Rectangular weir discharge equation (neglecting end contractions): choose the correct expression for Q as a function of crest length L and head H.

Introduction / Context:
Sharp-crested (rectangular) weirs are standard flow-measurement devices in open channels. The discharge depends on crest length L and head H above the crest, with a coefficient of discharge C_d accounting for contraction and viscosity effects.

Given Data / Assumptions:

• Sharp-crested rectangular weir, fully contracted effects embedded in C_d.
• Neglect velocity of approach or include it within an adjusted head if necessary (here neglected).
• Hydrostatic pressure distribution at the crest; inviscid, steady flow assumption near crest.

Concept / Approach:

Integrate the elementary discharge over the depth from y = 0 to y = H: dQ = C_d * L * v(y) * dy with v(y) = sqrt(2 * g * (H − y)). The integral yields the standard result.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Dimensional analysis: L * (g)^(1/2) * H^(3/2) gives L^3/T, the correct dimension of discharge; C_d is dimensionless.

Why Other Options Are Wrong:

(b) and (e) have incorrect exponents on H; (c) uses 1/2 instead of 3/2; (d) corresponds to a different profile (e.g., triangular/V-notch formula has 5/2 with tan terms), not a rectangular weir.

Common Pitfalls (misconceptions, mistakes):

Forgetting velocity of approach corrections; confusing rectangular and V-notch (triangular) weir formulas.

Final Answer:

Q = (2/3) * C_d * L * sqrt(2 * g) * H^(3/2)