The heights of water on the upstream and downstream side of a submerged weir of length 3.5m are 300mm and 150mm respectively. If Cd for free and drowned portion is 0.6 and 0.8 respectively, find the discharge over the weir.

Discharge Over a Submerged Weir

Problem Statement

The heights of water on the upstream and downstream sides of a submerged weir are 300 mm and 150 mm, respectively. Given a weir length of 3.5 m, and the coefficients of discharge for the free and drowned portions as 0.6 and 0.8 respectively, determine the discharge over the weir.

Given Data

Upstream Water Height (H)	300 mm = 0.3 m
Downstream Water Height (h)	150 mm = 0.15 m
Length of Weir (L)	3.5 m
Coefficient of Discharge (Free Flow, Cd₁)	0.6
Coefficient of Discharge (Drowned Flow, Cd₂)	0.8
Acceleration due to Gravity (g)	9.81 m/s²

Calculation

The total discharge (Q) over a submerged weir is the sum of the discharge through the free (non-submerged) and drowned portions.

Q = (2/3) Cd₁ L √(2g) (H – h)^3/2 + Cd₂ L h √[2g(H – h)]

Substituting the given values:

Q = (2/3) × 0.6 × 3.5 × √(2 × 9.81) × (0.3 – 0.15)^3/2 + 0.8 × 3.5 × 0.15 × √(2 × 9.81 × (0.3 – 0.15))

Evaluating the above expression results in:

Q ≈ 1.08 m³/s

Detailed Explanation

The solution is divided into two parts corresponding to the free flow and the drowned flow conditions:

Free Flow Portion: The term (2/3) Cd₁ L √(2g) (H – h)^3/2 represents the discharge when the water flows freely over the weir without submergence. Here, (H – h) is the effective head driving the flow. The exponent 3/2 indicates that the discharge increases with the head in a nonlinear fashion.
Drowned Flow Portion: The term Cd₂ L h √[2g(H – h)] accounts for the additional flow when the weir is submerged (or drowned) on the downstream side. This term uses the downstream water depth (h) and the same effective head (H – h), reflecting the impact of submergence on the discharge.

Both terms are derived from principles of fluid mechanics and energy conservation, modified by empirical coefficients (Cd) to account for losses and flow conditions.

Physical Meaning

The discharge equations used in the calculation have a strong foundation in fluid dynamics:

Effective Head (H – h): This represents the driving force for the flow over the weir. It is the difference in water surface elevation between the upstream and downstream sides.
Square Root Term (√(2g(H – h))): This term is derived from the Bernoulli equation and represents the velocity that water would have under free-fall conditions due to the gravitational force acting over the effective head.
Coefficients of Discharge (Cd₁ and Cd₂): These are empirical factors that adjust the theoretical discharge to account for real-world inefficiencies such as viscosity, flow separation, and turbulence.
Non-linear Relationship (Exponent 3/2): The non-linear exponent in the free flow discharge equation reflects how small changes in head can lead to larger changes in flow rate, a common characteristic in open channel flows.

In summary, the equations balance theoretical predictions with observed behavior, providing a practical method for estimating flow rates under different flow conditions.

Conclusion

The total discharge over the submerged weir is 1.08 m³/s.