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Heat Transfer Theoretical Analysis Experimental Investigations and Industrial Systems Part 17

Tham khảo tài liệu 'heat transfer theoretical analysis experimental investigations and industrial systems part 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 630 Heat Transfer - Theoretical Analysis Experimental Investigations and Industrial Systems m m Fl XWi Fr XCrs . i 1 i 1 We expect on physical grounds Fl and Fr to be rather smooth so r e N could be assumed to be at least 2. This indicates that the Fourier coefficients of the unknown functions JFl x cos nnx dx JFr x cos nnx dx 0 0 as well as of the other dependent quantities involved should decay at least as 0 1 n2 or faster. As a consequence only a small part of the infinite system is expected to be significant for Fl and Fr. Thus for a particular choice of m e N only the equations n 0 1 . m are taken into account. This gives a system of 2 m 1 nonlinear equations for the unknown coefficients Cl Clì m 0 CR Cr í m 0 written in short as fÍCLị g C CR MN.2 f cR g2 cL Cr . The structure of the system MN.2 follows from 23 but with MN.1 added to each equations block. There are two important steps to be considered. The first is the choice of the subspace Sm. We decided to try first perhaps the simplest approach by choosing the subspace Sm as the space of polynomials Sm Pm of degree m. The numerical results turned out satisfactory. Alternatively we could always switch to a proper spline space. The second step regards the efficient numerical solution of the system MN. 2 . Inspection of the equations 23 reveals that the function f depends linearly on the unknowns. Since the functions gi are much more complicated the direct iteration seems to be a cheap shortcut. So with the starting choice incorporating the conditions MN.1 c cR0 i i 1 2 m MN.3a 2 m 1 cL 0 cRo0 1 - l x X- MN.3b L 0 R 0 2 m 1 i 1 i the direct iteration reads c k 1 c k m f c L g1 c L c R fíc _k 1 Ì a-.ir k c k f k 0 1 f c R g2 c L c R k 0 1 . The Rate of Heat Flow through Non-Isothermal Vertical Flat Plate 631 This approach was quite satisfactory for some parameter values but failed to converge for others. Clearly the map involved in this case ceases to be a contraction. However the Newton method turned out to .