heat_3_88 - PROBLEM 3.88 KNOWN Radii and thermal...

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PROBLEM 3.88 KNOWN: Radii and thermal conductivities of reactor fuel element and cladding. Fuel heat generation rate. Temperature and convection coefficient of coolant. FIND: (a) Expressions for temperature distributions in fuel and cladding, (b) Maximum fuel element temperature for prescribed conditions, (c) Effect of h on temperature distribution. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction, (3) Negligible contact resistance, (4) Constant properties. ANALYSIS: (a) From Eqs. 3.49 and 3.23, the heat equations for the fuel (f) and cladding (c) are ( ) ( ) c f 1 1 2 dT 1 d dT q 1 d 0 r r r 0 r r r r dr dr r dr dr f k = = Λি Ξ৯ Λি Ξ৯ Μ৏ Ο৿ Μ৏ Ο৿ Νয় Πਏ Νয় Πਏ Hence, integrating both equations twice, 2 f 1 1 f 2 f f f f dT qr C qr C T ln r C dr 2k k r 4k k = + = + + (1,2) c 3 3 c 4 c c dT C C T ln r C dr k r k = = + (3,4) The corresponding boundary conditions are: ) ( ) ( ) f f 1 c 1 r 0 dT dr 0 T r T r = = = (5,6) ( ) [ ] 1 1 2 c c f c c c 2 r r r r r r dT dT dT f k k k h T r T dr dr dr = = = = =
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