problem4-37 - PROBLEM 4.37 KNOWN Two-dimensional...

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PROBLEM 4.37 KNOWN: Two-dimensional cylindrical configuration with prescribed radial ( Δ r) and angular ( Δφ ) pacings of nodes. s F IND: Finite-difference equations for nodes 2, 3 and 1. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) Two-dimensional conduction in cylindrical oordinates (r, φ ), (3) Constant properties. c ANALYSIS: The method of solution is to define the appropriate control volume for each node, to dentify relevant processes and then to perform an energy balance. i (a) Node 2. This is an interior node with control volume as shown above. The energy balance is Using Fourier’s law for each process, find in a b c d E q q q q = + + + = ± 0. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 2 3 2 i i i 2 1 2 i i T T T T 3 k r r k r 2 r r r T T T T 1 k r r k r 0. 2 r r r φ φ φ φ + Δ Δ + Δ + Δ + Δ Δ + + Δ Δ + Δ = Δ + Δ Δ C anceling terms and regrouping yields, ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 2 2 i 2 i 5 3 1 i i 2 2 i i r r 1 3 1 2 r r T r r T T T r r T 0. r r 2 2 r r φ φ Δ Δ + Δ + + + Δ + + + + Δ = + Δ Δ + Δ Δ (b) Node 3.
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