I am trying to answer the following question…
Consider a wall made of brick $10$ centimeters thick, which separates a room in a house from the outside. The room is kept at $20$ degrees. Initially the outside temperature is $10$ degrees and the temperature in the wall has reached steady state. Then there is a sudden cold snap and the outside temperature drops to $-10$ degrees. Find the temperature in the wall as a function of position and time.
I am okay executing the separation of variables technique, but I can't really reason through how to model this scenario. The solution manual states that the Initial/Boundary Value Problem is…$$u_t = ku_{xx},\ u(0,\ t) = 20,\ u(10,\ t) = -10,\ u(x,\ 0) = 20 – x$$
This question comes in a section before higher dimensional heat equations are introduced, but to me, it seems that this should be modeled as a three-dimensional heat equation, because thin walls are two-dimensional, and the bricks are prescribed thickness. How can I intuitively reason through this word problem to model it correctly?
Best Answer
Strictly speaking, you are correct. But if you stay away from the corners, then you just have 1d heat conduction across the brick only - the transverse gradients can be neglected. Then, the equations are just as you decribe with an error function solution. HTH