Discharge coefficients in orifice flow – relationship among Cd, Cc, and Cv In classical hydraulics for a sharp-edged orifice, the overall coefficient of discharge Cd is related to the coefficient of contraction Cc and the coefficient of velocity Cv by which expression?

Introduction:
The coefficient of discharge Cd for an orifice accounts for both jet contraction and departure from ideal velocity. This question checks whether you remember the fundamental decomposition linking Cd to the coefficient of contraction Cc and the coefficient of velocity Cv, a staple identity in fluid mechanics labs and exams.

Given Data / Assumptions:

• Sharp-edged orifice issuing into the atmosphere (free jet) or similar condition where standard coefficients apply.
• Steady flow; head measured appropriately so Torricelli’s law is usable with coefficients.
• Neglect compressibility; liquid is Newtonian.

Concept / Approach:

Ideal jet velocity is V_ideal = sqrt(2 * g * H). Real jets contract to a smaller effective area A_e = Cc * A and leave with a velocity V = Cv * V_ideal. The actual discharge is Q = A_e * V = (Cc * A) * (Cv * V_ideal). Comparing with the ideal discharge Q_ideal = A * V_ideal identifies Cd = Q / Q_ideal = Cc * Cv.

Step-by-Step Solution:

Verification / Alternative check:

Typical values for sharp-edged orifices: Cc ≈ 0.62, Cv ≈ 0.97, giving Cd ≈ 0.60. This matches widely reported data, supporting Cd = Cc * Cv.

Why Other Options Are Wrong:

Cc x Cr or Cv x Cr: Coefficient of resistance Cr is not multiplied this way to obtain Cd.Cc/Cr or Cc + Cv: Do not reflect the physical composition of area and velocity effects.

Common Pitfalls:

Confusing Cv with Cd, or forgetting that contraction and velocity effects are independent phenomena whose influences multiply in discharge.

Final Answer:

Cc x Cv