Notches and weir

Brief introduction about notches and weir and discharge in various types of notches and weir.

1. Notches and Weir

2. Introduction A notch is a device used for measuring the rate of flow of a liquid through a small channel or a tank. It may be define as an opening in the side of a tank or a small channel in such a way that the liquid surface in the tank or a channel is below the top edge of the opening.

3. A weir is a concrete or a masonry structure, placed in an open channel over which the flow occurs. It is generally in the form of vertical wall, with a sharp edge at the top, running all the way across the open channel. The notch is of small size while the weir is the bigger size. The notch is generally made of metallic plate while weir is made of concrete or masonry structure.

4. Nappe or Vein: The sheet of water flowing through a notch or over a weir is called Nappe or Vein. Crest or Sill: The bottom edge of a notch or a top of weir over which the water flows, is known as the sill or crest.

5. Classification of Notches and Weir The notches are classified as; According to the shape of the opening: Rectangular Notch Triangular Notch Trapezoidal Notch Stepped Notch According to the effect of the sides of the nappe: Notch with end contraction Notch without end contraction or suppressed notch

6. Weir are classified according to the shape of the opening the shape of the crest, the effect of the sides on the nappe and nature of discharge According to the shape of the opening: Rectangular weir Triangular weir Trapezoidal weir

7. According to the shape of the crest: Sharp-crested weir Broad-crested weir Narrow-crested weir Ogee-shaped weir According to the effect of sides on the emerging nappe: Weir with end contraction Weir without end contraction

8. Discharge Over a Rectangular Notch or Weir The expression for discharge over a rectangular notch or weir is the same. Consider a rectangular notch or weir provided in a channel carrying water as shown in fig. Let, H = head of water over the crest and L = Length of the notch or weir. For finding a discharge of water flowing over the weir or notch, consider an elementary horizontal strip of water of thickness dh and Length L at a depth of h from free surface of water.

9. The area of strip, = L x dh and, Theoretical velocity of water flowing through strip = √2gh The discharge dQ , through strip is dQ = Cd x Area x Velocity = Cd x L x dh x √ 2gh The total discharge , Q for the whole notch or weir is determined by integrating the above equation between the limits 0 to H .

10. Q = C d x L x √ 2gh dh Q = (2/3) C d x L x √ 2g [H] 3/2

11. Discharge Over a Triangular Notch or Weir The expression for the discharge over a triangular notch or weir is the same. Let, H = head of water above V-notch and θ = angle of notch Consider a horizontal strip of water of thickness “ dh ” at a depth of h from the free surface of water as shown in Fig. From Fig (b), we have tan θ /2 = AC/OC = AC/(H-h)

12. AC = (H – h) tan θ /2 Width of strip, AB = 2AC = 2(H – h) tan θ /2 Area of strip = 2 (H – h) tan θ /2 dh The theoretical velocity of water through strip = √ 2gh Discharge, dQ through the strip is = Cd x Area of Strip x Velocity = 2 Cd (H – h) tan θ /2 dh √2gh Total Discharge, Q is will get by integrating the strip discharge between 0 to H.

13. ∴ Q = 8/15 Cd tan θ /2 √2g H (5/2) For a right-angled V-notch, if C d = 0.60 ∴ Discharge Q = 1.417 H (5/2)

14. Advantages of Triangular Notch or Weir: A triangular notch or weir is preferred to a rectangular weir or notch due to following reasons: The expression for discharge for a right –angled V-notch or weir is very simple. For measuring low discharge, a triangular notch gives more accurate results than a rectangular. In case of triangular notch, only one reading, i.e. (H) is required for the computation of discharge. Ventilation of triangular notch is not necessary.

15. Discharge Over a Trapezoidal Notch or Weir As shown in fig., a trapezoidal notch or weir is a combination of a rectangular and triangular notch or weir. Thus the total discharge will be equal to the sum of discharge through rectangular weir or notch and discharge through a triangular notch or weir. Let, H = Height of water over notch, L = Length of the crest of the notch. Total Discharge Q = (2/3) C d x L x √2g [H] 3/2 + 8/15 Cd tan θ /2 √2g H (5/2 )

16. Effect on Discharge Over a Notch or Weir due to Error in the Measurement of Head For an accurate value of the discharge over a weir or notch, an accurate measurement of head over a weir or notch is very essential as the discharge over a triangular notch is proportional H (5/2 ) to and in case of rectangular notch it is proportional to H (3/2) . A small error in the measurement of head, will affect the discharge considerably.

17. For rectangular Weir or Notch: The discharge for a rectangular weir or notch is given by Q = ( 2/3) C d x L x √2g [ H] 3/2 Differentiating the above equation, we get dQ = k (3/2) H (½) dH Dividing, dQ /Q = (3/2) dH /H Equation show that an error of 1% in measuring H will produce 1.5% error in discharge over a rectangular weir or notch.

18. For Triangular Weir or Notch: The discharge for a triangular weir or notch is given by Q = 8/15 Cd tan θ /2 √2g H (5/2 ) Differentiating the above equation, we get dQ = k (5/2 ) H (3/2) dH Dividing, dQ /Q = (5/2 ) dH /H Equation show that an error of 1% in measuring H will produce 2.5 % error in discharge over a triangular weir or notch.

19. Velocity of Approach Velocity of approach is defined as the velocity with which the water approaches or reaches the weir or notch before it flows over it. Thus if V a is the velocity of approach, than an additional head ha equal to V a 2 /2g due to velocity of approach, is acting on the water flowing over the notch. Then initial height of water over the notch becomes (H + h a ) and final height becomes equal to h a . Then all the formulae are changed in taking into consideration of velocity approach.

20. The velocity of approach, Va is determined by finding the discharge over the notch or weir, neglecting velocity approach. Then dividing the discharge by the c/s area of the channel on the u/s side of the weir or notch, the velocity of approach is obtained. Mathematically, Va = Q / area of channel This velocity of approach is used to find an additional head. Again the discharge calculate and above process is repeated for more accurate discharge. Discharge, over a rectangular weir, with velocity approach = (2/3) C d x L x √2g [(H 1 +h a ) 3/2 - h a 3/2 ]

21. Empirical Formulae for Discharge Over Rectangular Weir The discharge over a rectangular weir is given by, Q = (2/3) Cd √2g L (H) 3/2 without velocity approach = (2/3) C d x L x √2g [( H 1 +h a ) 3/2 - h a 3/2 ] with velocity of approach. Both equation are applicable to the weir or notch for which the crest length is equal to the width of the channel. This type of weir is called Suppressed weir. But if the weir is not suppressed, the effect of end contraction will be taken into account.

22. Francis Formula Francis on the basis of his experiments established that end contraction decreases the effective length of the crest of weir and hence decreases the discharge. Each end contraction reduces the crest length by 0.1 x H, where H is the head over the weir. For rectangular weir there are two end contractions only and hence effective length, L = (L – 0.2 H) and Q = (2/3) Cd √2g L (H) 3/2 If Cd = 0.623, g = 9.81 m/sec2, then Q = 1.84 (L – 0.2 H)

23. If end contractions are suppressed, then H = 1.84 L H 3/2 If velocity of approach is consider, then Q = 1.84 L [(H 1 +h a ) 3/2 - h a 3/2 ]

24. Bazin’s Formula On the basis of results of a series of experiments, Bazin proposed the following formula for the discharge over a rectangular weir as, Q = m L √2g H 3/2 where, m = (2/3) Cd = 4.05 + (0.003/H) If velocity of approach is considered, then Q = m 1 L √2g [(H 1 +h a ) 3/2 - h a 3/2 ] where, m 1 = 0.405 + 0.003 / (H + h a )

25. Discharge Over a Broad Crested Weir A weir having a wide crest is known as broad crested weir. Let, H = height of water above the crest and L = length of the crest If 2L > H, the weir is called broad crested weir If 2L < H, the weir is called narrow crested weir As shown in fig. let, h = head of water at the middle of weir which is constant, v = velocity of flow over the weir.

26. By applying Bernoulli’s equation to the still water surface on the u/s side and running water at the end of weir, 0 + 0 + H = 0 + v 2 /2g + h ∴ v 2 /2g = H – h ∴ v = √[2g (H - h)] The discharge over a weir, Q = C d x Area x Velocity ∴ Q = C d x L x h x √[2g (H - h)] ∴ Q = C d x L x √[2g (Hh 2 – h 3 )]

27. The discharge will be maximum, if Hh 2 – h 3 is maximum or d/dh Hh 2 – h 3 = 0 or 2h x H + 3h = 0 or 2H = 3h, ∴ h = 2/3 H Q max will obtained by substituting this value of h in the equation, Q max = C d x L x √[2g (Hh 2 – h 3 )] Q max = 1.705 x C d x L x H 3/2

28. Discharge Over a Narrow Crested Weir For narrow crested weir, 2L < H. it is similar to a rectangular weir or notch hence, Q is given by, ∴Q = 2/3 x C d x L x √2g x H 3/2

29. Discharge Over an Ogee Weir As shown in fig. an Ogee weir, in which the crest of the weir rises up to maximum height of 0.115 x H, and then falls. The discharge for an Ogee weir is same as that of rectangular weir, and it is given by, ∴ Q = 2/3 x C d x L x √2g x H 3/2

30. Discharge Over Submerged or Drowned Weir When the water level on the d/s side of a weir is above the crest of the weir, then the weir is called to be submerged or drowned weir. The total discharge, over the weir is obtained by the dividing the weir into two parts. The portion between u/s and d/s water surface may be treated as free weir and portion between d/s water surface and crest of weir as drowned weir. Let, H = height of water on the u/s of the weir, h = height of water on the d/s side of weir.

31. Then, Q 1 = discharge over upper portion = 2/3 C d1 x L x √2g x (H – h) 3/2 Q 2 = discharge through drowned portion = C d2 x Area of flow x Velocity of flow = C d2 x L x h x √[2g (H – h)] ∴Total discharge, Q = Q1 + Q2 Q = (2/3) C d1 x L x √2g x (H – h) 3/2 + C d2 x L x h x √[2g (H – h)]