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Extra info for Advanced Light Source - Users Handbook
The Inverse Laplace Transformation of Eq. 95 results: I1 pﬃﬃﬃﬃ x 1 2 atierfc pﬃﬃﬃﬃ À eÀdx Tðx; tÞ ¼ 2d k 2 at pﬃﬃﬃﬃ 1 ad2 tþdx x þ e erfc d at þ pﬃﬃﬃﬃ 2d 2 at ! p ﬃﬃﬃﬃ 1 ad2 tÀdx x p ﬃﬃﬃﬃ þ e erfc d at À 2d 2 at ð2:95Þ ð2:96Þ where erf is the error function, erfc is the complementary error function, and ierfc is the integral of complementary error function, which are: Z x 2 2 eÀv dv erf ðvÞ ¼ pﬃﬃﬃ p 0 ð2:97Þ erfcðvÞ ¼1 À erf ðvÞ 1 Àv2 ierfcðvÞ ¼ pﬃﬃﬃ e À verfcðvÞ p Introducing dimensionless quantities as: s ¼ ad2 t : x0 ¼ xd : T 0 ¼ kd T I1 Substituting the dimensionless quantities in Eq.
With respect to t, the Laplace transformation of Eq. 86 yields: Ã o2 T I1 d Àdx 1 Â e þ ¼ sT À Tðx; 0Þ 2 ox ks a ð2:89Þ where T ¼ Tðx; sÞ. Using the initial condition, Tðx; 0Þ ¼ 0; Eq. 87 yields: o2 T I1 d Àdx e À q2 T ¼ À ox2 ks ð2:90Þ where q2 ¼ as . 89 has a solution: T ¼ Aeqx þ BeÀqx þ I1 ad À Á eÀdx ks s À ad2 ð2:91Þ where A and B are the constants and they are calculated through the boundary conditions. Substituting boundary condition, ooxT ¼ 0 at the surface (x ¼ 0), it gives: 20 2 Equilibrium Laser Pulse Heating and Thermal Stress Analysis A¼Bþ I1 ad2 À Á qks s À ad2 ð2:92Þ The boundary condition, T ¼ 0 at x ¼ 1, results A ¼ 0 in Eq.
1 x 7 6 h2 t Â 4qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ À Á þ h e Erf h t À c 5 Á U½1 1 p t À x =c 1 ð2:287Þ 52 2 Equilibrium Laser Pulse Heating and Thermal Stress Analysis and À Á rx 212 ¼ À ðh þ 1Þc3 1 hÀ Á 2 ðb þ 1Þðh þ b =t Þ c2 1 þ b =t 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 b b x 7 6 Àb þ 5 Á U½1 Â 4qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ À eÀb Erf À Á t c1 t p t À x =c =t 1 ð2:288Þ and À Á ðh þ 1Þc3 1Àh Á À Á Á rx 312 ¼ À 2 h À c2 1 À c2 c2 1 1 1 þ b =t 2 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!