25-Nav-A1 Fundamentals of Naval Architecture · May-98-Mar-A1 2017

Question 6 of 8: Oil Heated by Hot Gas in Crossflow

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

Notes on this paper

National Examinations, May 2017 — 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 Heated by Hot Gas in Crossflow (20 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. Thin-walled tube, $D=50$ mm, $L=6$ m; oil $\dot m=0.025$ kg/s, $T_{m,i}=23\,{}^{\circ}\text{C}$; hot gas (air properties) $T_\infty=300\,{}^{\circ}\text{C}$, $V=10$ m/s crossflow; wall limit $100\,{}^{\circ}\text{C}$.

Oil flowing inside a thin-walled tube, heated by hot gas in crossflow.

Find. Whether the tube wall exceeds 100°C anywhere along its length.

Approach. Evaluate the external (gas, Churchill–Bernstein crossflow) and internal (oil, thermally-developing laminar, Hausen) convection coefficients using standard property tables (Incropera Tables A.4/A.5), then combine them in a series resistance network to estimate the wall temperature at the tube inlet and outlet, where oil is respectively coldest and (after warming) still well below the gas temperature.

Internal flow regime (oil, properties ≈ 300 K: $\mu=0.486\text{ Pa}\!\cdot\!\text{s}$, $k=0.145\text{ W/m}\!\cdot\!\text{K}$, $c_p=1909\text{ J/kg}\!\cdot\!\text{K}$, $\text{Pr}\approx6400$). $$\text{Re}_D=\dfrac{4\dot m}{\pi D\mu}=\dfrac{4(0.025)}{\pi(0.05)(0.486)}=1.31\quad(\text{deeply laminar})$$
Internal (oil-side) Nusselt number — thermally-developing laminar flow, high Pr (Hausen correlation). Graetz number $\text{Gz}=(D/L)\text{Re}_D\text{Pr}=69.9$, $$\overline{\text{Nu}}_D=3.66+\dfrac{0.0668\,\text{Gz}}{1+0.04\,\text{Gz}^{2/3}}=3.66+2.78=6.44\ \Rightarrow\ h_{oil}=\dfrac{\overline{\text{Nu}}_D k_{oil}}{D}=18.7\text{ W/m}^2\text{K}$$
External flow (gas/air, properties ≈ 500 K film temp.: $\nu=38.8\times10^{-6}\text{ m}^2/\text{s}$, $k=0.0407\text{ W/m}\!\cdot\!\text{K}$, $\text{Pr}=0.684$). $\text{Re}_D=VD/\nu=10(0.05)/38.8\times10^{-6}=12{,}890$; Churchill–Bernstein gives $\text{Nu}_D=60.9$, so $$h_{gas}=\dfrac{\text{Nu}_D k}{D}=\dfrac{60.9(0.0407)}{0.05}=49.5\text{ W/m}^2\text{K}$$
Overall coefficient and oil outlet temperature. $\dfrac1U=\dfrac1{h_{gas}}+\dfrac1{h_{oil}}\Rightarrow U=13.6\text{ W/m}^2\text{K}$. With surface area $A=\pi DL=0.943\text{ m}^2$ and $\dot m c_p=47.7\text{ W/K}$, treating the gas as an isothermal reservoir, $$T_{m,o}=T_\infty-(T_\infty-T_{m,i})e^{-UA/\dot mc_p}=300-277\,e^{-0.268}=88.1\,{}^{\circ}\text{C}$$
Wall temperature check (series-resistance split of the local heat flux, at inlet and outlet). $T_s=T_m+\dfrac{U(T_\infty-T_m)}{h_{oil}}$: $$T_{s,inlet}=23+\dfrac{13.6(300-23)}{18.7}\approx\boxed{224\,{}^{\circ}\text{C}},\qquad T_{s,outlet}=88.1+\dfrac{13.6(300-88.1)}{18.7}\approx\boxed{242\,{}^{\circ}\text{C}}$$ Both are far above the $100\,{}^{\circ}\text{C}$ limit, so yes — there is a problem: the oil-side convection is so weak (Re$_D\approx1.3$, deep laminar) relative to the gas-side that the tube wall runs at $\sim$200–240°C along essentially the whole tube, well above the decomposition limit, even though the oil bulk itself only warms to $88\,{}^{\circ}\text{C}$.

Question 6 — final results
Quantity	Value
Oil-side coefficient	$h_{oil} = 18.7\text{ W/m}^2\text{K}$
Gas-side coefficient	$h_{gas} = 49.5\text{ W/m}^2\text{K}$
Oil outlet temperature	$T_{m,o} = 88.1\,{}^{\circ}\text{C}$
Estimated wall temperature, inlet / outlet	$\approx224\,{}^{\circ}\text{C}\ /\ 242\,{}^{\circ}\text{C}$
Verdict	Yes — wall exceeds the 100°C limit

Check: engine-oil properties are taken from the standard Incropera Table A.5 (unused engine oil, evaluated near 300 K) since the paper does not append an oil property table; the qualitative conclusion (wall temperature roughly double the 100°C limit) is insensitive to the exact property set because $h_{oil}\ll h_{gas}$ by more than a factor of two regardless.

Topic: Combined internal (laminar, thermally-developing) and external (crossflow) forced convection.

Key relations: $\text{Re}_D=4\dot m/(\pi D\mu)$ for internal flow; Hausen entry-length correlation for laminar, high-Pr internal flow; Churchill–Bernstein for external crossflow over a cylinder; series thermal resistance $1/U=1/h_1+1/h_2$.

Why this works: High-viscosity liquids like oil reach very low Reynolds numbers even at modest mass flow rates, driving deep laminar flow with an internal coefficient that can be an order of magnitude below a "merely" gaseous external coefficient — counter-intuitive until the huge Prandtl number (thermal diffusivity swamped by momentum diffusivity) is recognized as the root cause.

Common pitfall: Assuming a gas-side (external) convection coefficient must be smaller than a liquid-side (internal) one "because it's just air" — here the opposite holds because the oil's Reynolds number is so low.

Source: Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, Ch. 7 (external flow — cylinder in crossflow) and Ch. 8 (internal flow — laminar thermal entry length).

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