Physics quiz question: horizontal pipe with varying cross-sectional areas

Physics 205A Quiz 5, fall semester 2019
Cuesta College, San Luis Obispo, CA

Assume ideal fluid flow for water through this horizontal pipe with different cross-sectional areas.


The greatest pressure is at:
(A) point [1].
(B) point [2].
(C) point [3].
(D) (There is a tie.)

Correct answer (highlight to unhide): (B)

From applying the continuity equation:

A₁·v₁ = A₂·v₂ = A₃·v₃,

where the fluid volume flow rate is the same throughout each of these three sections of pipe. As the cross-sectional area of the pipe is smallest at point [3] and largest at point [2], then:

A₃ < A₁ < A₂,

such that the speeds at each section of pipe can be ordered accordingly, where the fastest speed occurs where the cross-sectional area is the narrowest:

v₂ < v₁ < v₃.

Then from Bernoulli's equation:

0 = ∆P + (1/2)·ρ·∆(v²) + ρ·g·∆y,

because the pipe is horizontal, then ∆y = 0, and we can neglect the last term, such that:

0 = ∆P + (1/2)·ρ·∆(v²).

Comparing points [1] and [2] gives us:

0 = P₂ – P₁ + (1/2)·ρ·((v₂)² – (v₁)²),

P₁ + (1/2)·ρ·(v₁)² = P₂ + (1/2)·ρ·(v₂)²,

and similarly comparing points [2] and [3] gives us:

0 = P₃ – P₂ + (1/2)·ρ·((v₃)² – (v₂)²),

P₂ + (1/2)·ρ·(v₂)² = P₃ + (1/2)·ρ·(v₃)².

Thus we can now compare the pressure and (1/2)·ρ·v² terms for all three points:

P₁ + (1/2)·ρ·(v₁)² = P₂ + (1/2)·ρ·(v₂)² = P₃ + (1/2)·ρ·(v₃)²,

where the location with the smallest (1/2)·ρ·v² term would correspond to having the greatest pressure. Earlier, from the continuity equation, since v₂ < v₁ < v₃, then:

P₂ > P₁ > P₃,

such that location [3] (having the largest area and slowest speed) would have the greatest pressure.

Sections 70854, 70855
Exam code: quiz05Gu1L
(A) : 1 student
(B) : 35 students
(C) : 10 students
(D) : 6 students

Success level: 67%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.66