# Hydrology, weirs and open flow (fluids.open_flow)¶

fluids.open_flow.Q_weir_V_Shen(h1, angle=90)[source]

Calculates the flow rate across a V-notch (triangular) weir from the height of the liquid above the tip of the notch, and with the angle of the notch. Most of these type of weir are 90 degrees. Model from [1] as reproduced in [2].

Flow rate is given by:

$Q = C \tan\left(\frac{\theta}{2}\right)\sqrt{g}(h_1 + k)^{2.5}$
Parameters: h1 : float Height of the fluid above the notch [m] angle : float, optional Angle of the notch [degrees] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

angles = [20, 40, 60, 80, 100] Cs = [0.59, 0.58, 0.575, 0.575, 0.58] k = [0.0028, 0.0017, 0.0012, 0.001, 0.001]

The following limits apply to the use of this equation:

h1 >= 0.05 m h2 > 0.45 m h1/h2 <= 0.4 m b > 0.9 m

$\frac{h_1}{b}\tan\left(\frac{\theta}{2}\right) < 2$

Flows are lower than obtained by the curves at http://www.lmnoeng.com/Weirs/vweir.php.

References

 [1] (1, 2) Shen, John. “Discharge Characteristics of Triangular-Notch Thin-Plate Weirs : Studies of Flow to Water over Weirs and Dams.” USGS Numbered Series. Water Supply Paper. U.S. Geological Survey : U.S. G.P.O., 1981
 [2] (1, 2) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

>>> Q_weir_V_Shen(0.6, angle=45)
0.21071725775478228

fluids.open_flow.Q_weir_rectangular_Kindsvater_Carter(h1, h2, b)[source]

Calculates the flow rate across rectangular weir from the height of the liquid above the crest of the notch, the liquid depth beneath it, and the width of the notch. Model from [1] as reproduced in [2].

Flow rate is given by:

$Q = 0.554\left(1 - 0.0035\frac{h_1}{h_2}\right)(b + 0.0025) \sqrt{g}(h_1 + 0.0001)^{1.5}$
Parameters: h1 : float Height of the fluid above the crest of the weir [m] h2 : float Height of the fluid below the crest of the weir [m] b : float Width of the rectangular flow section of the weir [m] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

The following limits apply to the use of this equation:

b/b1 ≤ 0.2 h1/h2 < 2 b > 0.15 m h1 > 0.03 m h2 > 0.1 m

References

 [1] (1, 2) Kindsvater, Carl E., and Rolland W. Carter. “Discharge Characteristics of Rectangular Thin-Plate Weirs.” Journal of the Hydraulics Division 83, no. 6 (December 1957): 1-36.
 [2] (1, 2) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

>>> Q_weir_rectangular_Kindsvater_Carter(0.2, 0.5, 1)
0.15545928949179422

fluids.open_flow.Q_weir_rectangular_SIA(h1, h2, b, b1)[source]

Calculates the flow rate across rectangular weir from the height of the liquid above the crest of the notch, the liquid depth beneath it, and the width of the notch. Model from [1] as reproduced in [2].

Flow rate is given by:

$Q = 0.544\left[1 + 0.064\left(\frac{b}{b_1}\right)^2 + \frac{0.00626 - 0.00519(b/b_1)^2}{h_1 + 0.0016}\right] \left[1 + 0.5\left(\frac{b}{b_1}\right)^4\left(\frac{h_1}{h_1+h_2} \right)^2\right]b\sqrt{g}h^{1.5}$
Parameters: h1 : float Height of the fluid above the crest of the weir [m] h2 : float Height of the fluid below the crest of the weir [m] b : float Width of the rectangular flow section of the weir [m] b1 : float Width of the full section of the channel [m] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

The following limits apply to the use of this equation:

b/b1 ≤ 0.2 h1/h2 < 2 b > 0.15 m h1 > 0.03 m h2 > 0.1 m

References

 [1] (1, 2) Normen für Wassermessungen: bei Durchführung von Abnahmeversuchen an Wasserkraftmaschinen. SIA, 1924.
 [2] (1, 2) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

>>> Q_weir_rectangular_SIA(0.2, 0.5, 1, 2)
1.0408858453811165

fluids.open_flow.Q_weir_rectangular_full_Ackers(h1, h2, b)[source]

Calculates the flow rate across a full-channel rectangular weir from the height of the liquid above the crest of the weir, the liquid depth beneath it, and the width of the channel. Model from [1] as reproduced in [2], confirmed with [3].

Flow rate is given by:

$Q = 0.564\left(1+0.150\frac{h_1}{h_2}\right)b\sqrt{g}(h_1+0.001)^{1.5}$
Parameters: h1 : float Height of the fluid above the crest of the weir [m] h2 : float Height of the fluid below the crest of the weir [m] b : float Width of the channel section [m] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

The following limits apply to the use of this equation:

h1 > 0.02 m h2 > 0.15 m h1/h2 ≤ 2.2

References

 [1] (1, 2) Ackers, Peter, W. R. White, J. A. Perkins, and A. J. M. Harrison. Weirs and Flumes for Flow Measurement. Chichester ; New York: John Wiley & Sons Ltd, 1978.
 [2] (1, 2) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.
 [3] (1, 2, 3) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

Example as in [3], matches. However, example is unlikely in practice.

>>> Q_weir_rectangular_full_Ackers(h1=0.9, h2=0.6, b=5)
9.251938159899948

fluids.open_flow.Q_weir_rectangular_full_SIA(h1, h2, b)[source]

Calculates the flow rate across a full-channel rectangular weir from the height of the liquid above the crest of the weir, the liquid depth beneath it, and the width of the channel. Model from [1] as reproduced in [2].

Flow rate is given by:

$Q = \frac{2}{3}\sqrt{2}\left(0.615 + \frac{0.000615}{h_1+0.0016}\right) b\sqrt{g} h_1 +0.5\left(\frac{h_1}{h_1+h_2}\right)^2b\sqrt{g} h_1^{1.5}$
Parameters: h1 : float Height of the fluid above the crest of the weir [m] h2 : float Height of the fluid below the crest of the weir [m] b : float Width of the channel section [m] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

The following limits apply to the use of this equation:

0.025 < h < 0.8 m b > 0.3 m h2 > 0.3 m h1/h2 < 1

References

 [1] (1, 2) Normen für Wassermessungen: bei Durchführung von Abnahmeversuchen an Wasserkraftmaschinen. SIA, 1924.
 [2] (1, 2, 3) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

Example compares terribly with the Ackers expression - probable error in [2]. DO NOT USE.

>>> Q_weir_rectangular_full_SIA(h1=0.3, h2=0.4, b=2)
1.1875825055400384

fluids.open_flow.Q_weir_rectangular_full_Rehbock(h1, h2, b)[source]

Calculates the flow rate across a full-channel rectangular weir from the height of the liquid above the crest of the weir, the liquid depth beneath it, and the width of the channel. Model from [1] as reproduced in [2].

Flow rate is given by:

$Q = \frac{2}{3}\sqrt{2}\left(0.602 + 0.0832\frac{h_1}{h_2}\right) b\sqrt{g} (h_1 +0.00125)^{1.5}$
Parameters: h1 : float Height of the fluid above the crest of the weir [m] h2 : float Height of the fluid below the crest of the weir [m] b : float Width of the channel section [m] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

The following limits apply to the use of this equation:

0.03 m < h1 < 0.75 m b > 0.3 m h2 > 0.3 m h1/h2 < 1

References

 [1] (1, 2) King, H. W., Floyd A. Nagler, A. Streiff, R. L. Parshall, W. S. Pardoe, R. E. Ballester, Gardner S. Williams, Th Rehbock, Erik G. W. Lindquist, and Clemens Herschel. “Discussion of ‘Precise Weir Measurements.’” Transactions of the American Society of Civil Engineers 93, no. 1 (January 1929): 1111-78.
 [2] (1, 2) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

>>> Q_weir_rectangular_full_Rehbock(h1=0.3, h2=0.4, b=2)
0.6486856330601333

fluids.open_flow.Q_weir_rectangular_full_Kindsvater_Carter(h1, h2, b)[source]

Calculates the flow rate across a full-channel rectangular weir from the height of the liquid above the crest of the weir, the liquid depth beneath it, and the width of the channel. Model from [1] as reproduced in [2].

Flow rate is given by:

$Q = \frac{2}{3}\sqrt{2}\left(0.602 + 0.0832\frac{h_1}{h_2}\right) b\sqrt{g} (h_1 +0.00125)^{1.5}$
Parameters: h1 : float Height of the fluid above the crest of the weir [m] h2 : float Height of the fluid below the crest of the weir [m] b : float Width of the channel section [m] Q : float Volumetric flow rate across the weir [m^3/s]

Notes

The following limits apply to the use of this equation:

h1 > 0.03 m b > 0.15 m h2 > 0.1 m h1/h2 < 2

References

 [1] (1, 2) Kindsvater, Carl E., and Rolland W. Carter. “Discharge Characteristics of Rectangular Thin-Plate Weirs.” Journal of the Hydraulics Division 83, no. 6 (December 1957): 1-36.
 [2] (1, 2) Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

>>> Q_weir_rectangular_full_Kindsvater_Carter(h1=0.3, h2=0.4, b=2)
0.641560300081563

fluids.open_flow.V_Manning(Rh, S, n)[source]

Predicts the average velocity of a flow across an open channel of hydraulic radius Rh and slope S, given the Manning roughness coefficient n.

Flow rate is given by:

$V = \frac{1}{n} R_h^{2/3} S^{0.5}$
Parameters: Rh : float Hydraulic radius of the channel, Flow Area/Wetted perimeter [m] S : float Slope of the channel, m/m [-] n : float Manning roughness coefficient [s/m^(1/3)] V : float Average velocity of the channel [m/s]

Notes

This is equation is often given in imperial units multiplied by 1.49.

References

 [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.
 [2] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

Example is from [2], matches.

>>> V_Manning(0.2859, 0.005236, 0.03)
1.0467781958118971

fluids.open_flow.n_Manning_to_C_Chezy(n, Rh)[source]

Converts a Manning roughness coefficient to a Chezy coefficient, given the hydraulic radius of the channel.

$C = \frac{1}{n}R_h^{1/6}$
Parameters: n : float Manning roughness coefficient [s/m^(1/3)] Rh : float Hydraulic radius of the channel, Flow Area/Wetted perimeter [m] C : float Chezy coefficient [m^0.5/s]

References

 [1] Chow, Ven Te. Open-Channel Hydraulics. New York: McGraw-Hill, 1959.

Examples

Custom example, checked.

>>> n_Manning_to_C_Chezy(0.05, Rh=5)
26.15320972023661

fluids.open_flow.C_Chezy_to_n_Manning(C, Rh)[source]

Converts a Chezy coefficient to a Manning roughness coefficient, given the hydraulic radius of the channel.

$n = \frac{1}{C}R_h^{1/6}$
Parameters: C : float Chezy coefficient [m^0.5/s] Rh : float Hydraulic radius of the channel, Flow Area/Wetted perimeter [m] n : float Manning roughness coefficient [s/m^(1/3)]

References

 [1] Chow, Ven Te. Open-Channel Hydraulics. New York: McGraw-Hill, 1959.

Examples

Custom example, checked.

>>> C_Chezy_to_n_Manning(26.15, Rh=5)
0.05000613713238358

fluids.open_flow.V_Chezy(Rh, S, C)[source]

Predicts the average velocity of a flow across an open channel of hydraulic radius Rh and slope S, given the Chezy coefficient C.

Flow rate is given by:

$V = C\sqrt{S R_h}$
Parameters: Rh : float Hydraulic radius of the channel, Flow Area/Wetted perimeter [m] S : float Slope of the channel, m/m [-] C : float Chezy coefficient [m^0.5/s] V : float Average velocity of the channel [m/s]

References

 [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.
 [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.
 [3] Chow, Ven Te. Open-Channel Hydraulics. New York: McGraw-Hill, 1959.

Examples

Custom example, checked.

>>> V_Chezy(Rh=5, S=0.001, C=26.153)
1.8492963648371776