2-D temperature Distribution - Matlab or Excel

[Rieck2000]

Homework Statement

I am trying to find the temperature distribution of a triangular cross section (right triangle). The left side is insulated, the other two sides are exposed to an ambient temperature and heat transfer coefficient when t>0. Both sides of the triangle are 20 cm long and we are told to make the change in x = 4 cm, and to find the temperature distribution for one hour and to use an explicit difference scheme. The triangle is intially at 20 degrees C while the outside temperature is 400 degrees C.

Homework Equations

So for explicit difference:

Ti(n+1) = Ti(n) + Fo(Ti+1(n) -2Ti(n) + Ti-1(n))

The Attempt at a Solution

I have been using excel at first. I am confused at how to set up the equations for interior nodes, the corners and exterior nodes.

I just need a good kick start . I was told by my teacher to use [A][T]=[c]

 

[KataKoniK]

and to find [A] and [c], where [A] represents the coefficients of the temperature distribution equation and [c] represents the known temperatures on the boundaries.

Hi there!

To start off, I would recommend breaking down the problem into smaller parts. First, let's focus on the interior nodes. Since the left side is insulated, we can assume that there is no heat transfer occurring on that side. This means that the temperature distribution equation for the interior nodes will be:

Ti(n+1) = Ti(n) + Fo(Ti+1(n) -2Ti(n) + Ti-1(n))

where i represents the node number and n represents the time step.

Next, let's focus on the corners. Since they are exposed to the ambient temperature and heat transfer coefficient, we can use the following equation:

Tcorner(n+1) = Tcorner(n) + Fo(Ti+1(n) -2Tcorner(n) + Ti-1(n)) + Fo(2Tambient - 4Tcorner(n))

where Tcorner represents the temperature at the corner node and Tambient represents the ambient temperature.

Finally, for the exterior nodes, we can use the equation:

Texterior(n+1) = Texterior(n) + Fo(Ti+1(n) -2Texterior(n) + Ti-1(n)) + Fo(2Tambient)

Now, to find the coefficients for the temperature distribution equation, we can use the finite difference method. This involves setting up a matrix equation with the known temperatures on the boundaries as the right-hand side, and the temperature coefficients as the elements of the matrix. We can then solve this matrix equation to find the temperature distribution at each time step.

I hope this helps to kick start your solution! Let me know if you have any further questions.