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 Eq q q q ′′′′ =+++= ± 0 . () ( ) 52 32 i i i2 12 i i TT 3 kr r 2r r r 1 k r r k r 0. r r φ −− ⎡⎤ +ΔΔ + Δ + ⎢⎥ Δ+ Δ Δ ⎣⎦ ++ Δ Δ + Δ = Δ Δ C anceling terms and regrouping yields, ( ) 22 i 2 i5 3 1 ii i i rr 13 1 2r r T r rT T T r rT 0 . 2 2 φφ ΔΔ −+ Δ + + + + + + Δ = Δ Δ (b) Node 3. The adiabatic surface behaves as a symmetry surface. We can utilize the result of Part (a) o write the finite-difference equation by inspection as
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