Correct use for internalHeatSource and thermalBC in 2D and 3D problems

Hello,

I am using Matlab 2022a. I am using the pde toolbox to try to solve a heat transfer problem.

I am uncertain about the use of internalHeatSource and thermalBC.

For internalHeatSource in 3D, I believe that one uses heat per unit volume (J/m3) as the input and for thermalBC heat per unit area (J/m2) is used. I assume that for the 2D problem I should use J/m2 for internalHeatSource and J/m for thermalBC.

I am simulating a thin free standing film where the heat source is located in the centre of the film. I was thinking that the result should be very similar if I restrict the internalHeatSource or the thermalBC to be in a small volume (in 3D) or area (in 2D) since the total amount of heat entering the system at this point should be similar. However, I get different results in 2D (between internalHeatSource and thermal BC) and in 3D (also between internalHeatSource and thermal BC and the 2D results).

The code for the 2D problem (currently setup for the internalHeatSource) is listed below. I can provide the 3D code if needed.

Many thanks in advance

Chris

close all

clear all

steadyState = false;

if steadyState

thermalmodel = createpde('thermal','steadystate-axisymmetric');

else

thermalmodel = createpde('thermal','transient-axisymmetric');

end

filmDepth = 0.000005;

beamWidth = 0.0001;

R1 = [3 4 .001 .001 .002 .002 0 .001 .001 0]';

R2 = [3 4 beamWidth beamWidth .001 .001 0 filmDepth filmDepth 0]';

R3 = [3 4 0 0 beamWidth beamWidth 0 filmDepth filmDepth 0]';

gm = [R1,R2,R3];

sf = 'R1+R2+R3';

ns = char('R1','R2','R3');

ns = ns';

g = decsg(gm,sf,ns);

geometryFromEdges(thermalmodel,g);

figure

pdegplot(thermalmodel,'FaceLabels','on','EdgeLabels','on')

axis equal

thermalmodel.StefanBoltzmannConstant = 5.670373E-8;

thermalProperties(thermalmodel,'Face', 1, 'ThermalConductivity',170,...

'MassDensity',19250,...

'SpecificHeat',134);

% Properties for Titanium in freestanding thin film where beam hits film.

thermalProperties(thermalmodel,'Face', 2, 'ThermalConductivity', @conductivityTi,...

'MassDensity',16690,...

'SpecificHeat',140);

% Properties for Titanium in freestanding thin film

thermalProperties(thermalmodel,'Face', 3, 'ThermalConductivity', @conductivityTi,...

'MassDensity',16690,...

'SpecificHeat',140);

ATval = 25; %ambient temperature Celsius

thermalBC(thermalmodel,'Edge',2,'Temperature',ATval);

lengthEdge9 = beamWidth;

areaFace3 = beamWidth*filmDepth;

volumeFace3 = areaFace3*filmDepth;

%thermalBC(thermalmodel,'Edge',9,'HeatFlux',60/lengthEdge9);

internalHeatSource(thermalmodel,60/areaFace3,'Face',3);

thermalIC(thermalmodel,ATval);

msh = generateMesh(thermalmodel);

figure

pdeplot(thermalmodel)

axis equal

if steadyState

result = solve(thermalmodel);

else

tfinal = 0.1;

tlist = 0:tfinal/100:tfinal;

result = solve(thermalmodel, tlist);

end

T = result.Temperature;

figure

axis equal

if steadyState

pdeplot(thermalmodel,'XYData',T,'Contour','on')

title 'Steady-State Temperature'

else

pdeplot(thermalmodel,'XYData',T(:,end),'Contour','on')

title(sprintf(['Transient Temperature' ...

' at Final Time (%g seconds)'],tfinal))

end

if ~steadyState

Tcentre = interpolateTemperature(result,[0.0;0.0],1:numel(tlist));

figure

plot(tlist,Tcentre)

title 'Temperature at the Centre of film as a Function of Time'

xlabel 'Time, s'

ylabel 'Temperature, C'

grid on

legend('Center Axis','Location','SouthEast')

end

function thermalVal = conductivityTi(location,state)

thermalVal = 57.53743136+0.003683833*state.u;

end

[Torsten]

Units are W/m^3 resp. W/m^2.

[Christopher]

Thankyou Torsten, I should (of course) have said W/m^3 and W/m^2 etc. The problem remain with obtaining different results when using internalHeatSource and thermalBC when I would have thought that (almost) the same result would be obtained.

Chris

[Torsten]

Why do you think that a heat source at the boundary and a heat source in the interior can be equivalent or even compared with each other ? Imagine a tank warmed up from its walls and with an immersion heater. I think it should make a big difference.

[Christopher]

Hello Torsten,

I am injecting heat into a small region and then letting that heat conduct away along a thin film. I was hoping that by injecting the same amount of heat into that small region, I would be simulating (reasonably closely) the same thing - but perhaps I am mistaken.

Thanks for responding.

Chris

[Torsten]

You are not injecting heat. The heat also has to conduct from the wall into the small region and from there away along the film. So you should make a very fine grid for the small region to resolve the process if you want to use the boundary condition and not the internal heat source for the process.

From my feeling, the heat flux from the wall in W/m^2 must be much bigger than the volumetric heat source in W/m^3 for that both have the same effect.

[Christopher]

Hi Torsten,

Thanks for your input. My actual heat source are electrons striking the thin film - and I was intending to extend the simulation to include this. Depending on the electron energy, the electrons may deposit their energy quite near the surface or at some depth (a couple of microns or maybe a bit more). From what you say, I think I need to simulate the how the heat source is distributed in the film as it is neither on the surface, nor completley distributed in the whole film (which is about 5 microns thick). The film is about 1 mm long, so I was imagining that the heat conduction across the film (in depth) would occur much faster than the conduction across the film (in length) - so the temperature at the top of the film and the bottom would always be roughly the same - but 1 mm away the film will remain much cooler. I need to think about this some more !

Best Wishes

Chris