A unified integral theory of laminar natural convection over surfaces at arbitrary inclination from horizontal to vertical

Similarity analysis shows that Nux varies as Grx1/4 for natural convection on an isothermal vertical surface but Nux varies as Grx1/5 for isothermal horizontal surfaces. It is thus difficult to develop a rigorously-derived, closed-form solution for Nux on a surface with arbitrary inclination. In the present study we have formulated, for the first time, a unified integral theory for laminar natural convection on an arbitrarily inclined surface, both for specified variation in surface temperature (Tw(x) = T∞ + f1(x)) and surface heat flux (qw = f2(x)), such that the Nusselt number matches with results obtained from the similarity analysis in the limiting cases of vertical and horizontal surfaces. The predictions of the present formulation also agree well with previous computational and experimental results at intermediate angles of inclination between the vertical and the horizontal. f1(x) or f2(x) can be any arbitrary function, including power law variation, and represents a differentially heated surface. Another important feature of the present integral theory is that the developed generalized equations can accommodate arbitrary orders of polynomials (λ and χ) representing the velocity and temperature profiles, and optimum values for λ and χ have been systematically determined for various boundary conditions (i.e. λ = 4, χ = 2 for isothermal case and λ = 3, χ = 2 for constant-heat-flux case). Because of the simplicity of the present theory, it is easy to generate results for combinations of Grashof number, Prandtl number and inclination angle not presented here. The different physical mechanisms for natural convection on vertical and horizontal surfaces (buoyancy versus indirect pressure difference) are explained with the help of the present analysis. It is shown that for moderate to high Prandtl number fluids, the natural convection mechanism for vertical surface is the dominating factor for a large range of inclination angles except for near horizontal configurations. The range of inclination angles for which the vertical solution predominates decreases as the Prandtl number decreases. For very low Prandtl number fluids at low Grashof number, the vertical mechanism applies only to nearly vertical surfaces. A physical explanation for such behaviour is discovered here, for the first time, in terms of the relative magnitudes of the buoyancy and indirect pressure difference. Compact scaling laws for significant data reduction are proposed and explained. New algebraic correlations have been developed that give Nusselt number as explicit functions of Grashof number, Prandtl number and inclination angle. A new methodology for the representation of the results brings out more powerfully the role of inclination angle in determining the heat transfer rate as well as the mechanism of natural convection.