heat_2_22 - PROBLEM 2.22 KNOWN: Steady-state temperature...

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PROBLEM 2.22 KNOWN: Steady-state temperature distribution in a cylindrical rod having uniform heat generation of ± . q W / m 1 3 = × 5 10 7 FIND: (a) Steady-state centerline and surface heat transfer rates per unit length, ʹ q r . (b) Initial time rate of change of the centerline and surface temperatures in response to a change in the generation rate from ±± . q to q = 10 W / m 12 83 SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in the r direction, (2) Uniform generation, and (3) Steady-state for ± . q = 5 10 W / m 1 73 × ANALYSIS: (a) From the rate equations for cylindrical coordinates, ʹʹ = qk T r q = -kA T r rr . Hence, 2 r L T r r = π 16 or ʹ = q kr T r r 2 where T/ r may be evaluated from the prescribed temperature distribution, T(r). At r = 0, the gradient is ( T/ r) = 0. Hence, from Eq. (1) the heat rate is ʹ = q r 00 . < At r = r o , the temperature gradient is T r K m rm T r K / m. r=r 2 o o o " $ # = × ± ! " $ # = × " $ # = × 2 4167 10 2 4167 10 0025 0 208 10 55 5 ..
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heat_2_22 - PROBLEM 2.22 KNOWN: Steady-state temperature...

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