PROBLEM 4.36 KNOWN:Conduction in a one-dimensional (radial) cylindricalcoordinate system with volumetric eneration. gFIND:Finite-difference equation for (a) Interior node, m, and (b) Surface node, n, with convection. SCHEMATIC:(a) Interior node, m (b) Surface node with convection, n ASSUMPTIONS:(1) Steady-state, one-dimensional (radial) conduction in cylindricalcoordinates, (2) Constant properties. ANALYSIS:(a) The network has nodes spaced at equal Δr increments with m = 0 at the center; hence, r = mΔr (or nΔr). The control volume is ( )V2rr2 mππ.=⋅Δ ⋅ =Δ Δ ⋅AAThe energy balance is ingabEEqqqV0+=+±±±()m-1mm+1mrTTTk2 r+q2 mr r0.2rπππ⎡⎤⎡⎤Δ−Δ−⎡⎤−++Δ⎢⎥⎢⎥⎣⎦⎢⎥ΔΔ⎣⎦⎣⎦±Δ=ARecognizing that r = mΔr, canceling like terms, and regrouping find 2m-1m+1m11qmrmTm+T2mT0.22kΔ⎡⎤−+⎢⎥⎣⎦±=<(b) The control volume for the surface node is ( )r r/2π.=⋅Δ⋅AThe energy balance is
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