PROBLEM 4.42 KNOWN:One-dimensional fin of uniform cross section insulated at one end with prescribed base emperature, convection process on surface, and thermal conductivity. tFIND:Finite-difference equation for these nodes: (a) Interior node, m and (b) Node at end of fin, n, here x = L. wSCHEMATIC:ASSUMPTIONS:(1) Steady-state conditions, (2) One-dimensional conduction. ANALYSIS:(a) The control volume about node m is shown in the schematic; the node spacing and control volume length in the x direction are both Δx. The uniform cross-sectional area and fin perimeter are Acand P, respectively. The heat transfer process on the control surfaces, q1and q2, represent conduction while qcis the convection heat transfer rate between the fin and ambient fluid. erforming an energy balance, findP()inout12cm-1mm+1mccEE0 qqq0TTTTkAkAhP x TT0.xx∞−=++=−−++Δ−ΔΔ±±m=Multiply the expression by Δx/kAcand regroup to obtain22m-1m+1mcchPhPTTx T2xT0 1<m<nkAkA∞⎡⎤++⋅Δ
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