problem4-72 - PROBLEM 4.72 KNOWN Thin metallic foil of...

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PROBLEM 4.72 KNOWN : Thin metallic foil of thickness, t, whose edges are thermally coupled to a sink at temperature T sink is exposed on the top surface to an ion beam heat flux, ¢¢ q s , and experiences radiation exchange with he vacuum enclosure walls at T t sur . F IND : Temperature distribution across the foil. SCHEMATIC : ASSUMPTIONS : (1) One-dimensional, steady-state conduction in the foil, (2) Constant properties, (3) Upper and lower surfaces of foil experience radiation exchange, (4) Foil is of unit length normal to the age. p ANALYSIS : The 10-node network representing the foil is shown below. From an energy balance on node n, in out E E 0 = ± ± , for a unit depth, a b c d e q q q q q + + + + = 0 ( ) ( )( ) ( ) ( )( ) s r,n sur n n 1 n r,n sur n n 1 n q x h x T T k t T T x h x T T k t T T x 0 + ′′ Δ + Δ + Δ + Δ + Δ = (1) where the linearized radiation coefficient for node n is ( ) ( ) 2 2 r,n sur n sur n h T T T εσ = + + T (2) Solving Eq. (1) for T n find, ( ) ( ) ( ) ( ) 2 2 2 n n 1 n 1 r,n sur s r,n T T T 2h x kt T x kt q h x kt 2 + ′′ = + +
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