In Section 5.5, the one-term approximation to the series solution for the temperature distribution was developed for a plane wan of thickness 2L that is initially at a uniform temperature and suddenly subjected to convection heat transfer. If Bi

(a) Determine the mid plane, T(0. t), and surface, T(L, t), temperatures at t = 100, 200, and 500 s using the one-term approximation to the series solution, Equation 5.40, what is the Biot number for the system?

(b) Treating the wan as a lumped capacitance, calculate the temperatures at t = 50, 100, 200, and 500 s. Did you expect these results to compare favorably with those from part (a)? Why are the temperatures considerably higher?

(c) Consider the 2- and 5-node networks shown schematically. Write the implicit form of the finite-difference equations for each network, and determine the temperature distributions for t = 50, 100, 200, and 500 s using a time increment of ∆t = 1 s. You may use IHT to solve the finite-difference equations by representing the rate of change of the nodal temperatures by the intrinsic function. Der(T, t). Prepare a table summarizing the results of parts (a). (b), and (c) Comment on the relative differences of the predicted temperatures.

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-Tx,t). Tx,0) = T; = 250°C p = 7800 kg/m3 c = 440 J/kg-K k = 15 W/m-K T = 25°C h = 500 W/m?-K Ler L= 20 mm # Nodes Δε 2 3 5 L14 1: 2 L/2 (At = 1s) Lex