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04-BS-7 · May 2017

Question 12 of 13: Duct Bend Head Loss — Orientation of a Rectangular Bend

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examination, 2017-May. 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 capillarity (Ch. 2), integral analysis and buoyancy (Ch. 3), viscous flow in ducts, pipe friction and the Moody chart (Ch. 6), flow past immersed bodies and drag (Ch. 7), open-channel flow and the hydraulic jump (Ch. 10).

Check — assumptions used across this paper:
  • Q2's figure shows the vertical leaf of the gate rising above the free surface, so water acts over the full depth x = 2.0 m; both "0.3 m" dimensions locate the centre of gravity (0.3 m right of the vertical leaf, 0.3 m above the arm). Reservoir water also fills the space beneath the 1.2 m arm (the vertical lines below O are dimension extension lines, not a wall), so the arm carries uplift at pressure ρgx, and the arm tip bears up against a lip on the spillway crest — the gate can only open by rotating clockwise, vertical leaf toward the spillway.
  • Q4's flow coefficient K ≈ 0.69 is read from the attached VDI chart on the Do/D1 = 0.50 curve (third from the top, the one that levels off at 0.62) at the approach Reynolds number R ≈ 2550. The printed curve gives 0.695 at R = 2000 and 0.684 at R = 3000; a reading of 0.68–0.70 moves the answer by only ±0.3 kPa.
  • Q7 and Q8 friction factors are obtained from the Colebrook–White/Haaland equation the Moody chart itself plots (smooth-wall case, as both problems specify or imply smooth surfaces).
  • Q9's sphere drag coefficient vs. Reynolds number is obtained from the Schiller–Naumann correlation $C_D=\tfrac{24}{Re}\left(1+0.15\,Re^{0.687}\right)$, which reproduces the published sphere-drag curve (the same curve reprinted in the attachment) to within a few percent for $Re<1000$ — used here because the resulting Reynolds number (≈19–20) is well above the range where the simple Stokes'-law formula ($C_D=24/Re$) alone is valid.

Question 12: Duct Bend Head Loss — Orientation of a Rectangular Bend (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. The exam sketches (checked against the printed paper) show two radiused 90° bends in the same flat rectangular duct, roughly 3:1 in aspect, carrying the same volumetric flow. Neither bend has a sharp corner. In A the short side of the rectangle lies in the plane of the bend: the passage is narrow in the radial direction (small $W$) and deep across it (large $H$), so the duct turns "the easy way". In B the same duct is turned about its other axis, so the long side lies in the plane of the bend (large $W$, small $H$) and it turns "the hard way". Both are drawn with a similarly small inside radius.

W small attached flow A (narrow side in plane) separation eddy W large B (wide side in plane)
Both bends seen in the plane of the turn, with the same inside radius. A: narrow radial width, so the bend is gentle relative to W and the flow stays attached. B: wide radial width with the same small inside radius, so the relative curvature is tight, a large pressure difference builds across the bend, and the flow separates from the inner wall downstream, leaving a recirculating eddy that blocks part of the duct.

Find. Which bend has the greater head loss, with justification.

Answer: bend B has the greater head loss. Both bends turn the same flow through 90°, so the difference comes from how sharply each one turns relative to the width of its passage. The relevant parameter is the ratio of centreline radius to the passage width in the plane of the bend, $R_c/W$.

1. Pressure difference across the bend. Turning the flow requires a pressure gradient toward the centre of curvature, $\partial p/\partial r\approx\rho V^2/r$. The pressure difference from outer to inner wall is therefore roughly $\Delta p\approx\rho V^2\,W/R_c$. In A the radial width $W$ is small and $R_c/W$ is comparatively large (about 1 or more on the sketch), so the difference is modest. In B, $W$ is about three times larger while the inside radius is just as small, so $R_c/W$ is well below 1 and the cross-bend pressure difference is several times larger.

2. Separation on the inside wall. Along the inner wall the flow speeds up into the low-pressure region at the start of the turn. It then has to slow down against that same large pressure rise as it leaves the bend. In B this adverse pressure gradient is too strong for the wall boundary layer, which separates just downstream of the inner corner. The result is a recirculating eddy (shaded in the sketch) that occupies a large part of the passage, a contracted jet hugging the outer wall, and violent mixing as the jet re-expands to fill the duct. That mixing is where the kinetic energy is lost. In A the gentler relative curvature keeps the boundary layer attached, or separates it only slightly, and the full passage stays in use.

3. Secondary flow. The same cross-bend pressure gradient pushes the slow fluid in the boundary layers on the two flat walls parallel to the plane of the bend toward the inside of the bend. This sets up a pair of counter-rotating secondary vortices that carry on well downstream and dissipate energy. In B those walls are the wide faces and $H/W$ is small, so the secondary cells are large and strong. In A they are the narrow faces, and most of the deep passage is far from them.

Published loss data for smooth-radius rectangular elbows (for example the ASHRAE duct-fitting tables) show exactly this: the loss coefficient climbs steeply as $R_c/W$ falls below about 1, and it is higher for small depth-to-width ratios $H/W$. B combines both unfavourable effects. For the same cross-section, radius and flow, B has the greater head loss, and in practice a flat duct should be bent about its narrow dimension, as in A, or fitted with turning vanes if it must be bent the hard way.