The one-dimensional system of mass M with constant

Chapter , Problem 2.54

(choose chapter or problem)

The one-dimensional system of mass M with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti . The electrical heater is suddenly energized, providing a uniform heat flux at the surface x 0. The boundaries at x L and elsewhere are perfectly insulated. (a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the system. (b) On T x coordinates, sketch the temperature distributions for the initial condition (t 0) and for several times after the heater is energized. Will a steady-state temperature distribution ever be reached? (c) On q x t coordinates, sketch the heat flux q x (x, t) at the planes x 0, x L/2, and x L as a function of time. (d) After a period of time te has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature Tf . Derive an expression that can be used to determine Tf as a function of the parameters , te, Ti , and the system characteristics M, cp, and As (the heater surface area).