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heat_3_82 - PROBLEM 3.82 KNOWN Distribution of volumetric...

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PROBLEM 3.82 KNOWN: Distribution of volumetric heating and surface conditions associated with a quartz window. FIND: Temperature distribution in the quartz. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction, (3) Negligible radiation emission and convection at inner surface (x = 0) and negligible emission from outer surface, (4) Constant properties. ANALYSIS: The appropriate form of the heat equation for the quartz is obtained by substituting the prescribed form of q ± into Eq. 3.39. ( ) 2 o - x 2 1 q d T e 0 k dx α α β ʹஒʹஒ + = Integrating, ( ) ( ) o - x - x 1 o 1 2 1 q 1 dT e C T q e C x+C dx k k α α β β α ʹஒʹஒ ʹஒʹஒ = + + = + Boundary Conditions: ( ) x=o o x=L k dT/dx) q k dT/dx) h T L T β ʹஒʹஒ = Ρਟ Τ੏ = Σਿ Φ੯ Hence, at x = 0: ( ) o 1 o 1 o 1- k q C q k C q / k β β Ρਟ Τ੏ ʹஒʹஒ ʹஒʹஒ + = ΢ਯ Υ੟ Σਿ Φ੯ ʹஒʹஒ = At x = L: ( ) ( ) - L - L o 1 o 1 2 1- 1- k q e C h - q e C L+C
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