Small Rectangular Orifice – Discharge Formula What is the discharge through a small rectangular orifice in a thin plate in terms of Cd, area a, and head h above the orifice centerline?

Introduction:
Flow through small sharp-edged orifices is estimated using Torricelli's theorem corrected by a discharge coefficient Cd to account for contraction and viscous effects.

Given Data / Assumptions:

• Small, sharp-edged rectangular orifice on a vertical tank wall.
• Head h is measured above the orifice center.
• a is the geometric area of the orifice opening.
• Cd is the coefficient of discharge accounting for contraction and friction.

Concept / Approach:

Ideal velocity from Torricelli: V_ideal = sqrt(2 * g * h). Ideal discharge would be a * V_ideal. Real discharge requires multiplying by Cd. Hence Q = Cd * a * sqrt(2 * g * h).

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check confirms units of Q are volume per unit time. Empirical values of Cd for sharp-edged orifices (about 0.60 to 0.65) support the corrected formula.

Why Other Options Are Wrong:

Cd * a * (2 * g * h): Missing the square root; dimensionally incorrect.Cd / a * sqrt(2 * g * h): Inverts area, giving wrong dimensions.a * sqrt(2 * g * h) / Cd: Dividing by Cd overestimates flow; Cd < 1.Cd * h * sqrt(2 * g * a): Mixes variables improperly; dimensionally wrong.

Common Pitfalls:

Forgetting the square root on 2 * g * h or using head to the wrong datum. Also confusing Cd with coefficients of contraction or velocity individually.

Final Answer:

Q = Cd * a * sqrt(2 * g * h)