Chapter 3, Problem 3.131P

In Section 5.5, the one-term approximation to the series solution for the temperature distribution was developed for a plane wall of thickness 2L that is initially at a uniform temperature and suddenly subjected to convection heat transfer. If $Bi<0.1$, the wall can be approximated as isothermal and represented as a lumped capacitance (Equation 5.7). For the conditions shown schematically, we wish to compare predictions based on the one-term approximation, the lumped capacitance method, and a finite difference solution.

(a) Determine the midplane, $T\left(0,t\right)$, and surface, $T\left(L,t\right)$, temperatures at $t=100$, 200 and 500 s using the one-term approximation to the series solution, Equation 5.43, What is the Biot number for the system?

(b) Treating the wall 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 $\Delta t=1s$. You may use IHT to solve the finite-difference equations by representing the rate of change of the nodal temperatures by the intrinsic function, $\text{Der}\left(T,t\right)$. Prepare a table summarizing the results of parts (a), (b), and (c). Comment on the relative differences of the predicted temperatures. Hint : See the Solver/Intrinsic Functions section of IHT/Help or the IHT Examples menu (Example 5.2) for guidance on using the $\text{Der}\left(T,t\right)$ function.