A flat body of thickness h and thermal conductivity λ has the shape of a right angled triangle whose short sides are of length l. The plane temperature profile of this body is given as ϑ(x, y) = ϑ 0 +...

A flat body of thickness h and thermal conductivity λ has the shape of a right angled triangle whose short sides are of length l. The plane temperature profile of this body is given as ϑ(x, y) = ϑ₀
+ ϑ₁
(y/l)²
− (x/l)²
; 0 ≤ x ≤ l, 0 ≤ y ≤ x.

a) Where do the highest and lowest temperatures ϑ_max
and ϑ_min
appear? How are they related to the given temperatures ϑ₀
and ϑ₁? 0
_1
0
can be assumed to be valid.

b) The values for grad ϑ and the vector q˙ for the heat flux have to be calculated. At which point is |q˙ | largest?

c) Calculate the heat flow through the three boundary surfaces indicated by y = 0, x = l and y = x, and show that as much heat flows into the triangle as flows out.