Heat transfer into constant temp sink

[iceriver500]

I have a simplified model (see attached picture) which is part of a bigger model but for now the problem is as follows.
I have a electropolished metal sheet which has spots on it which act as heat sinks because they are kept at constant temperature of 100K. The sheet is at room temperature initially (300K) and will loose heat to the white spots to get to 100K over time. I need to come up with some equation which can describe how heat will flow by conduction through the sheet.
I know it's very easy in COMSOL, and in the COMSOL solution, the areas near the white spots will be blueish (indicating low temp) and away from the white spot will be reddish.
Basically, i need to come up with an equation that shows the Comsol's solution.

 

[bucher]

Gosh, I know the equations for a 1-dimensional heat sink (like down the length of and object). But once you get into anything above 1-dimensional temperature variations, especially with multiple heat sink spots, the governing equations blow up on you and a lot of times go into the non-linear state (very bad for solving on paper). If this is for a project and you absolutely need to know the correct equations, then I would suggest changing the problem a bit by simplifying it. You could make a sink somewhere in the middle and have it be all along a vertical or horizontal line; that would make it a 1-dimensional problem if you were to look at it from on of the sides. I hope that this helps...though it's not what you were looking for.

 

[ank_gl]

here it is

 

[iceriver500]

Hey ang_gl ,
That equation might not work. It doesn't have the time component in it. I know the equation that will be used but am not sure how to implement it. It is the equation you wrote +having q"'/k on LHS equated with (1/alpha)(dT/dt) on the RHS, where q"' is the rate of internal energy conversion (heat generation)

 

[ank_gl]

Yea, the equation isn't complete. It applies for the case of steady state conduction with no heat generation. I didn't read carefully, "to get to 100K over time", so its an unsteady situation. Yes you are right about the term on right, but heat generation term is zero in the case described.

 

[Nick Bruno]

maybe after running the simulation you can use the line graph along the plate and export the data to matlab... then use curve fitting? I am not sure if you would be able to export the points on the graph tho.