The goal of this paper is to present a versatile framework for solution verification of PDE＇s. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that construc...The goal of this paper is to present a versatile framework for solution verification of PDE＇s. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W. Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are separately an incompressible back step flow test case and a heat transfer problem for a battery. Our error estimate might be ultimately verified with a near by manufactured solution. While our pro- cedure is systematic and requires numerous computation of residuals, one can take advantage of distributed computing to get quickly the error estimate.展开更多
A combined conduction and radiation heat transfer model was used to simulate the heat transfer within wafer and investigate the effect of thermal transport properties on temperature non-uniformity within wafer surface...A combined conduction and radiation heat transfer model was used to simulate the heat transfer within wafer and investigate the effect of thermal transport properties on temperature non-uniformity within wafer surface. It is found that the increased conductivities in both doped and undoped regions help reduce the temperature difference across the wafer surface. However, the doped layer conductivity has little effect on the overall temperature distribution and difference. The temperature level and difference on the top surface drop suddenly when absorption coefficient changes from 104 to 103 m-1. When the absorption coefficient is less or equal to 103 m-1, the temperature level and difference do not change much. The emissivity has the dominant effect on the top surface temperature level and difference. Higher surface emissivity can easily increase the temperature level of the wafer surface. After using the improved property data, the overall temperature level reduces by about 200 K from the basis case. The results will help improve the current understanding of the energy transport in the rapid thermal processing and the wafer temperature monitor and control level.展开更多
The flow of pseudoplastic power-law fluids with different flow indexes at a microchannel plate was studied using computational fluid dynamic simulation.The velocity distribution along the microchannel plate and especi...The flow of pseudoplastic power-law fluids with different flow indexes at a microchannel plate was studied using computational fluid dynamic simulation.The velocity distribution along the microchannel plate and especially in the microchannel slits,flow pattern along the outlet arc and the pressure drop through the whole of microchannel plate were investigated at different power-law flow indexes.The results showed that the velocity profile in the microchannel slits for low flow index fluids was similar to the plug flow and had uniform pattern.Also the power-law fluids with lower flow indexes had lower stagnation zones near the outlet of the microchannel plate.The pressure drop through the microchannel plate showed huge differences between the fluids.The most interesting result was that the pressure drops for power-law fluids were very smaller than that of Newtonian fluids.In addition,the heat transfer of the fluids through the microchannel with different channel numbers in a wide range of Reynolds number was investigated.For power-law fluid with flow index(n=0.4),the Nusselt number increases continuously as the number of channels increases.The results highlight the potential use of using pseudoplastic fluids in the microheat exchangers which can lower the pressure drop and increase the heat transfer efficiency.展开更多
基金Sandia Nat.Lab.Sandia is a multiprogram laboratory operated by Sandia Corporation,a Lockheed Martin Company,for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000
文摘The goal of this paper is to present a versatile framework for solution verification of PDE＇s. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W. Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are separately an incompressible back step flow test case and a heat transfer problem for a battery. Our error estimate might be ultimately verified with a near by manufactured solution. While our pro- cedure is systematic and requires numerous computation of residuals, one can take advantage of distributed computing to get quickly the error estimate.
基金Project(N110204015)supported by the Fundamental Research Funds for the Central Universities,ChinaProject(2012M510075)supported by the China Postdoctoral Science Foundation
文摘A combined conduction and radiation heat transfer model was used to simulate the heat transfer within wafer and investigate the effect of thermal transport properties on temperature non-uniformity within wafer surface. It is found that the increased conductivities in both doped and undoped regions help reduce the temperature difference across the wafer surface. However, the doped layer conductivity has little effect on the overall temperature distribution and difference. The temperature level and difference on the top surface drop suddenly when absorption coefficient changes from 104 to 103 m-1. When the absorption coefficient is less or equal to 103 m-1, the temperature level and difference do not change much. The emissivity has the dominant effect on the top surface temperature level and difference. Higher surface emissivity can easily increase the temperature level of the wafer surface. After using the improved property data, the overall temperature level reduces by about 200 K from the basis case. The results will help improve the current understanding of the energy transport in the rapid thermal processing and the wafer temperature monitor and control level.
文摘The flow of pseudoplastic power-law fluids with different flow indexes at a microchannel plate was studied using computational fluid dynamic simulation.The velocity distribution along the microchannel plate and especially in the microchannel slits,flow pattern along the outlet arc and the pressure drop through the whole of microchannel plate were investigated at different power-law flow indexes.The results showed that the velocity profile in the microchannel slits for low flow index fluids was similar to the plug flow and had uniform pattern.Also the power-law fluids with lower flow indexes had lower stagnation zones near the outlet of the microchannel plate.The pressure drop through the microchannel plate showed huge differences between the fluids.The most interesting result was that the pressure drops for power-law fluids were very smaller than that of Newtonian fluids.In addition,the heat transfer of the fluids through the microchannel with different channel numbers in a wide range of Reynolds number was investigated.For power-law fluid with flow index(n=0.4),the Nusselt number increases continuously as the number of channels increases.The results highlight the potential use of using pseudoplastic fluids in the microheat exchangers which can lower the pressure drop and increase the heat transfer efficiency.
