A plane wall (4000 kg/m3, cp500 J/kgK,k10 W/mK) of

Chapter 5, Problem 5.112

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A plane wall (??4000 kg/m3, cp?500 J/kg?K,k?10 W/m?K) of thickness L?20 mm initially hasa linear, steady- state temperature distribution withboundaries maintained at T1?0C and T2?100C.Suddenly, an electric current is passed through thewall, causing uniform energy generation at a rate. The boundary conditions T1and T2remain fixed.(a) On T xcoordinates, sketch temperature distrib-utions for the following cases: (i) initial condi-tion (t?0); (ii) steady-state conditions (tl?),assuming that the maximum temperature inthe wall exceeds T2; and (iii) for two intermedi-ate times. Label all important features of thedistributions.(b) For the system of three nodal points shownschematically (1, m, 2), define an appropriatecontrol volume for node mand, identifying all rel-evant processes, derive the corresponding finite-difference equation using either the explicitorimplicitmethod.(c) With a time increment of ?t?5 s, use the finite-difference method to obtain values of Tmfor thefirst 45 s of elapsed time. Determine the corre-sponding heat fluxes at the boundaries, that is, (0, 45 s) and qx(20 mm, 45 s).(d) To determine the effect of mesh size, repeat youranalysis using grids of 5 and 11 nodal points(?x? .0 and 2.0 mm, respectively).