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 T_i. The electrical heater is suddenly energized providing a uniform heat flux q₀ at the surface x = O. 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 t e has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature T_f. Derive an expression that can be used to determine Tf as a function of the parameters , and the system characteristics M, c_p ' and A_s (the heater surface area).

Transcribed Image Text:

-Insulation System, mass M Electrical heater