Answer Happy

A very thick solid wall is initially at the temperature To. At time t = 0, a constant heat flux qo is applied to one surface of the wall (at y = 0), and this heat flux is maintained. Find the time-dependent temperature profiles T(y, t) for small times. Since the wall is very thick it can be safely assumed that the two wall surfaces are an infinite distance apart in obtaining the temperature profiles. (1) (2 pts) Write down the equation of energy (in terms of temperature) and omit zero terms. (2) (2 pts) Differentiate the equation of energy with respect to y and multiply by -k to obtain a differential equation of q instead of T. (3) (4 pts) Use the method of combination of variables n = √4at to solve the differential equation of q and then prove that the temperature profile is given by: T(y, t) - To=. (₁ exp(-y²/4at) - exp(-u²³) du) 90 4at k TT Ә дz 2y VTT y/V4at Note: The following Leibniz rule for differentiation under integration sign might be useful for you. 00 Ja(z) rb (z) ab f (x, 2) dx = fotof z) (b) of dx + f (b (2), z) Ja(z) əz əz да - f (a (z), z). дz