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?
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AQ = Cd * a * sqrt(2 * g * h)
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BQ = Cd * a * (2 * g * h)
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CQ = Cd / a * sqrt(2 * g * h)
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DQ = a * sqrt(2 * g * h) / Cd
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EQ = Cd * h * sqrt(2 * g * a)
Answer
Correct Answer: Q = Cd * a * sqrt(2 * g * h)
Explanation
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:
Step 1: Start with Torricelli: V = sqrt(2 * g * h).Step 2: Multiply by area to get ideal discharge: Q_ideal = a * sqrt(2 * g * h).Step 3: Apply coefficient of discharge: Q = Cd * Q_ideal = Cd * a * sqrt(2 * g * h).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)