The height of the weir at time t, depending on drop mechanism.
+* <math>\tau_w</math> = The [[Weir target level (Water Overlay)|WEIR_TARGET_LEVEL]] of the the weir.
-* <math>z_w</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir.
+* <math>T_{wm}</math> = Moment in time in seconds after which the weir can adjust its height again.
-* <math>z_b</math> = The water bottom elevation at the weir.
+* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
-* <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>z^{*}_{w,t}</math> = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
-* <math>t_{wd,t}</math> = The remaining time the weir is dropped.
+* <math>z_{b,l}</math> = The water bottom elevation at left side of the weir the weir.
-* <math>f_{w,d}</math> = Dutch weir factor, set to 1.7.
+* <math>z_{b,r}</math> = The water bottom elevation at the right side of the weir.
-* <math>c_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
+* <math>z_{min,w}</math> = The minimum allowed height for weir w.
-* <math>w_w</math> = The [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
+* <math>z_{max,w}</math> = The maximum allowed height for weir w.
+* <math>\mu</math> = The [[Weir move step m (Water Overlay)|WEIR_MOVE_STEP_M]] of the water overlay, applicable to all weirs.
+* <math>\rho</math> = The [[Weir move range m (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> = The [[Weir move interval s (Water Overlay)|WEIR_MOVE_INTERVAL_S]] of the water overlay, applicable to all weirs.
+* <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>r_h</math> = The ratio of water heights on either side of the culvert.
 * <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>Q_f</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_s</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>f_{dw}</math> = Dutch weir factor, set to 1.7.
-* <math>U_{loss}</math> = Loss coefficient for submerged weirs, set to 0.9.
+* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
+* <math>b</math> = The breadth of weir crest, adjustable using the [[Weir width (Water Overlay)|WEIR_WIDTH]].
+* <math>f_{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>g</math> = Acceleration factor of gravity, set to 9.80665.
+* <math>g</math> = {{gravity}}.
-* <math>\Delta w</math> = The water flow which takes place.
+* <math>\Delta w</math> = The water level change in meters, which takes place.
-* <math>\Delta t</math> = Computational timestep.
+* <math>\Delta t</math> = Computational [[Timestep formula (Water Overlay)|timestep]] in seconds.
-* <math>\Delta x</math> = Cell size.
+* <math>\Delta x</math> = Cell size in meters.
+===Free flow coefficients===
+To clarify the free flow formula in comparison with the ISO{{ref|ISO}} standard definitions:
+When <math>h \approx 0.01</math> meters, the following holds.
+:<math>f_{dw} \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>
+where:
+* <math>f_{dw}</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.
+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.
 ==Related==
 The following topics are related to this formula.
 ; Structures
-: [[Weir_(Water_Overlay)|Weir]]
+: [[Weir (Water Overlay)|Weir]]
+: [[Breach (Water Overlay)|Breach]]
 ; Models
 : [[Surface model (Water Overlay)|Surface model]]
+; Formulas
+: [[Breach flow formula (Water Overlay)|Breach flow formula]]
+; Test cases
+: [[Weir height test case (Water Module)|Weir height test case]]
+{{Article end
+| references=<references>
+  {{ref|ISO
+    |name=ISO FDIS 4360, 2020 Edition, March 3, 2020 - HYDROMETRY - OPEN CHANNEL FLOW MEASUREMENT USING TRIANGULAR PROFILE WEIRS
+    |author=Technical Committee: ISO/TC 113/SC 2 Flow measurement structures
+    |page=
+    |source=
+    |link=https://www.iso.org/standard/70915.html
+    |lastvisit=28-11-2022
+  }}
+</references>
+}}
 {{WaterOverlay formula nav}}

Weir formula (Water Overlay): Difference between revisions

Latest revision as of 13:21, 5 March 2024

Free flow coefficients

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