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problem4-53

# problem4-53 - PROBLEM 4.53 KNOWN Volumetric heat generation...

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PROBLEM 4.53 K NOWN: Volumetric heat generation in a rectangular rod of uniform surface temperature. FIND: (a) Temperature distribution in the rod, and (b) With boundary conditions unchanged, heat eneration rate causing the midpoint temperature to reach 600 K. g SCHEMATIC: ASSUMPTIONS: (1) Steady-state, two-dimensional conduction, (2) Constant properties, (3) Uniform volumetric heat generation. ANALYSIS: (a) From symmetry it follows that six unknown temperatures must be determined. Since all nodes are interior ones, the finite-difference equations may be obtained from Eq. 4.35 written in the form ( ) ( ) i neighbors T 1 4 T 1 4 q x y k = + Δ ± Δ . With ( ) q x y 4k Δ Δ ± = 62.5 K, the system of finite-difference equations is (1) ( ) 1 s 2 4 s T 0.25 T T T T 15.625 = + + + + (2) ( ) 2 s 3 5 1 T 0.25 T T T T 15.625 = + + + + (3) ( ) 3 s 2 6 2 T 0.25 T T T T 15.625 = + + + + (4) ( ) 4 1 5 1 s T 0.25 T T T T 15.625 = + + + + (5)
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