Forced Convection Heat Transfer From Microstructures

For many microstructures, which utilize forced convection cooling, the average thickness of the thermal boundary layer is of the same order as the length of the heated element. For these cases, thermal boundary layer theory is invalid. The elliptic energy equation for steady, two-dimensional incompressible flow over a finite flat plate with insulated starting and ending lengths is analyzed utilizing matched asymptotic expansions. A conventional Blasius technique transforms the energy equation into an elliptic-to-parabolic equation. A new technique is used that treats the boundary layer solution as the outer expansion of the elliptic-to-parabolic equation. The inner expansion, or leading-edge equation, is found by stretching to independent variables simultaneously. Trailing-edge effects are considered using superposition methods. A first-order composite formula is constructed based on the outer and inner expansions, which is uniformly valid over the entire surface of the plate. With the aid of statistics, a correlation is developed for the average Nusselt number where x₀ and l represent the lengths of the insulated starting section and the heated element, respectively. This correlation is accurate to within 2 percent as compared with the entire composite solution.