Discharge through an external mouthpiece — practical formula Let a be the cross-sectional area of the external mouthpiece and H the head above its centerline. The discharge Q is approximately:

Introduction / Context:
Mouthpieces are short tubes fitted at tank orifice openings. For a well-designed external mouthpiece running full, empirical studies give a practical coefficient close to 0.855 for the discharge relation.

Given Data / Assumptions:

• External (protruding) mouthpiece of short length.
• Steady incompressible flow; fully aerated jet; negligible approach velocity.
• Coefficient of discharge C_d ≈ 0.855 (typical value).

Concept / Approach:
Ideal discharge from Bernoulli/continuity is Q_ideal = a * sqrt(2 * g * H). Real discharge is Q = C_d * Q_ideal with C_d < 1 to account for contraction and losses through the mouthpiece.

Step-by-Step Solution:

Verification / Alternative check:
Compare to sharp-edged orifice (C_d ≈ 0.62): the mouthpiece tends to have a higher C_d when running full because it controls contraction and reduces separation.

Why Other Options Are Wrong:

• 1.855: too high; implies C_d > 1.
• 0.585: too low for a full, well-formed mouthpiece.
• 0.855 * a * (2 g H): missing the square root, wrong dimensions.
• Coefficient = 1: assumes ideal flow without losses.

Common Pitfalls:
Mixing orifice and mouthpiece coefficients; forgetting the square-root dependence on head.

Final Answer:
Q = 0.855 * a * sqrt(2 * g * H)