Unformatted text preview: ü Problem 1.5.13 The spherically symmetric heat equation is given by
∑ k∑ ∑ ÅÅÅÅuÅ = ÅÅÅÅÅÅ ÅÅÅÅÅÅ Hr2 ÅÅÅÅuÅ L Å Å ∑t r 2 ∑r ∑r d d ÅÅÅÅÅ Hr2 ÅÅÅÅÅuÅ L = 0, Å dr dr (1) If the system is at steady state, we can set the time derivative to zero and then solve the following BVP u H1L = 0, u H4L = 80
du r2 ÅÅÅÅÅÅ = c1 dr (2) Integrating once gives (3) Integrating a second time, after dividing through by r2 , gives u = - c1 ê r + c2 (4) Applying the BCs gives Solving for c1 and c2 gives Thus 320 1 uHrL = ÅÅÅÅ3ÅÅÅÅ H1 - ÅÅrÅÅ L Å c1 = c2 = 320 ê 3 uH4L = - c1 ê 4 + c2 uH1L = 0 = - c1 + c2 , (5) (6) (7) ...
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- Fall '09
- Thermodynamics, c1 + c2, Å dr dr