A broad-crested weir of length 40m, has 400mm height of water above its crest. Find the maximum discharge neglecting velocity of approach. If the velocity of approach is taken into consideration, find the maximum discharge when the channel has a cross-sectional area of 40m2 on the upstream side. Take Cd =0.6.

Discharge Over a Broad-Crested Weir

Problem Statement

A broad-crested weir of length 40 m has a head of water 400 mm above its crest. Find the maximum discharge neglecting velocity of approach. If velocity of approach is considered, find the maximum discharge when the channel has a cross-sectional area of 40 m² on the upstream side.
Take C_d = 0.6.

Given Data

Length of Weir (L)	40 m
Head of Water (H)	0.4 m
Coefficient of Discharge (C_d)	0.6
Cross-Sectional Area of Channel (A)	40 m²
Acceleration due to Gravity (g)	9.81 m/s²

1. Maximum Discharge Without Velocity of Approach

The maximum discharge over a broad-crested weir without considering velocity of approach is given by:

Q_max = 1.705 C_d L H^3/2

Substituting values:

Q_max = 1.705 × 0.6 × 40 × (0.4)^3/2

Calculation gives:

Q_max ≈ 10.352 m³/s

2. Velocity of Approach

Velocity of approach is given by:

V_a = Q_max / A = 10.35 / 40 = 0.25875 m/s

Velocity head is calculated as:

h_a = V_a² / (2g) = (0.25875)² / (2 × 9.81) = 0.0034 m

3. Maximum Discharge Considering Velocity of Approach

The corrected discharge equation is:

Q_max = 1.705 C_d L [(H + h_a)^3/2 – h_a^3/2]

Substituting values:

Q_max = 1.705 × 0.6 × 40 × [(0.4 + 0.0034)^3/2 – (0.0034)^3/2]

Calculation gives:

Q_max ≈ 10.476 m³/s

Conclusion

The maximum discharge over the broad-crested weir without velocity of approach is 10.352 m³/s.
Considering velocity of approach, the corrected discharge is 10.476 m³/s.