Natural Convection Cooling of a Tank Wall: 22-Agric-A3 Heat Engineering, December 2014, Q4

Problem 4: Natural Convection Cooling of a Tank Wall (25 points)

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.

Quantity	Value
Tank fluid temperature, T_fluid	40°C
Ambient air temperature, T∞	14°C
Volumetric expansion coefficient, β	0.00348 K⁻¹
Wall (side) height, L	0.4 m
Kinematic viscosity of air, ν (at 27°C)	1.556×10⁻⁵ m²/s
Thermal diffusivity of air, α (at 27°C)	2.203×10⁻⁵ m²/s
Prandtl number, Pr	0.711

Find. The average convection coefficient h, the average heat flux q, and the thermal boundary-layer thickness δ at the top of the wall (x = L).

Approach. Because the internal coefficient is very large, the outer wall surface sits essentially at the tank fluid temperature (40°C), so this is an external natural-convection problem on a vertical isothermal plate. Form the Grashof and Rayleigh numbers at the given film-temperature properties, apply the laminar vertical-plate correlation for the average Nusselt number, then use the companion integral-solution result for the boundary-layer thickness at the top of the plate.

Check: air's thermal conductivity k is not printed in the source data (only ν, α and Pr are given). k ≈ 0.0262 W/m·K is the standard air-property-table value at 300 K, consistent with the given ν and α (their ratio reproduces the stated Pr to within rounding) — this is the value an open-book candidate would read from the same reference table.

Natural-convection boundary layer on the tank's vertical side: warm air rises from the bottom leading edge, thickening to δ ≈ 17.1 mm by the top (x = L = 0.4 m).

Film temperature and driving ΔT. The wall runs at the tank fluid temperature (internal h very large), so $T_s = 40\,{}^{\circ}\text{C}$, $T_\infty=14\,{}^{\circ}\text{C}$, $\Delta T = 26\,\text{K}$; the given properties are already evaluated at the correct film temperature, $T_f=(40+14)/2=27\,{}^{\circ}\text{C}$.
Grashof and Rayleigh numbers. Using the full height as the characteristic length ($L=0.4\ \text{m}$), $$Gr_L=\frac{g\beta\Delta T L^3}{\nu^2}=\frac{9.81(0.00348)(26)(0.4)^3}{(1.556\times10^{-5})^2} = 2.35\times10^{8}$$ $$Ra_L = Gr_L\,Pr = 2.35\times10^{8}\times0.711 = \boxed{1.67\times10^{8}}$$ This falls inside the laminar range for a vertical plate, $10^4 \lt Ra_L \lt 10^9$.
Average Nusselt number and convection coefficient. With the Rayleigh number in the laminar range, McAdams' correlation applies, $$\overline{Nu}_L = 0.59\,Ra_L^{1/4} = 0.59(1.67\times10^{8})^{1/4} = 67.1$$ $$h = \frac{\overline{Nu}_L\,k}{L}$$ Air's thermal conductivity at 300 K (27°C), consistent with the given ν and α, is $k \approx 0.0262\ \text{W/m}\cdot\text{K}$ (standard air-property table), so $$h = \frac{67.1\times0.0262}{0.4} = \boxed{4.40\ \text{W/m}^2\cdot\text{K}}$$
Average heat flux. $$q = h\,\Delta T = 4.40\times26 = \boxed{114.4\ \text{W/m}^2}$$
Thermal boundary-layer thickness at the top (x = L). The companion integral-solution result for the laminar natural-convection boundary layer on a vertical isothermal plate gives $$\frac{\delta}{x} = 3.93\,Pr^{-1/2}(0.952+Pr)^{1/4}Gr_x^{-1/4}$$ Evaluated at $x=L=0.4\ \text{m}$ (using $Gr_L$ found above), $$\delta = 0.4\times3.93\times(0.711)^{-1/2}\times(0.952+0.711)^{1/4}\times(2.35\times10^{8})^{-1/4} = \boxed{0.0171\ \text{m} = 17.1\ \text{mm}}$$

Quantity	Result
Rayleigh number, Ra_L	1.67×10⁸
Average convection coefficient, h	4.40 W/m²·K
Average heat flux, q	114.4 W/m²
Boundary-layer thickness at top, δ	17.1 mm

Concept & Theory

Topic: Natural (free) convection on a heated vertical plate.

Key relations: $Gr_L=g\beta\Delta T L^3/\nu^2$; $Ra_L=Gr_L\,Pr$; laminar vertical-plate correlation $\overline{Nu}_L=0.59\,Ra_L^{1/4}$ for $10^4 \lt Ra_L \lt 10^9$; boundary-layer thickness $\delta/x=3.93\,Pr^{-1/2}(0.952+Pr)^{1/4}Gr_x^{-1/4}$.

Why this works: Buoyancy driven by the density difference between the heated air near the wall and the cooler free stream replaces the free-stream velocity of forced convection; the Grashof number is natural convection's Reynolds-number analogue, comparing buoyant to viscous forces. The boundary layer starts at zero thickness at the bottom leading edge and grows with height exactly as a forced-convection laminar boundary layer grows with distance, which is why both h and δ depend on the SAME characteristic length raised to the same 1/4 power, just with Gr in place of Re.

Common pitfall: Treating the very-large internal h as meaning h is also large on the OUTSIDE (air) face — it is the opposite: the large internal coefficient means the wall offers negligible thermal resistance on the tank-fluid side, so essentially all the temperature drop, and all of the (small) natural-convection resistance, is on the air side. A second common slip is using the LOCAL boundary-layer correlation instead of the AVERAGE one for h.

Source: J.P. Holman, Heat Transfer, Ch. 7 (natural convection systems, vertical plates, boundary-layer thickness).

Question 4 of 4: Natural Convection Cooling of a Tank Wall

Problem 4: Natural Convection Cooling of a Tank Wall (25 points)