PROBLEM 4.37 KNOWN:Two-dimensional cylindrical configuration with prescribed radial (Δr) and angular (Δφ) pacings of nodes. sFIND: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. cANALYSIS: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 interiornode with control volume as shown above. The energy balance is Using Fourier’s law for each process, find inabcdEqqqq′′′′=+++=±0.()( )5232iii212iiTT3krr2rrr1krrkr0.rrφ−−⎡⎤+ΔΔ+ Δ+⎢⎥Δ+ΔΔ⎣⎦++ΔΔ+Δ=ΔΔCanceling terms and regrouping yields, ()22i2i531iiiirr1312r rT rrTT TrrT 0.22φφΔΔ−+Δ++++++Δ=+ΔΔΔ(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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