# Natural convection on inclined and horizontal surfaces

## Inclined Flat Surface

Depending on whether the wall is heated or cooled and tilted upward or downward, there are four different configurations for convection over an inclined flat plate (see Fig. 1). While the effect of inclination thickens the boundary layers for upper surface of case (a) and bottom surface of case (b), the boundary layer becomes thinner for bottom surface of case (a) and upper surface of case (b). When the inclination angle is moderate ( $-{{60}^{\circ }}<\phi <{{60}^{\circ }}$), the momentum equation for laminar natural convection over an inclined wall is , $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}+g\cos \phi \beta (T-{{T}_{\infty }})$

where the gravitational acceleration g has been replaced by the component of the gravitational acceleration along the direction that is parallel to the inclined plate, gcosφ. For laminar natural convection, the following empirical correlations ${{\overline{\text{Nu}}}_{L}}=0.68+\frac{\text{0}\text{.67Ra}_{L}^{1/4}}{{{[1+{{(0.492/\Pr )}^{9/16}}]}^{4/9}}}$

is still valid if the Rayleigh number is defined as: $\text{R}{{\text{a}}_{x}}=\frac{g\cos \phi \beta ({{T}_{w}}-{{T}_{\infty }}){{x}^{3}}}{\alpha \nu }$ (1)

For turbulent flow, Vliet (1969) suggested that the experimental results correlated better using g, instead of gcosφ, i.e. heat transfer in turbulent natural convection is not sensitive to the inclination angle and the following empirical correlations can be used. ${{\overline{\text{Nu}}}_{L}}={{\left\{ 0.825+\frac{\text{0}\text{.387Ra}_{L}^{1/6}}{{{[1+{{(0.492/\Pr )}^{9/16}}]}^{8/27}}} \right\}}^{2}}$

## Horizontal Flat Surface

Natural convection on a horizontal surface can find its application in nature such as natural convection on ground or water surfaces, as well as on the top of a building. It can also be applied to electronic cooling. Due to its complicated nature, analytical solutions are only available for limited cases. Rotem and Claasen (1969) studied natural convection over a semi-infinite isothermal flat plate with a single leading edge (see Fig. 2). The fluid near the flat plate is heated and becomes lighter, which results in a pressure difference that causes flow over the surface near the leading edge. The continuity and energy equations are the same as the case of the vertical plate: $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ $u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}$

The momentum equation is, however, different from that of the vertical plate because the direction of gravity is normal to the horizontal plate. The momentum equations are: $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}-\frac{1}{\rho }\frac{\partial {{p}_{d}}}{\partial x}$ (2) $g\beta (T-{{T}_{\infty }})=\frac{1}{\rho }\frac{\partial {{p}_{d}}}{\partial y}$ (3)

where pd is the dynamic pressure whose gradient is the driving force for the boundary layer flow. Defining the following similarity variables: $\eta =\frac{y}{x}{{\left( \frac{\text{G}{{\text{r}}_{x}}}{5} \right)}^{1/5}},\text{ }\psi =5\nu F(\eta ){{\left( \frac{\text{G}{{\text{r}}_{x}}}{5} \right)}^{1/5}},\text{ }{{p}_{d}}=\frac{5{{\nu }^{2}}\rho }{x}F(\eta ){{\left( \frac{\text{G}{{\text{r}}_{x}}}{5} \right)}^{4/5}}G(\eta )$ (4)

eqs. (15), (2), (3) and (16) become (Pera and Gebhart, 1973): ${F}'''+3{F}''F-2{{{f}'}^{2}}+\frac{2}{5}{G}'\eta -\frac{2}{5}G=0$ (5)
 G' = θ (6) ${\theta }''+3\Pr F{\theta }'=0$ (7)

which can be numerically solved to obtain the following local Nusselt number: $\text{N}{{\text{u}}_{x}}=0.394\text{Gr}_{x}^{1/5}{{\Pr }^{1/4}}$ (8)

Equation (8) is valid for an isothermal surface. For a horizontal surface with constant heat flux, the local Nusselt number is: $\text{N}{{\text{u}}_{x}}=0.501\text{Gr}_{x}^{1/5}{{\Pr }^{1/4}}$ (9)

While the above solution is for the case of a heated surface facing upward, it can also be applied to the case of a cold surface facing downward. However, the similarity solution cannot be applied for the case of a hot surface facing downward or a cold surface facing upward.

For most applications, the horizontal surface cannot be treated as a semi-infinite surface with only one leading edge. Two extreme types of flow situation arise for the horizontal surface with finite size. If the hot surface of the plate faces upwards or cold surface of the plate faces downwards, a plume is formed whereby the hotter fluid layers rise up or colder fluid mass goes down, respectively. For the cases that heated surface facing downward [bottom surface of Fig. 3 (a)] or cold surface facing upward [upper surface of Fig. 3. (b)], the flow leaves the boundary layer as a vertical plume at the central region. If the temperature difference happens to be large enough, then the flow occurs from all over the horizontal surface. On the contrary, when the hot surface faces downward [bottom surface of Fig. 3(a)] or the cold one faces upward [upper surface of Fig. 3(b)], the boundary layer covers the entire surface and the flow is spilled over the edges.

For the case of upward facing hot isothermal surfaces or downward facing cold isothermal surfaces, the following correlation can be applied (McAdams, 1954): ${{\overline{\text{Nu}}}_{L}}=\left\{ \begin{matrix} 0.54Ra_{L}^{1/4} & {{10}^{4}}<\text{R}{{\text{a}}_{L}}<{{10}^{7}} \\ 0.15Ra_{L}^{1/3} & {{10}^{7}}<\text{R}{{\text{a}}_{L}}<{{10}^{9}} \\ \end{matrix} \right.$ (10)

The characteristic length in the average Nusselt number and Rayleigh number isL = A / P, where A refers to the area of the plate and P is the perimeter. It should be pointed out that the average Nusselt numbers in eqs. (10) and (11) are proportional to $\text{Ra}_{L}^{1/4}\text{ or Ra}_{L}^{1/3}$, not $\text{Gr}_{L}^{1/5}\text{P}{{\text{r}}^{\text{1/4}}}$ as was suggested by the above similarity solution. Thus, the applicability of the similarity solution is limited to natural convection over a horizontal surface due to the complicated nature of the problem. The corresponding correlation for downward facing hot surfaces or upward facing cold surfaces is: ${{\overline{\text{Nu}}}_{L}}=\begin{matrix} 0.27Ra_{L}^{1/4} & {{10}^{5}}<\text{R}{{\text{a}}_{L}}<{{10}^{10}} \\ \end{matrix}$ (11)

While eqs. (10) and (11) are for isothermal surfaces, they can also be used for the case of uniform heat flux provided the average Nusselt and Rayleigh numbers are defined based on the average surface temperature of the flat surface.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

McAdams, W.H., 1954, Heat Transmission, 3rd Ed. McGraw-Hill, New York, NY.

Pera, L., and Gebhart, B., 1973, “Natural Convection Boundary Layer Flow over Horizontal and Slightly Inclined Surfaces,” Int. J. Heat Mass Transfer, Vol. 16, pp. 1131-1146.

Rotem, Z., and Claassen, L., 1969, “Natural Convection above Unconfined Horizontal Surface,” J. Fluid Mech., Vol. 39, pp. 173-192.

Vliet, G.C., 1969, “Natural Convection Local Heat Transfer on Constant-Heat-Flux Inclined Surface,” ASME J. Heat Transfer, Vol. 91, pp. 511-516.