Question 11 of 13: Capillary Rise Between Rods of Two Sizes
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 11: Capillary Rise Between Rods of Two Sizes (5 marks)
Array A (small rods) draws water up between the rods to about twice the height reached in Array B (rods of double the diameter).
Array A — the bundle of small-diameter rods — will show the greater rise of water between the rods.
The interstitial channel between four rods touching in a square array is a capillary passage in its own right, and for geometrically similar packings its characteristic opening (the gap between adjacent rod surfaces) scales linearly with the rod diameter $D$. The general capillary-rise relation on the paper's own reference sheet, $h=\dfrac{\sigma\cos\theta}{\rho g}\times\dfrac{P}{A}$, applies to any tube shape via its wetted perimeter-to-area ratio $P/A$; for a channel whose linear size scales with $D$, both $P$ and $A$ scale with $D$ and $D^2$ respectively in the same way a circular capillary of radius $r\propto D$ would, so $P/A\propto 1/D$ and hence $h\propto1/D$ — exactly as in the familiar circular-tube result $h=2\sigma\cos\theta/(\rho g r)$.
Since Array B's rods have double the diameter of Array A's ($D_B=2D_A$), the effective capillary radius of its interstitial channels is also double, so
Array A's water rises to roughly twice the height reached in Array B, for the same reason a fine capillary tube outperforms a wide one: surface tension pulls the liquid up around the wetted perimeter, but gravity must lift the entire cross-sectional volume of that column, and the volume grows with the square of the passage size while the lifting perimeter only grows linearly — so smaller passages always win on rise height.
