# How to calculate heat generation on 2D plate?

I have a 2D plate with heat source at one part and I need to calculate the heat generation. The plate is made of aluminum (k = 212 W/m°C). I have watched this example video, but I'm not sure if I can apply it to my case. Some data:

• Length of the plate: 1 m
• Width: 1 m
• Surface being heated: 0.16 m^2
• Tmax = 500 °C

The heat source is placed in the center of the plate and the boundaries of the plate are insulated.

I want to find the heat generated as in this example, equation (4). You can see Q is being divided by ρcp and thus the units of Q must be W/m3.

• What are the conditions at the boundaries of the plate. e.g. insulated, set at a certain temperature, convection cooled? – regdoug May 7 '15 at 0:32
• What do you mean by heat generation? Are you asking what the equilibrium temperature profile of the plate will be? – Chris Mueller May 7 '15 at 0:43
• BCs are insulation. I mean as in this example, equation (4) (geodynamics.usc.edu/~becker/teaching/557/problem_sets/…). You can see Q is being divided with rho and cp, and thus unit of Q must be W/m^3. – cvut May 7 '15 at 0:57
• I would model radiation as a black body, i.e. Q=sT^4. Then just apply the equations in the link in your comment. – nivag May 7 '15 at 8:30
• You have a problem with your boundary conditions. If all sides are perfectly insulated, as you say, and you are pumping 2kW into the plate, the maximum temperature will not be 500°C. The temperature of the plate will keep rising at a uniform rate forever. – regdoug May 9 '15 at 16:15

This is not an answer to your question. This is an answer to what information you need to determine in order to solve a heat problem such as this one.

In general, a heat conduction problem in 2 dimensions needs 4 constraints, 2 in each direction. I will use the example in section 2 of the PDF you linked (http://geodynamics.usc.edu/~becker/teaching/557/problem_sets/problem_set_fd_2dheat.pdf) to illustrate.

I drew a diagram of the 2D heat conduction that is described in the problem.

You have mentioned before that you wish to solve the problem using an explicit finite-difference method. The first step would be to discretize the problem area into a matrix of temperatures. For the example linked, elements 1.5km wide by 2km high were chosen.

The boundary conditions are:

1. Perfect insulation (zero heat flux) on the left and right sides. In an explicit solver, all Neumann (fixed first derivative) conditions are represented by creating an extra row of "fictitious" temperatures which are outside the problem area. Since the heat flux is zero, the fictitious temperatures are held equal to the temperatures at the edge of the problem zone.

2. Fixed temperature (Dirichlet BC) at the upper surface for all time. The surface of the problem area will always be 0°C.

3. Initial temperature at lower surface of 1300°C, with the central point begin held at 1500°C for all times >eq;0. Note I have denoted the Heaviside function as $u(t)$.

Once you get the boundary conditions set up correctly, explicit solvers become simple plug and chug to solve the temperature distribution.

If you want to find out how much heat is supplied by the constant temperature heat source, you will need to use Fourier's law with the approximate derivatives given by the explicit finite-difference method. Assuming the heat is applied at a point $T_{i,j}$, the drawing below shows how to compute the heat flux per unit depth. Obviously, you would need to know how far in an out of the page this problem extends to compute the heat flux.

• wow, thanks! just to make clear, heat flux is known(2kW/0.16m^2 = 12500 W/m^2). i'm not sure i understand how to calculate q'. should i assume that (as in this example) T_(i,j) = T_max, and T_(i+1,j)=T_(i,j+1) = T_min? because when i run the simulation, obviously the temperature is rising from T_min to T_max. are you familiar with Python? i might as well send you my code.. – cvut May 8 '15 at 17:14
• Your code won't help because I don't understand your question. Edit your question to include all of your boundary conditions and then maybe I can help. – regdoug May 9 '15 at 2:25