Forced convection heat transfer from microstructures

The objectives of this study are to examine the validity of integral methods used with boundary layer theory for forced convection over a flat plate with an unheated starting length, investigate the breakdown of thermal boundary layer theory near the leading and trailing edges of the heated section, and develop a new correlation to judge the validity of, and if necessary, replace conventional boundary layer theory. The applications of this work include modern microstructure design where boundary layer theory fails to predict the heat transfer rate from extremely small (on the order of 10{dollar}mu m{dollar}) heated elements.; A conventional Blasius technique transforms the energy equation for steady, two-dimensional incompressible flow into an elliptic-to-parabolic equation. A second-stage transformation is introduced to solve the low-order parabolic equation. It is shown that the error is the local Nusselt number from integral methods is approximately 2 percent for 0.5 {dollar}le{dollar} Pr {dollar}le{dollar} 100 compared with the present closed-form solution. This implies that integral solutions are very good approximations to the solutions of the boundary layer equations. Very close to the leading and trailing edges, however, boundary layer theory breaks down and the full elliptic equation must be solved. Therefore a new principle is developed for solving the elliptic equation, which treats the boundary layer solution as the outer expansion of the elliptic-to-parabolic equation. The inner expansion is found by stretching two independent variables simultaneously. Trailing-edge effects are considered using superposition methods. It is revealed that shear flow equations must be solved near the leading and trailing edges of the heated part of the plate. It is shown that the conventional boundary condition, a discontinuous temperature distribution at the leading and trailing edges, contains a singularity which is nonintegrable for average Nusselt number if the thermal diffusion along the x-direction is involved. A more practical mixed boundary condition is proposed which is solved using the Wiener-Hopf method. 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 in the regime of laminar flow, 100 {dollar}le{dollar} Re{dollar}sb{lcub}xsb0{rcub}{dollar} and 0.5 {dollar}le{dollar} Pr {dollar}le{dollar} 100; (UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}&overline{lcub}rm Nu{rcub}sb{lcub}l{rcub} = 0.6626 {lcub}rm Prsp{lcub}1/3{rcub} Re{rcub}sbsp{lcub}xsb0 + l{rcub}{lcub}1/2{rcub} lbrack 1 - ({lcub}xsb0over xsb0 + l{rcub})sp{lcub}3/4{rcub}rbracksp{lcub}2/3{rcub}crcr &sk{lcub}100{rcub} lbrack 1 + {lcub}0.3981 (xsb0/l)sp{lcub}0.5987{rcub}over{lcub}rm Pr{rcub}sp{lcub}0.3068{rcub}{lcub}rm Re{rcub}sbsp{lcub}xsb0{rcub}{lcub}0.4675{rcub}{rcub}rbrackcr{rcub}{dollar}{dollar}(TABLE/EQUATION ENDS)where the term containing 0.3981 is obtained from the present work. The error from this statistical correlation is less than 2 percent as compared with the original solution.