Submerged Orifices – Discharge for a wholly drowned rectangular orifice A rectangular orifice is wholly drowned (submerged) when both the upstream and downstream faces are under water. If the orifice has width b and height d (so area a = b * d), and the difference between the upstream and downstream free-surface levels is H, which expression gives the discharge Q?

Introduction:
Fully submerged (wholly drowned) orifices occur when both the upstream and the downstream faces are below their respective free surfaces. In such cases, the driving head for flow through the opening is the difference in free-surface elevations between the two sides, not the submergence below either face alone. This question asks for the correct discharge formula for a wholly drowned rectangular orifice.

Given Data / Assumptions:

• Rectangular orifice of width b and height d; area a = b * d.
• Upstream–downstream free-surface head difference is H.
• Coefficient of discharge Cd is constant over the head range considered.
• Steady conditions; neglect approach velocity and additional local losses beyond Cd.

Concept / Approach:

For orifice flow, Torricelli’s ideal speed is V_ideal = sqrt(2 * g * head). For a fully drowned orifice, the effective head is the difference in piezometric head across the opening, which is simply H (upstream free surface minus downstream free surface). The real discharge uses a coefficient Cd to account for contraction and viscous effects: Q = Cd * a * sqrt(2 * g * H) = Cd * b * d * sqrt(2 * g * H).

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis gives [Q] = L^3/T; since a has L^2 and sqrt(2 * g * H) has L/T, the product has L^3/T, confirming consistency. Empirical data show Cd typically in the range 0.6–0.65 for sharp-edged orifices, supporting the use of a single coefficient.

Why Other Options Are Wrong:

Cd * b * H * sqrt(2 * g * d) or Cd * b * H * sqrt(2 * g * H): misuse H inside area or double-count head.Cd * b * d * sqrt(2 * g * d): wrongly uses d as the driving head for a drowned case.(b * d / Cd) * sqrt(2 * g * H): dividing by Cd overestimates flow; Cd < 1.

Common Pitfalls:

Confusing the geometric height d (or submergence below faces) with the actual driving head across faces. For drowned flow, velocity depends on the free-surface difference H.

Final Answer:

Q = Cd * b * d * sqrt(2 * g * H)