Question 7 of 13: Pipe Diameter Selection Using the Moody Diagram
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 7: Pipe Diameter Selection Using the Moody Diagram (5 marks)
Gravity flow between two reservoirs; the 70 m elevation drop is entirely consumed by pipe friction.
Find. A commercial pipe diameter $D$ (in the range 0.2–0.4 m) that carries 0.3 m³/s with the available 70 m of head.
Approach. Express $V(D)$, $Re(D)$ and $e/D$ in terms of the unknown $D$, obtain $f$ from the same Colebrook–White relation the attached Moody chart plots (equivalent to reading the chart), and iterate $h_L=f(L/D)(V^2/2g)$ to the available head of 70 m.
Available head. $h_L=190-120=\boxed{70\ \text{m}}$ (reservoir surfaces, both open, so all of the elevation drop goes to pipe friction).
Trial values across the hinted range. For each guessed $D$: $V=Q/(\tfrac\pi4D^2)$, $Re=\rho VD/\mu$, $e/D$, then $f$ from Colebrook–White $\left(1/\sqrt f=-2\log_{10}\!\left(\tfrac{e/D}{3.7}+\tfrac{2.51}{Re\sqrt f}\right)\right)$, and $h_L=f(L/D)(V^2/2g)$:
D (m)
V (m/s)
Re
f
h_L (m)
0.20
9.55
31,800
0.0240
≈279
0.30
4.24
21,200
0.0260
≈40
0.40
2.39
15,900
0.0277
≈10
Plotting these three $(f,Re)$ points on the Moody chart and cross-plotting $h_L$ against $D$ shows the 70 m target falls between the $D=0.20$ and $D=0.30$ trials.
Converge on the root (interpolating/iterating the same relation).
$$D\approx\boxed{0.267\ \text{m}},\quad V=5.38\ \text{m/s},\quad Re=23{,}900,\quad e/D=2.25\times10^{-4},\quad f=0.0253,\quad h_L=70.0\ \text{m}$$
Select a commercial size. The calculated diameter (267 mm) is a minimum; the next standard commercial steel pipe size at or above it is selected so the achieved head loss is safely $\le 70$ m:
$$D_{selected}=\boxed{300\ \text{mm (12 in nominal)}}$$
