V-Notch Weir Flow Rate Analysis
Problem Statement
Compute the flow rate if the measured head above the bottom of the V-notch is 35cm, when θ = 60° and Cd = 0.6. If the flow is wanted within an accuracy of 2%, what are the limiting values of the head?
Given Data
| Measured head (H) | 35 cm = 0.35 m |
| V-notch angle (θ) | 60° |
| Discharge coefficient (Cd) | 0.6 |
| Required accuracy | ±2% |
| Acceleration due to gravity (g) | 9.81 m/s2 |
1. Flow Rate Calculation
For a V-notch weir, the discharge equation is:
Q = (8/15) × Cd × tan(θ/2) × H5/2 × √(2g)Substituting the values:
Q = (8/15) × 0.6 × tan(60°/2) × (0.35)5/2 × √(2 × 9.81)Evaluating step by step:
tan(60°/2) = tan(30°) = 0.577 √(2g) = √(2 × 9.81) = 4.429 (8/15) × 0.6 × 0.577 × 4.429 = 0.818 (0.35)5/2 = (0.35)2 × √(0.35) = 0.1225 × 0.592 = 0.0725 Q = 0.818 × 0.0725 = 0.059 m3/s = 59 liters/s2. Determining Head Limits for 2% Accuracy
With ±2% error in the flow rate:
Q1 = Q + 0.02Q = 0.059 + 0.02 × 0.059 = 0.0602 m3/s Q2 = Q - 0.02Q = 0.059 - 0.02 × 0.059 = 0.0578 m3/sRearranging the weir equation to solve for H:
H = (Q / (0.818))2/5For upper limit of head (H1):
H1 = (0.0602 / 0.818)2/5 = (0.0736)2/5 = 0.352 mFor lower limit of head (H2):
H2 = (0.0578 / 0.818)2/5 = (0.0707)2/5 = 0.346 mConclusion
Based on the analysis of the 60° V-notch weir:
1. The flow rate with a head of 0.35 m is 0.059 m3/s (59 liters/s).
2. To maintain a flow accuracy of ±2%:
- The upper head limit is 0.352 m (35.2 cm)
- The lower head limit is 0.346 m (34.6 cm)
This demonstrates the sensitivity of V-notch weirs to head measurements. A variation of only ±3 mm in the head measurement results in a 2% change in the calculated flow rate.