22-Mec-A1 Applied Thermodynamics and Heat Transfer · May 2016
Question 6 of 8: Oil-Heating Tube in Cross-Flow — Wall-Temperature Limit
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Reference texts: Çengel & Boles, Thermodynamics: An Engineering Approach (9th ed., McGraw-Hill) — closed- and open-system energy balances, steam tables, vapour and gas power cycles, and vapour-compression refrigeration; Çengel & Ghajar, Heat and Mass Transfer (6th ed.) and Incropera, DeWitt, Bergman & Lavine, Fundamentals of Heat and Mass Transfer (8th ed., Wiley) — cylindrical composite-wall conduction, internal-flow and cross-flow convection correlations, natural convection with radiation, and the ε–NTU heat-exchanger method. Freon-12 property data are taken from the appendix supplied with the exam; steam, air and engine-oil data from standard tables.
Paper format: National Examination 07-Mec-A1, 3 hours, open book. Part A — Thermodynamics (Q1–4); Part B — Heat Transfer (Q5–8). Each answer carries equal value; a complete paper is any five (three from one part, two from the other). All eight questions are solved in full below.
Figure 6 — Combined external cross-flow (gas side, $h_o$) and internal laminar oil flow ($h_i$). Because the oil-side coefficient is comparable to the gas-side one, the wall floats roughly midway between the oil and the 300 °C gas.
Given. Oil at 0.25 kg/s entering a 50 mm × 6 m tube at 23 °C, heated by a 300 °C gas (air properties) in cross-flow at 10 m/s, with a 100 °C wall-temperature limit. Find. Whether the wall stays below 100 °C anywhere along the tube.
Approach. Find the external cross-flow coefficient and the internal (laminar, thermally developing) oil coefficient, combine to an overall $U$, march the oil temperature down the tube, then evaluate the wall temperature from the two-film balance — the maximum sits at the oil exit.
External gas coefficient (cross-flow cylinder). At a film temperature ~200 °C, $\text{Re}_D=VD/\nu=1.43\times10^{4}$; the Churchill–Bernstein correlation gives $\text{Nu}_D=65$ and $$h_o=\frac{\text{Nu}_D\,k}{D}\approx50\ \text{W/m}^2\text{K}.$$
Internal oil coefficient (laminar, entry). For engine oil, $\text{Re}_D=4\dot m/\pi D\mu\approx13$ (deeply laminar). With Graetz number $\text{Gz}=(D/L)\text{Re}\,\text{Pr}\approx700$, the Hausen thermal-entry correlation gives $\text{Nu}_D=3.66+\dfrac{0.0668\,\text{Gz}}{1+0.04\,\text{Gz}^{2/3}}\approx14.9$, so $h_i\approx43\ \text{W/m}^2\text{K}.$
Overall coefficient and oil outlet. $1/U=1/h_i+1/h_o\Rightarrow U\approx23\ \text{W/m}^2\text{K}$; with $A_s=\pi DL=0.94$ m², $$\frac{T_\infty-T_{m,o}}{T_\infty-T_{m,i}}=e^{-UA_s/\dot m c_p}=e^{-0.046}\Rightarrow T_{m,o}\approx35\,{}^\circ\text{C}.$$ The oil bulk barely rises.
Wall temperature (hottest at exit). Equating internal and external fluxes, $T_w=\dfrac{h_oT_\infty+h_iT_m}{h_o+h_i}$. Even at the cold inlet $T_w\approx171\,{}^\circ\text{C}$; at the exit $$T_w=\frac{50(300)+43(35)}{50+43}\approx\boxed{177\,{}^\circ\text{C}}.$$
Conclusion. The wall runs at roughly 170–180 °C everywhere — far above the 100 °C limit — so yes, there is a serious problem: the oil film in contact with the tube wall will overheat and decompose, even though the bulk oil leaves at only ~35 °C.
Check — engine-oil properties assumed.
Engine-oil properties ($\mu\approx0.49$ Pa·s, $k\approx0.145$ W/m·K, $c_p\approx1910$ J/kg·K near 300 K) are taken from standard tables; the exact oil grade is unspecified. The wall temperature depends on the ratio $h_i/h_o$, but for any plausible viscous oil $h_i$ stays within a factor of ~2 of $h_o$, so the wall remains well above 100 °C and the qualitative conclusion (a decomposition problem) is robust.
