Conduction - Steady State Triangular Element

[minger]

Homework Statement

Obtain the temperature distribution T(x,y) of the triangular cross section, assuming constant thermal conductivity, k.

The triangle is a right-isosceles triangle with the right angle at (0,0). The triangle goes "up" to a, and to the right to a, then diagonally across. Imagine a square split from top left to bottom right.

Most importantly the bottom is held at T2, and the left and diagonal sides are held at T1.

Homework Equations

The relevant equation would be the general heat conduction equation.

The Attempt at a Solution

We had a similar problem where the diagonal was insulated. This let you actually pretend you had a square where the top was at T1 also, and the right at T2. Then you used a symmetry line and used the principle of super-position to solve the problem. You would first define a variable theta = T - T2. Then you have homogenous boundary conditions on the bottom and right. Then superimposed solutions for the top theta = T1 - T2 and the left theta = T1 - T2.

The problem with this one is that you cannot do this, since the diagonal is not insulated, so it's not a symmetry line, and I have no idea how to start.

 

[D Weber]

I am working on the exact problem you discribe where the diagonal is insulated. I under stand the method I need to use is superposition by modeling the triangle as a square. I am confussed on what the symmetry boundary condition is down the center? How do I model that there is no delta T on the diagonal line that is in terms of x and y?

 

[piercas]

I would approach this problem by first understanding the physical system and its boundary conditions. From the given information, we can infer that the triangular element has a right-isosceles shape with one side at a constant temperature of T2 and the other two sides at a constant temperature of T1. The thermal conductivity, k, is assumed to be constant throughout the element.

To solve for the temperature distribution, we can use the general heat conduction equation, which relates the temperature distribution to the thermal conductivity, the heat source, and the boundary conditions. However, since the element is triangular and not a regular shape, we cannot use standard analytical methods to solve for the temperature distribution.

One approach to solving this problem would be to use numerical methods, such as finite element analysis, to discretize the element into smaller elements and solve for the temperature distribution at each node. Another approach could be to use experimental methods, such as thermal imaging, to obtain the temperature distribution directly.

In conclusion, the solution to this problem may require advanced mathematical or experimental techniques, depending on the specific context and resources available.