Question 10 of 13: Duct Bend Head Loss — Plain vs. Vaned
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
04-BS-7 Mechanics of Fluids — National Examination, 2017-Dec. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Graphical & Analytical, 4 questions, do 3); every question is answered below regardless of the exam's "do N of M" instruction, so the set is a complete study resource.
Reference texts: White, F.M., Fluid Mechanics (8th ed.) — fluid statics and manometry (Ch. 2), Bernoulli and flow measurement (Ch. 3), viscous flow in ducts and the Moody chart (Ch. 6), flow past immersed bodies and drag (Ch. 7), turbomachinery and wind turbines (Ch. 11).
Check — assumptions used across this paper:
Q1's manometer is read from the printed figure as a benzene(hatched)–mercury(black)–carbon-tetrachloride(clear) chain: benzene fills pipe A up and over the first bend and down the U-tube's left arm to the benzene/mercury interface at the UPPER dimension line (2.0 m + 400 mm = 2.4 m above A); the mercury stands 400 mm lower in the right arm, at the mercury/CCl₄ interface on the LOWER line (2.0 m above A); CCl₄ then fills the rest of the run over the second bend down to B, 3.0 m below A. (The mercury is higher on the A side, so pA < pB.)
Q6's spillway/gate width is read from the drawing as the 8.76 m dimension (the question's own hint: "width of each gate is slightly greater than its height" – 8.76 m > 8.23 m); the closed gate's wetted height at F.S.L. is F.S.L. − Crest = 7.92 m (the gate's own 8.23 m height extends slightly above F.S.L., matching the paper's note that "the top of the gate is higher than F.S.L.").
Q7 and Q9's friction/drag coefficients come from the Colebrook–White equation and the plotted drag curve — the same relations the attached Moody and drag charts plot.
Q9's Reynolds number (≈5.9×107) is beyond the attached drag chart's plotted range (10−1 to 106); CD = 0.3 is taken from the right-hand end of the cylinder curve (minimum ≈0.3 past the drag crisis, ≈0.33 at 106) — the best available reading — and flagged here as an extrapolation.
Q8(c)'s fuel density (needed to convert a fuel mass into litres) is not stated in Q8 itself; the paper lists no fuel density on its Constants page, so the gasoline SG = 0.75 given in Q4 of this same paper is adopted (Q2's 0.72 would give 3.50 L/100 km instead of 3.36).
Question 10: Duct Bend Head Loss — Plain vs. Vaned (5 marks)
Duct A: unguided flow separates at the inner corner, forming a large recirculation zone. Duct B: turning vanes guide the flow smoothly around the bend in several attached passages.
Duct B, fitted with curved turning vanes, has the lesser head loss.
A 90° bend is a minor loss characterized by $h_L=K\dfrac{V^2}{2g}$, where the loss coefficient $K$ depends almost entirely on how well the flow can follow the curvature. In Duct A the flow approaches the corner at high velocity with no guidance; it cannot turn through 90° in the tight radius of the duct wall itself, so it separates from the inner wall immediately past the bend. This separation creates a large, energy-dissipating recirculation zone (secondary flow / Dean vortices) that blocks part of the cross-section, forces the "live" flow through a reduced effective area, and converts a significant fraction of the flow's kinetic energy into turbulence that is ultimately lost as heat. Typical unguided sharp (mitre) 90° bends have $K\approx1.2$–$1.5$.
Duct B's curved vanes subdivide the single sharp 90° turn into several smaller-radius passages, each vane acting like a mini guide-rail that keeps the local radius of curvature large relative to the passage width. A larger radius-to-width ratio keeps the adverse pressure gradient on the inner surface of each passage gentle enough that the boundary layer stays attached all the way around the bend — no separation bubble forms, and the flow exits the bend as a nearly uniform, low-turbulence stream. Vaned elbows typically achieve $K\approx0.2$–$0.3$, roughly a four- to six-fold reduction in loss coefficient (and hence head loss, at the same velocity) compared with the plain bend.
