[Mathematica] Workaround for NDSolve boundry value problem?

[ChalkChalkson]

Hey guys,

I am currently trying to upgrade my simulation of CPU coolers in Mathematica from simulating just a single fin to multiple fins ('cause my design has some fins not directly connected to the CPU).

When I hand my set of equations over to NDSolve it complains that "NDSolve is not currently able to solve boundary value problems with discrete variables".

I was hoping somebody maybe knows a work around  

Here is my code

Peqns = Table[D[power[i][h], h] == -Cv*Width[h]*temp[i][h] +
   If[i > 1, (temp[i - 1][h] - temp[i][h])*ConW[h]/ConL*Cd,0] + If[i < Fins, (temp[i + 1][h] - temp[i][h],0)*ConW[h]/ConL*Cd], {i, Fins}];
   
Teqns = Table[D[temp[i][h], h] == -power[i][h]/(Cd*Width[h]*thickness), {i, Fins}];

Ieqns = {Table[power[i][0] == CPUpower/Length[ConFins], {i, Fins}], Table[power[i][height] == 0, {i, Fins}]};

eqns = Join[Peqns, Teqns, Ieqns];

sol = NDSolve[eqns, Join[Table[temp[i], {i, 0, Fins}], Table[power[i], {i, Fins}]], {h,0, height}];
     
Evaluate[temp[1][0] /. sol]

All the other stuff is pretty much constant (ie read them as constants, makes more sense that way)

Here is a list of what the things are:

Spoiler

Cd is the conductivity 237 [W/mK]

Width[h] is the width of the fins at a given height currently set to 0.1 everywhere for debugging

thickness is the thickness of the fins 0.001

Cv is the convection 10.45 [W/m²K]

CPUPower is... well the CPU power, 90 [W]

Fins is the number of fins, currently set to 3 for debugging

ConFins is the set of all fins that are connected to the CPU, currently set to {1} for debugging

ConL is the connector (eg heatpipes)  length 0.005 [m]

ConW[h] is a function that gives the width of the connector(s) at the current height. It gives values of 0.01 to 0 and is mostly 0

height is the total height of the cooler/fins, currently 0.14

 

Note that I can verify, that the problem only arises, when there are terms involving the neighboring temps, if I set that term to 0 it works. But even if I choose my conditions in a way that guaranties that these terms will be 0 it refuses to even start to compute

[Brooksie359]

May I ask what the end goal is? What are you trying to solve for? There may be some CFD equations you could use to solve it more directly.

[ChalkChalkson]

Just now, Brooksie359 said:

May I ask what the end goal is? What are you trying to solve for? There may be some CFD equations you could use to solve it more directly.

I am solving for the temperature on the bottom of the fins

(Really I want the total temperature curves, but I guess these will be needed anyway)

[Brooksie359]

8 minutes ago, ChalkChalkson said:

I am solving for the temperature on the bottom of the fins

(Really I want the total temperature curves, but I guess these will be needed anyway)

Well then this seems like something CFD was created for. Do you know much about CFD?

[Brooksie359]

if you want to look up more about CFD and how it can help you just search for Patankar's Book and it should be the first result. chapter 4 goes over how to do continuations for conduction while chapter 5 goes over convection. 

[ChalkChalkson]

1 hour ago, Brooksie359 said:

if you want to look up more about CFD and how it can help you just search for Patankar's Book and it should be the first result. chapter 4 goes over how to do continuations for conduction while chapter 5 goes over convection. 

Thanks, I'll look into CFD, but the issue is, that I am sitting on a system that has both conduction and convection on several objects influencing each other.

The topology of the object sadly is purely 2D, so I can't reduce it to 1 continuous function either 

[Brooksie359]

38 minutes ago, ChalkChalkson said:

Thanks, I'll look into CFD, but the issue is, that I am sitting on a system that has both conduction and convection on several objects influencing each other.

The topology of the object sadly is purely 2D, so I can't reduce it to 1 continuous function either 

I believe you would just use one as the others boundary condition and and solve that way. 

[Brooksie359]

i do believe the book goes into that as well so if you look at it you will probably be able to figure it out.