sec2-1 - ChE 120B Shell Balances Spherical Geometry S.S...

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ChE 120B 2 - 1 Shell Balances — Spherical Geometry S.S. Heat conduction with source term: Try spherical geometry using a shell balance: Input — Output + Source = 0 2 | | 4 | 0 r r r r r r r Q Q S r r π + Δ = Remember, we can substitute Fourier's Laws anytime to get: r dT Q k A dr = − A = 4 π r 2 ( ) 2 2 2 4 | 4 | 4 | 0 r r r r r dT dT k r k r S r r dr dr π π π + + Δ = , rearranging we get: 2 2 2 0 | | lim r r r r r dT dT r r S dr dr r r k Δ → + = Δ or 2 2 r S d T r r dr r k = − Integrating once we get 2 3 1 3 r S dT r r C dr k = − + or 1 2 3 r rS C dT dr k r = − + We can use the obvious boundary condition that nothing real goes to : as 0 dT r dr → ∞ unless 1 0 C = 3 r rS dT dr k = − Alternative: We can also say that the center of a sphere is a point of symmetry, so 0 0 | r T r = = ; we get the same result.
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