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?

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 C_d = 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 (C_d)	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) × C_d × tan(θ/2) × H^5/2 × √(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/s

The flow rate through the V-notch weir is 0.059 m³/s (59 liters/s)

2. 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/s

Rearranging the weir equation to solve for H:

H = (Q / (0.818))^2/5

For upper limit of head (H₁):

H₁ = (0.0602 / 0.818)^2/5 = (0.0736)^2/5 = 0.352 m

For lower limit of head (H₂):

H₂ = (0.0578 / 0.818)^2/5 = (0.0707)^2/5 = 0.346 m

For 2% accuracy in flow measurement, the head should be kept between 0.346 m and 0.352 m

Conclusion

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.