Question 12 of 13: Steam Condenser — Small Tubes vs. Large Tubes
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 12: Steam Condenser — Small Tubes vs. Large Tubes (5 marks)
Same total flow area and flow rate, so the same mean velocity passes through many small tubes (A) or few large tubes (B, double the diameter).
(a) Design B (large tubes) has the lesser pressure drop. (b) Design A (small tubes) has the better heat transfer.
(a) Pressure drop. Because the total flow area and total flow rate are identical in both designs, the mean water velocity $V=Q/A_{total}$ is the same in Design A and Design B — only the diameter of the individual tubes differs. The Darcy–Weisbach relation for friction head loss per unit tube length is
$$\frac{h_L}{L}=f\,\frac{V^2}{2gD}$$
so, for the same $f$ and $V$, head loss is inversely proportional to tube diameter, $h_L\propto 1/D$. With $D_B=2D_A$,
Design A's small tubes lose roughly twice the head (and hence pressure) to friction that Design B's large tubes do, for the same throughput — and because Design A also has a smaller $D$ at the same $V$, its Reynolds number is lower, which (for turbulent flow) makes $f$ itself slightly higher too, reinforcing the same conclusion.
(b) Heat transfer. Equal total flow area with $D_B=2D_A$ means Design A needs four times as many tubes as Design B to carry the same flow ($N\propto1/D^2$, so $N_A=4N_B$). Total heat-transfer surface area per unit length is $S=N\pi D$, so
Design A therefore exposes twice the cooling surface area of Design B. In addition, the convective film coefficient from the Dittus–Boelter correlation, $h\propto V^{0.8}/D^{0.2}$, is itself slightly higher for the smaller tubes at the same velocity. Both effects — more surface area and a higher film coefficient — push the overall heat-transfer product $UA$ higher for Design A, so the small-tube condenser condenses the steam more effectively for the same water flow and temperature conditions. This is precisely why real tube-bundle condensers favour many small tubes despite the pressure-drop penalty found in (a).
