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Heat Analysis and Thermodynamic Effects Part 10

Tham khảo tài liệu 'heat analysis and thermodynamic effects part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Integral Transform Method Versus Green Function Method in Electron Hadron or Laser Beam - Water Phantom Interaction 259 ểí k dxỊ dx x2 Kij x 0 4 Equation 4 provides a series of positive eigenvalues z j j 6 N and eigenfunctions . X . . . ổ í Ổ Kij x x. of the differential operator 2 I k x -f õx 0x 7 . The eigenfunctions were necessary for solving equation 2 . The solution of equation 4 has the form Kij x x. Al 2Z jự hi k xi mix hl . iY 2Z j-7 hi k xi mix IwJ I 1 J 5 j 6 N Where h. PC and Jo and Yo are the Bessel and Weber functions respectively. After the application of the integral operator Ki x equation 4 becomes -Z2Wi z j t -PiCi dTi x j t Ct f zj t 6 where Ui xj t J T x t Ki x zj dx C Zj x. ỵ __ -1 xl 1 f zj t Cjxj -J f x t Kj x zj dx and C xj is a normalization factor. Here we have 7 C xj fj1 Ki2 x. x j dx . 7 i 0 xt In the same manner one can apply the functions Kk hk y and Kl eị z which satisfy the equations c K hk y h2Ã hk y 0 Jy 8 Ổ2K g z l g 2 K1 g z 0 Cz2 l M l Ì This next gives K g z cos g z h kg sin g z with k being the thermal conductivity and h the heat transfer coefficient . 260 Heat Analysis and Thermodynamic Effects Then the following equation is inferred - X j X Sí t U T. X j X Sí t SỈT X j hk S t dTi Xj qk el t _ f x y z t 9 õt _ Pici Where from it follows T X J qk sl t _ i X o Ị -- I I C Xj C hk C El x 1 yj 2 z3 2 _ _ _ 10 xJ J J T x y z t Kij Xj x Kk qk y Kl sl z dxdydz x -yj j-z3 j In order to eliminate the time parameter t we apply the direct and inverse Laplace transform to equation 9 . If we have like in most cases f x y z t _ f x y z h t - h t - to one can get the solution XXX T x y z t _ 22 j_1k_1l_1 1 1 -yU ej -X t Xj ej qj L - 1 -. q t-to h t -10 X 11 g X j hk sl x Kij X j x x Kk hk y x Kl el z where g X j hk Sl C Xj C hk C Bl x n-1 x 1 yj 2 z3 2 _ _ 12 2 J J J fi x y z t Kij Xj x Kk hk y Kl sl z dxdydz i_0 x -yj j-Ỉ3 1 Y stands here for the thermal diffusivity. We point out that our semi-analytical solution .