25-Nav-A1 Fundamentals of Naval Architecture · May-98-Mar-A1 2016

Question 6 of 8: Oil-Heating Tube in Cross-Flow Hot Gas

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

Notes on this paper

National Examinations, May 2016 — 98-Mar-A1 Applied Thermodynamics and Heat Transfer, 3 hours, open book (Part A: Thermodynamics, Part B: Heat Transfer; 5 of 8 questions required, all 8 answered below for full study coverage).

Reference texts: Cengel & Boles, Thermodynamics: An Engineering Approach; Sonntag, Borgnakke & Van Wylen, Fundamentals of Thermodynamics; Incropera & DeWitt, Fundamentals of Heat and Mass Transfer.

It is solved as the thermodynamics/heat-transfer exam it actually is.

Question 6: Oil-Heating Tube in Cross-Flow Hot Gas

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. Unused engine oil is warmed inside a thin-walled tube by a hot gas (air properties) in cross-flow outside.

Given data
Quantity	Symbol	Value
Oil mass flow rate	$\dot{m}$	0.25 kg/s
Tube diameter, length	$D,\,L$	50 mm, 6 m
Oil inlet temperature	$T_{m,i}$	23°C
Gas (crossflow) temperature, velocity	$T_\infty,\,V$	300°C, 10 m/s
Maximum allowable wall temperature	$T_t$	100°C

Find. Whether the tube-wall temperature exceeds $100^\circ\text{C}$ anywhere along its length.

Oil inside a thin-walled tube, heated by hot gas in cross-flow outside — two convective resistances in series through a negligible tube wall.

Approach. Compute the oil-side (internal, laminar-entry) and gas-side (external, cross-flow cylinder) convection coefficients, combine into an overall $U$ to get the oil outlet temperature, then find the local wall temperature from the two resistances in parallel-balance — and check it against every point along the tube, not just the outlet.

Oil-side coefficient (internal, laminar, combined entry). At a mean bulk oil temperature near 30°C, unused-engine-oil properties (Incropera Table A.5) give $Re_D=4\dot m/(\pi D\mu)\approx16$ (deeply laminar) and $Re_D Pr_D D/L\approx706$ (strong entry effects over the full 6 m). The Sieder–Tate combined-entry correlation, $$Nu_D=1.86\left(Re_DPr_D\frac{D}{L}\right)^{1/3}\left(\frac{\mu}{\mu_s}\right)^{0.14}$$ gives $$\boxed{h_i\approx60.4\text{ W/m}^2\text{K}}$$
Gas-side coefficient (external cross-flow over a cylinder). At the gas film temperature ($\approx500\text{ K}$, consistent with the converged wall temperature below), $Re_D=VD/\nu\approx1.32\times10^4$; the Churchill–Bernstein correlation gives $Nu_D\approx61.7$, so $$\boxed{h_o\approx50.2\text{ W/m}^2\text{K}}$$
Oil outlet temperature. With the tube wall thin (negligible conduction resistance), the overall coefficient is $U=(1/h_i+1/h_o)^{-1}\approx27.4\text{ W/m}^2\text{K}$ on area $A_s=\pi DL=0.9425\text{ m}^2$. Treating the hot gas as an effectively constant-temperature reservoir: $$\frac{T_\infty-T_{m,o}}{T_\infty-T_{m,i}}=\exp\!\left(-\frac{UA_s}{\dot mc_p}\right)\ \Rightarrow\ \boxed{T_{m,o}\approx37.5^\circ\text{C}}$$
Local wall temperature (inlet and outlet). With the wall thin, the two convective fluxes balance locally: $h_o(T_\infty-T_w)=h_i(T_w-T_m)$, so $T_w=\dfrac{h_oT_\infty+h_iT_m}{h_o+h_i}$. Since $h_o$ and $h_i$ are comparable in magnitude, $T_w$ sits roughly *midway* between the gas and oil temperatures, not close to the oil: $$T_w(\text{inlet}, T_m=23^\circ\text{C})\approx148.7^\circ\text{C},\qquad T_w(\text{outlet}, T_m=37.5^\circ\text{C})\approx156.6^\circ\text{C}$$ Both are far above the $100^\circ\text{C}$ decomposition limit — and since $T_w$ increases monotonically with $T_m$ along the tube, the wall exceeds $100^\circ\text{C}$ at every point from inlet to outlet, not merely at the ends. $$\boxed{\text{Yes: the wall temperature (}\approx149\text{--}157^\circ\text{C) exceeds }100^\circ\text{C everywhere along the tube.}}$$

Question 6 — final results
Quantity	Value
Oil-side coefficient	$h_i\approx60.4\text{ W/m}^2\text{K}$
Gas-side coefficient	$h_o\approx50.2\text{ W/m}^2\text{K}$
Oil outlet temperature	$T_{m,o}\approx37.5^\circ\text{C}$
Wall temperature, inlet / outlet	$\approx148.7^\circ\text{C}$ / $\approx156.6^\circ\text{C}$
Verdict	Wall exceeds 100°C limit everywhere — overheating problem confirmed

Topic: Coupled internal/external convection through a thin tube wall — local wall temperature from two resistances in series.

Key relations: Sieder–Tate combined-entry laminar correlation; Churchill–Bernstein cross-flow-cylinder correlation; local wall balance $h_o(T_\infty-T_w)=h_i(T_w-T_m)$.

Why this works: A thin wall carries the same heat flux on both sides, so its local temperature is a convection-coefficient-weighted average of the two fluid temperatures — it is NOT close to whichever fluid has the lower temperature unless that fluid's convection coefficient dominates. Here the oil's low thermal conductivity/high viscosity gives it a coefficient of the same order as the gas side, so the wall sits far closer to the hot-gas temperature than intuition (based on the oil being "the cold fluid") would suggest.

Common pitfall: assuming the wall temperature tracks close to the internal fluid because "that's the fluid being protected" — it only does so if $h_i\gg h_o$; checking only the tube inlet or only the outlet instead of the whole length (here BOTH ends already exceed the limit).

Source: Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, Ch.7 (external flow, cross-flow over a cylinder) & Ch.8 (internal laminar flow, combined entry length).

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