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Open channel flow equation

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Flow Equations for sluice gate.Introduces different flow equations to students which are widely utilized for the design of sluice gates connected to open channel.This tutorial will help to understand and articulate the basic flow equation utilized by designers all over the world.

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Open channel flow equation

  1. 1. Open Channel Flow Equation Through Sluice Gate By Dr.Mrinmoy Majumder, Founnding and Honorary Editor www.baipatra.ws
  2. 2. Content › Sluice Gates › Flow Equation through sluice gate
  4. 4. Flow Equation through Sluice Gate The sluice gate flow rate measurement is based on the Bernoulli Equation and can be expressed as: 1 2 𝜌𝑣1 2 + 𝜌𝑔ℎ1 = 1 2 𝜌𝑣2 2 + 𝜌𝑔ℎ2 (1) where h = elevation height (m) ρ = density (kg/𝑚3) v = flow velocity (m/s) The pressure components in the equation are in general irrelevant since pressure upstream and downstream are the same (p1 - p2 = 0).
  5. 5. Assuming uniform upstream and downstream velocity profiles - the Continuity Equation gives: q = v1 A1= v2 A2 (2) where q = flow rate (m3/s), A = flow area (m2) (2) can be modified to: q = v1 h1 b = v2 h2 b (3) Where b = width of the sluice (m),h1 = upstream height (m),h2 = downstream height (m)
  6. 6. CONTD. Combining (1) and (3), gives the "ideal" equation: 𝑞 = ℎ2 𝑏 2𝑔 ℎ1 − ℎ2 1 − ℎ2 ℎ1 (4) Assuming h1 >> h2 (4) can be modified to: Q = h2 b [2 g h1]1/2 (5)
  7. 7. This is approximately true when the depth ratio h1 / h2 is large, the kinetic energy upstream is negligible (v1 is small) and the fluid velocity after it has fallen the distance (h2 - h1) ≈ h1 - is: v2 = [2 g h1]1/2 (6) The ideal equation (3) can be modified with a discharge coefficient:: q = cd h2 b [2 g h1]1/2 (7) where cd = discharge coefficient The discharge coefficient depends on different parameters - such as upstream and tail-water depths, gate opening, contraction coefficient of the gate and the flow condition. In practice the typical discharge coefficient is approximately 0.61 for free flow conditions and depth ratios ho / h1 < 0.2.
  8. 8. Sluice Gate Specifications The most commonly used specification for sluice gates in water and wastewater treatment plants is ANSI/AWWA C560-00. This specification should be used as a guidance for gates selection and operating equipment and associated hardware. Example - Flow Rate through a Sluice Gate Water flows under a sluice gate with an opening height of 0.4 m. The width of the sluice is 3 m and the height from the water surface to the bottom of the sluice is 10 m. Since ℎ1 >> ℎ2 and the depth ratio 0.4/10 < 0,2 - the contraction coefficient can be set to 0.61 - and equation (7) can be used for flow calculation: q = 0.61 (0.4 m) (3 m) [2 (9.81 m/𝑠2) (10 m)] = 10.25 𝑚3/s Specification and Examples
  9. 9. Flow Rate Calculated with the Velocity-Area Principle
  10. 10. Calculate flow rate or discharge in an open conduit, channel or river based on the velocity-area principle Velocities and depths across the stream are measured as indicated in the figure above. A partial discharge in a section of the stream can be calculated as qn = vn an (1) where qn = flow rate or discharge in section n (m3/s, ft3/s) vn = measured velocity in section n (m/s, ft/s) an = area of section n (m2, ft2) One simple way to express the section area is an = dn (ln+1 - ln-1) / 2 (2) The total flow in the stream can be summarized to Q = Σ1 n vn an (3) where Q = summarized flow rate or discharge in the conduit (m3/s, ft3/s) The accuracy of estimate depends on the profile of the conduit and the number of measurements. For conduits with regular shapes like rectangular channels a limited number of measurements are required. For irregular shapes - like natural rivers or similar - higher accuracy requires more measurements both horizontal and vertical.
  11. 11. Measured Values Calculated Values n v (m/s) d (m) l (m) a (m2) q (m3/s) 0 0 0 0 1 3 1 2 2 6 2 4 1.5 4 3 12 3 3 0.9 6 1.8 5.4 4 0 0 8 Summarized 23.4 From a conduit we have three measurements:
  12. 12. Example - Computing Flow Rate in a Channel The section areas can be calculated like a1 = (1 m) ((4 m) - (0 m)) / 2 = 2 m2 a2 = (1.5 m) ((6 m) - (2 m)) / 2 = 3 m2 a3 = (0.9 m) ((8 m) - (4 m)) / 2 = 1.8 m2 q1 = (3 m/s) (2 m2) = 6 m3/s q2 = (4 m/s) (3 m2)= 12 m3/s q3 = (3 m/s) (1.8 m2) = 5.4 m3/s The total flow can be summarized as Q = (6 m3/s) + (12 m3/s) + (5.4 m3/s) = 23.4 m3/s Note - there are alternative ways to calculate the section flow rates.
  13. 13. Simple Average Method Using the simple average of two successive vertical depths, their mean velocity, and the distance between them can be expressed as: qn to n+1 = [(vn + vn+1) / 2] [(dn + dn+1 ) / 2] (ln+1 - ln) (4) Midsection Method With the midsection method, the depth and mean velocity are measured for each number of verticals along the cross section. The depth at a vertical is multiplied by the width, which extends halfway to the preceding vertical and halfway to the following vertical, to develop a cross-sectional area. The section flow rate can be expressed as qn = vn [((ln - ln-1) + (ln+1 - ln)) / 2] dn (5)
  14. 14. Weirs - Open Channel Flow Rate Measurement Weirs can be used to measure flow rates in open channels and rivers - common for water supply and sewage plants
  15. 15. Weirs are structures consisting of an obstruction such as a dam or bulkhead placed across the open channel with a specially shaped opening or notch. The flow rate over a weir is a function of the head on the weir. Common weir constructions are the rectangular weir, the triangular or v-notch weir, and the broad-crested weir. Weirs are called sharp-crested if their crests are constructed of thin metal plates, and broad-crested if they are made of wide timber or concrete. If the notch plate is mounted on the supporting bulkhead such that the water does not contact or cling to the downstream weir plate or supporting bulkhead, but springs clear, the weir is a sharp- crested or thin-plate weir. Water level-discharge relationships can be applied and meet accuracy requirements for sharp-crested weirs if the installation is designed and installed consistent with established ASTM and ISO standards.
  16. 16. Rectangular weirs and triangular or v-notch weirs are often used in water supply, wastewater and sewage systems. They consist of a sharp edged plate with a rectangular, triangular or v-notch profile for the water flow. Broad-crested weirs can be observed in dam spillways where the broad edge is beneath the water surface across the entire stream. Flow measurement installations with broad-crested weirs will meet accuracy requirements only if they are calibrated. Rectangular weirs
  17. 17. Example The flow rate measurement in a rectangular weir is based on the Bernoulli Equation principles and can be expressed as: q = 2/3 cd b (2 g)1/2 h3/2 (1) where q = flow rate (m3/s) h = elevation head on the weir (m) b = width of the weir (m) g = 9.81 (m/s2) - gravity cd = discharge constant for the weir - must be determined cd must be determined by analysis and calibration tests. For standard weirs - cd - is well defined or constant for measuring within specified head ranges. The lowest elevation (h = 0) of the overflow opening of the sharp-crested weirs or the control channel of broad-crested weirs is the head measurement zero reference elevation.
  18. 18. The Francis Formula - Imperial Units Flow through a rectangular weir can be expressed in imperial units with the Francis formula q = 3.33 (b - 0.2 h) h3/2 where q = flow rate (ft3/s) h = head on the weir (ft) b = width of the weir (ft)
  19. 19. Triangular or V-Notch Weir The triangular or V-notch, thin-plate weir is an accurate flow measuring device particularly suited for small flows. For a triangular or v-notch weir the flow rate can be expressed as: q = 8/15 cd (2 g)1/2 tan(θ/2) h5/2 where θ = v-notch angle
  20. 20. Broad-Crested Weir For the broad-crested weir the flow rate can be expressed as: q = cd h2 b ( 2 g (h1 - h2) )1/2 (3)
  21. 21. Measuring the Levels For measuring the flow rate it's obviously necessary to measure the flow levels, then use the equations above for calculating. It's common to measure the levels with: •ultrasonic level transmitters, or •pressure transmitters Ultrasonic level transmitters are positioned above the flow without any direct contact with the flow. Ultrasonic level transmitters can be used for all measurements. Some of the transmitters can even calculate a linear flow signal - like a digital pulse signal or an analog 4 - 20 mA signal - before transmitting it to the control system. Pressure transmitters can be used for the sharp-crested weirs and for the first measure point in broad-crested weir. The pressure transmitter outputs a linear level signal - typical 4-20 mA - and the flow must be calculated in the transmitter or the control system.
  22. 22. Ultrasonic level transmitters
  23. 23. Pressure Indicator