NivaarExam PrepOfficial exam papers ↗

23-Chem-A2 Unit Operations and Separation Processes · December 2013

Question 5 of 6: Finned-Pipe Heat Transfer

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

Notes on this paper

Paper format. National Exam 04-Chem-A2 Mechanical and Thermal Operations, December 2013 — open-book, 3 hours, any non-communicating calculator. Two sections: Section A (Mechanical Operations, A1–A3) and Section B (Thermal Operations, B1–B3); every problem is 25 marks. The rubric asks candidates to attempt two problems per section, but all six are solved in full below.

Reference texts: McCabe, Smith & Harriott, Unit Operations of Chemical Engineering (7th ed., McGraw-Hill) — pipe friction, loss coefficients, packed beds and centrifugal separation (Ch. 5–7); de Nevers, Fluid Mechanics for Chemical Engineers and Brodkey & Hershey, Transport Phenomena: A Unified Approach — the mechanical-energy balance and the appended friction-factor chart and fitting table; Geankoplis, Transport Processes and Separation Process Principles — tubular-centrifuge neutral-zone analysis; Incropera & DeWitt, Fundamentals of Heat and Mass Transfer (7th ed., Wiley) and Çengel, Heat and Mass Transfer — conduction with generation, annular-fin efficiency (Fig. B1) and LMTD/correction-factor exchanger design (Fig. B2). Loss-coefficient, friction-factor, fin-efficiency and correction-factor data are read from the appended Table A1, Fig. A1, Table B1 and Figs. B1–B2.

Section A — Mechanical Operations

Question B2: Finned-Pipe Heat Transfer (25 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. Pipe outer diameter $30$ mm ($r_1=15$ mm), wall $130\ \mathrm{^\circ C}$, air $20\ \mathrm{^\circ C}$ ($\Delta T=110\ \mathrm{^\circ C}$), $h=60\ \mathrm{W/(m^2\cdot ^\circ C)}$. Fins: aluminium $k=200$, thickness $t=2$ mm, outer diameter $60$ mm ($r_2=30$ mm), gap 3 mm (pitch 5 mm → 200 fins/m).

Find. (a) bare-pipe heat rate per metre; (b) the increase in heat rate per metre once the fins are added.

pipe wall 130°C, D=30 mm circular Al fins: t=2 mm, OD=60 mm, pitch 5 mm (200/m), $\eta\approx0.96$ 5 mm
Figure B2 — Annular aluminium fins on the pipe. Because the efficiency parameter $\xi\propto\sqrt{h/kt}$ is small, the high-conductivity fins stay nearly isothermal with the base ($\eta\approx0.96$), so almost all of the added area is effective.

Approach. Compute bare-pipe convection; then read the annular-fin efficiency from Fig. B1 (confirmed from the Bessel-function solution), add up the fin and inter-fin duty per metre, and subtract the bare value.

  1. Bare pipe (part a). Convection from the plain outer surface per metre: $$q_{\text{bare}}=h(\pi D)\Delta T=60(\pi\times0.030)(130-20)=\boxed{622\ \mathrm{W/m}}.$$
  2. Fin efficiency. With $r_1=15$, $r_2=30$ mm, $t=2$ mm the abscissa of Fig. B1 is $$\xi=\Bigl(L+\tfrac{t}{2}\Bigr)\sqrt{\tfrac{h}{kt}}=(0.016)\sqrt{\tfrac{60}{200(0.002)}}=0.196,\qquad \frac{r_2+t/2}{r_1}=2.07,$$ giving $\eta_{\text{fin}}\approx0.96$ (Fig. B1, confirmed from the annular-fin Bessel solution).
  3. Fin surface area. Two faces plus the rim: $$A_{\text{fin}}=2\pi\bigl(r_2^2-r_1^2\bigr)+2\pi r_2 t=2\pi(0.030^2-0.015^2)+2\pi(0.030)(0.002)=0.004618\ \mathrm{m^2}.$$
  4. Heat from a finned metre (part b). With 200 fins/m occupying $0.40$ m and $0.60$ m of bare pipe between them, $$q_{\text{fins}}=n\eta h A_{\text{fin}}\Delta T=200(0.965)(60)(0.004618)(110)=5.88\times10^{3}\ \mathrm{W},$$ $$q_{\text{between}}=h(\pi D)(1-nt)\Delta T=60(\pi\times0.030)(0.60)(110)=373\ \mathrm{W},$$ so $q_{\text{finned}}=6.25\times10^{3}\ \mathrm{W/m}$ and $$\boxed{\Delta q=q_{\text{finned}}-q_{\text{bare}}=6253-622\approx5.63\times10^{3}\ \mathrm{W/m}\ (\approx10\times)}.$$
QuantityResult
(a) Bare-pipe heat rate622 W/m
Fin efficiency $\eta$≈ 0.96
Finned-tube heat rate≈ 6.25 kW/m
(b) Increase $\Delta q$≈ 5.63 kW/m (≈10×)