Section 4: Soil and Water Conservation Engineering
Fluid Mechanics
Subtopic: Flow through orifices, weirs and notches
Question Statement:
Regarding water measurement using weirs and notches in agricultural channels, which of the following statements are CORRECT?
Options:
(A) The theoretical discharge through a rectangular weir is proportional to $H^{3/2}$, where $H$ is head over the crest
(B) The theoretical discharge through a triangular $90^\circ$ V-notch weir is proportional to $H^{5/2}$
(C) A triangular V-notch weir provides greater measurement accuracy than a rectangular weir for small discharge rates
(D) Francis formula for a rectangular weir with two end contractions is $Q = 1.84 (L - 0.2 H) H^{3/2}$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. Rectangular weir discharge: $Q = \frac{2}{3} C_d \sqrt{2g} L H^{3/2} \propto H^{3/2}$. (Statement A is True)
2. Triangular notch discharge: $Q = \frac{8}{15} C_d \sqrt{2g} \tan(\theta/2) H^{5/2} \propto H^{5/2}$. (Statement B is True)
3. For low discharges, the head $H$ on a V-notch is substantially larger than on a rectangular weir, minimizing relative reading errors and surface tension distortion. (Statement C is True)
4. Francis formula accounting for $n = 2$ end contractions reduces effective length by $0.1 n H = 0.2 H$: $Q = 1.84(L - 0.2H) H^{3/2}$. (Statement D is True)