The Numerical Optimization of a Neumann-Type Boundary for the Graetz Problem

REFERENCES

• A. Bejan , Convection Heat Transfer , , 3rd ed. , John Wiley & Sons, Inc. , New York , 2004 .
   Google Scholar
• E. Papoutsakis , D. Ramkrishna , and H. C. Lim , The Extended Graetz Problem with Prescribed Wall Flux , Aiche J. , vol. 26 , no. 5 , pp. 779 – 787 , 1980 .
   Web of Science ®Google Scholar
• P. A. Ramachandran , Boundary Integral Solution Method for the Graetz Problem, Numer. Heat Transfer B , vol. 23, no. 2, pp. 257–268, 1993.
   Web of Science ®Google Scholar
• B. Weigand , M. Kanzamar , and H. Beer , The Extended Graetz Problem with Piecewise Constant Wall Heat Flux for Pipe and Channel Flows , Int. J. Heat and Mass Transfer , vol. 44 , no. 20 , pp. 3941 – 3952 , 2001 .
   Web of Science ®Google Scholar
• B. Weigand and D. Lauffer , The Extended Graetz Problem with Piecewise Constant Wall Temperature for Pipe and Channel Flows , Int. J. Heat and Mass Transfer , vol. 47 , no. 24 , pp. 5303 – 5312 , 2004 .
   Web of Science ®Google Scholar
• M. A. Ebadian and H. Y. Zhang , An Exact Solution of Extended Graetz Problem with Axial Heat-Conduction , Int. J. Heat and Mass Transfer , vol. 32 , no. 9 , pp. 1709 – 1717 , 1989 .
   Web of Science ®Google Scholar
• S. Bilir , Numerical-Solution of Graetz Problem with Axial Conduction , Numer. Heat Transfer A , vol. 21 , no. 4 , pp. 493 – 500 , 1992 .
   Web of Science ®Google Scholar
• X. Yin and H. H. Bau , The Conjugate Graetz Problem with Axial Conduction , J. Heat Transfer-Trans. of the ASME , vol. 118 , no. 2 , pp. 482 – 485 , 1996 .
   Web of Science ®Google Scholar
• J. Lahjomri and A. Oubarra , Analytical Solution of the Graetz Problem with Axial Conduction , J. Heat Transfer––Trans. of the ASME , vol. 121 , no. 4 , pp. 1078 – 1083 , 1999 .
   Web of Science ®Google Scholar
• F. P. Incropera , J. S. Kerby , D. F. Moffatt , and S. Ramadhyani , Convection Heat-Transfer from Discrete Heat-Sources in a Rectangular Channel , Int. J. Heat and Mass Transfer , vol. 29 , no. 7 , pp. 1051 – 1058 , 1986 .
   Web of Science ®Google Scholar
• A. B. McEntire and B. W. Webb , Local Forced Convective Heat-Transfer from Protruding and Flush-Mounted 2-Dimensional Discrete Heat-Sources , Int. J. Heat and Mass Transfer , vol. 33 , no. 7 , pp. 1521 – 1533 , 1990 .
   Web of Science ®Google Scholar
• H. J. Shaw and W. L. Chen , Laminar Forced-Convection in a Channel with Arrays of Thermal Sources , Warme Und Stoffubertragung––Thermo and Fluid Dynamics , vol. 26 , no. 4 , pp. 195 – 201 , 1991 .
   Web of Science ®Google Scholar
• C. Y. Wang , Optimum Placement of Heat-Sources in Forced-Convection , J. Heat Transfer—Trans. of the ASME , vol. 114 , no. 2 , pp. 508 – 510 , 1992 .
   Web of Science ®Google Scholar
• A. M. Morega and A. Bejan , Optimal Spacing of Parallel Boards with Discrete Heat-Sources Cooled by Laminar Forced-Convection , Numer. Heat Transfer A , vol. 25 , no. 4 , pp. 373 – 392 , 1994 .
   Web of Science ®Google Scholar
• A. Gupta and Y. Jaluria , Forced Convective Liquid Cooling of Arrays of Protruding Heated Elements Mounted in a Rectangular Duct , J. Electronic Packaging , vol. 120 , no. 3 , pp. 243 – 252 , 1998 .
   Web of Science ®Google Scholar
• A. K. da Silva , S. Lorente , and A. Bejan , Optimal Distribution of Discrete Heat Sources on a Plate with Laminar Forced Convection , Int. J. Heat and Mass Transfer , vol. 47 , no. 10–11 , pp. 2139 – 2148 , 2004 .
   Web of Science ®Google Scholar
• A. K. da Silva , and L. Gosselin , Evolutionary Placement of Discrete Heaters in Forced Convection , Numer. Heat Transfer A , vol. 54 , no. 1 , pp. 20 – 33 , 2008 .
   Web of Science ®Google Scholar
• M. Tye-Gingras , L. Gosselin , and A. K. da Silva , Syncronizing Time-Varying Discrete Heating Elements in Forced Convection for Minimal Thermal Resistance , Int. J. Heat and Mass Transfer , vol. 53 , no. 21–22 , pp. 4917 – 4926 , 2010 .
   Web of Science ®Google Scholar
• E. M. Sparrow and C. Prakash , Enhancement of Natural-Convection Heat-Transfer by a Staggered Array of Discrete Vertical Plates , J. Heat Transfer—Trans. of the ASME , vol. 102 , no. 2 , pp. 215 – 220 , 1980 .
   Web of Science ®Google Scholar
• T. J. Heindel , F. P. Incropera , and S. Ramadhyani , Conjugate Natural Convection from an Array of Discrete Heat Sources, 2. A Numerical Parametric Study , Int. J. Heat and Fluid Flow , vol. 16 , no. 6 , pp. 511 – 518 , 1995 .
   Web of Science ®Google Scholar
• T. J. Heindel , S. Ramadhyani , and F. P. Incropera , Conjugate Natural Convection from an Array of Discrete Heat Sources, 1. Two- and Three-Dimensional Model Validation , Int. J. Heat and Fluid Flow , vol. 16 , no. 6 , pp. 501 – 510 , 1995 .
   Web of Science ®Google Scholar
• M. Fujii , S. Gima , T. Tomimura , and X. Zhang , Natural Convection to Air from an Array of Vertical Parallel Plates with Discrete and Protruding Heat Sources, Int. J. Heat and Fluid Flow , vol. 17, no. 5, pp. 483–490, 1996.
   Web of Science ®Google Scholar
• T. J. Heindel , F. P. Incropera , and S. Ramadhyani , Enhancement of Natural Convection Heat Transfer from an Array of Discrete Heat Sources , Int. J. Heat and Mass Transfer , vol. 39 , no. 3 , pp. 479 – 490 , 1996 .
   Web of Science ®Google Scholar
• H. V. Mahaney , S. Ramadhyani , and F. P. Incropera , Numerical-Simulation of 3-Dimensional Mixed Convection Heat-Transfer from an Array of Discrete Heat-Sources in a Horizontal Rectangular Duct , Numer. Heat Transfer , Part A , vol. 16 , no. 3 , pp. 267 – 286 , 1989 .
   Web of Science ®Google Scholar
• H. V. Mahaney , F. P. Incropera , and S. Ramadhyani , Comparison of Predicted and Measured Mixed Convection Heat-Transfer from an Array of Discrete Sources in a Horizontal Rectangular Channel , Int. J. Heat and Mass Transfer , vol. 33 , no. 6 , pp. 1233 – 1245 , 1990 .
   Web of Science ®Google Scholar
• C. G. Rao , C. Balaji , and S. P. Venkateshan , Conjugate Mixed Convection with Surface Radiation from a Vertical Plate with a Discrete Heat Source , J. Heat Transfer––Trans. of the ASME , vol. 123 , no. 4 , pp. 698 – 702 , 2001 .
   Web of Science ®Google Scholar
• R. N. Mathews , and C. Balaji , Numerical Simulation of Conjugate , Turbulent Mixed Convection Heat Transfer in a Vertical Channel with Discrete Heat Sources , Int. Comm. in Heat and Mass Transfer , vol. 33 , no. 7 , pp. 908 – 916 , 2006 .
   Web of Science ®Google Scholar
• P. M. Guimaraes and G. J. Menon , Combined Free and Forced Convection in an Inclined Channel with Discrete Heat Sources , Int. Comm. in Heat and Mass Transfer , vol. 35 , no. 10 , pp. 1267 – 1274 , 2008 .
   Web of Science ®Google Scholar
• A. K. da Silva , S. Lorente , and A. Bejan , Optimal Distribution of Discrete Heat Sources on a Wall with Natural Convection , Int. J. Heat and Mass Transfer , vol. 47 , no. 2 , pp. 203 – 214 , 2004 .
   Web of Science ®Google Scholar
• A. Bejan , Shape and Structure: From Engineering to Nature , Cambridge University Press , Cambridge , 2000 .
   Google Scholar
• A. Bejan , Advanced Engineering Thermodynamics , , 2nd ed. , John Wiley & Sons, Inc. , New York , 1997 .
   Google Scholar
• Comsol Multiphysics®: Heat Transfer Module—User's Guide, v. 3.4 , Burlington , MA , 2007 .
   Google Scholar
• Comsol Multiphysics®: Modeling Guide, v. 3.4 , Burlington , MA , 2007 .
   Google Scholar
• Comsol Multiphysics®: Reference Guide, v. 3.4 , Burlington , MA , 2007 .
   Google Scholar
• http://www.cise.ufl.edu/research/sparse/umfpack (accessed 18 February 2011) .
   Google Scholar
• Matlab®, Genetic Algorithm and Direct Search Toolbox™ 2 User's Guide , The MathWorks, Inc. , Natick , MA , 2008 .
   Google Scholar
• L. Gosselin , M. Tye-Gingras , and F. Mathieu-Potvin , Review of Utilization of Genetic Algorithms in Heat Transfer Problems , Int. J. Heat and Mass Transfer , vol. 52 , no. 9–10 , pp. 2169 – 2188 , 2009 .
   Web of Science ®Google Scholar
• R. K. Shah , Thermal Entry Lenght Solutions for the Circular Tube and Parallel Plates, In Proc. 3rd National Heat and Mass Transfer Conference, Bombay , 1975 .
   Google Scholar
• R. K. Shah and A. L. London , Laminar Flow Forced Convection in Ducts , Academic Press , New York , 1978 .
   Google Scholar