Mouthpieces – discharge for an internal mouthpiece running free For an internal (Borda’s) mouthpiece discharging freely, the approximate discharge Q is best represented by which expression (a = area of mouthpiece, H = head over the mouthpiece)?

Introduction / Context:
Mouthpieces are short tubes attached to an orifice. An internal (Borda’s) mouthpiece may run free or full depending on downstream conditions. Each condition has a characteristic coefficient of discharge, which directly affects the flow rate.

Given Data / Assumptions:

• Internal mouthpiece (Borda’s type).
• Running free (vena-contracta formed and jet not filling the tube).
• H is the static head over the centre of the mouthpiece.

Concept / Approach:
For a free-running internal mouthpiece, the coefficient of discharge Cd is approximately 0.5. Thus, the discharge is Q ≈ Cd * a * √(2gH) ≈ 0.5 * a * √(2gH). When the mouthpiece runs full, Cd increases to about 0.707, and for well-designed external mouthpieces values can be higher.

Step-by-Step Solution:

Verification / Alternative check:
Textbook tables list typical Cd: internal free ≈ 0.5; internal full ≈ 0.707; external free ≈ 0.62–0.65; values corroborate the selection.

Why Other Options Are Wrong:

• 0.707 corresponds to internal mouthpiece running full.
• 0.855 and 1.0 are too high for this condition; 1.33 is nonphysical for a coefficient.

Common Pitfalls:
Confusing “free” with “full” running; the flow behavior inside the mouthpiece changes contraction and thus Cd.

Final Answer:
0.5 · a · √(2gH)