In Section 5.5, the one-term approximation to theseries

Chapter 5, Problem 5.129

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In Section 5.5, the one-term approximation to theseries solution for the temperature distribution wasdeveloped for a plane wall of thickness 2Lthat is ini-tially at a uniform temperature and suddenly sub-jected to convection heat transfer. If Bi?0.1, the wall can be approximated as isothermal and repre- sented as a lumped capacitance (Equation 5.7). Forthe conditions shown schematically, we wish tocompare predictions based on the one-term approxi-mation, the lumped capacitance method, and a finite-difference solution.(a) Determine the midplane, T(0, t), and surface, T(L,t), temperatures at t?100, 200, and 500 s usingthe one-term approximation to the series solu-tion, Equation 5.43, What is the Biot number forthe system?(b) Treating the wall as a lumped capacitance, calcu-late the temperatures at t?50, 100, 200, and500 s. Did you expect these results to comparefavorably with those from part (a)? Why are thetemperatures considerably higher?(c) Consider the 2- and 5-node networks shownschematically. Write the implicit form of the finite-difference equations for each network, and deter-mine the temperature distributions for t?50, 100,200, and 500 s using a time increment of ?t?1s.You may use IHTto solve the finite-differenceequations by representing the rate of change of thenodal temperatures by the intrinsic function, Der(T, t). Prepare a table summarizing the resultsof parts (a), (b), and (c). Comment on the relativedifferences of the predicted temperatures. Hint:See the Solver/Intrinsic Functionssection ofIHT/Helpor the IHT Examplesmenu (Example5.2) for guidance on using the Der(T, t) function.