Hydraulic coefficients of an orifice – correct relation Given the standard hydraulic coefficients of an orifice (C_v: coefficient of velocity, C_c: coefficient of contraction, C_d: coefficient of discharge, C_r: coefficient of resistance), which relation is correct?

Introduction / Context:
Discharge through a sharp-edged orifice differs from the ideal due to jet contraction and frictional/viscous effects. Engineers summarize these effects with three widely used coefficients that multiply to the net discharge coefficient.

Given Data / Assumptions:

• C_v accounts for the actual jet speed at the vena contracta relative to ideal velocity.
• C_c accounts for the reduced jet area at the vena contracta relative to the orifice area.
• C_d accounts for the overall discharge reduction: C_d = Q_actual / Q_ideal.

Concept / Approach:

The actual discharge Q_actual = A_c * V_actual, where A_c = C_c * A_orifice and V_actual = C_v * V_ideal. Therefore, Q_actual = (C_c * A_orifice) * (C_v * V_ideal) = (C_c * C_v) * (A_orifice * V_ideal). Hence, C_d = Q_actual / Q_ideal = C_c * C_v.

Step-by-Step Solution:

Verification / Alternative check:

Typical sharp-edged values: C_c ≈ 0.62, C_v ≈ 0.97, giving C_d ≈ 0.60, consistent with laboratory measurements.

Why Other Options Are Wrong:

(b) C_r is not defined by that formula; it is sometimes related to head-loss but not as shown. (c) and (e) are dimensionally inconsistent with definitions. (d) Inverts the correct product relation.

Common Pitfalls:

Confusing coefficients at the vena contracta with discharge coefficient at the orifice plane; mixing coefficients from different orifice geometries.

Final Answer:

C_d = C_c * C_v