How do I make an object in heat transfer transparent?

Hello,

I am trying to model heat transfer between two slabs that are laser-heated. The laser-heating is accounted for by a Gaussian heat equation and I am using the pde toolbox. However, the slab at which the laser acts is transparent. Right now I have a model where the laser heating source acts on the slab as if it WERE NOT transparent. How would I account for the fact that it is transparent so that the heating source acts on the second slab?

Thanks,

%set parameters----------------------------------------------------
samp_thick = 5e-6;              % sample thickness (m)
samp_radius = 20e-6;            % sample thickness (m)
sam_k = 85;                     % sample themral condcutivty (W/mK)
sam_cp = 440;                   % sample specific heat (J/kgK)
sam_rho = 7800;                 % sample density (kg/m^3)

ins_thick = 4.5e-6;              % insulation thickness (m)
ins_radius = 20e-6;             % sample thickness (m)
ins_k = 18;                     % insulation themral condcutivty (W/mK)
ins_cp = 880;                   % insulation specific heat (J/kgK)
ins_rho = 370;                  % insulation density (kg/m^3)


P0 = 30;                        % (W)
R = 10e-6;                      % beam waist (m)                    
%B = P0/(pi*samp_thick*R^2);    % heat flux (W/m^2)
B = P0/(pi*R^2);     % heat flux (W/m^2)
r_center = 0;       % Center of the Gaussian distribution (m)

alpha = 0.1*1e5;                % absorption coeff (m^-1)
%absorp = (1-exp(-1*alpha*samp_thick));
absorp = 1;

T0 = 300;                       % initial temp (K)



%set geometry-----------------------------------------------------
r1 = [3 4 0 samp_radius samp_radius 0 -1*samp_thick/2 -1*samp_thick/2 samp_thick/2 samp_thick/2];
r2 = [3 4 0 ins_radius ins_radius 0 samp_thick/2 samp_thick/2 ins_thick ins_thick];



gdm = [r1; r2]';
[g, bt] = decsg(gdm,'R1+R2',['R1'; 'R2']');

%build model---------------------------------------------------------

model = createpde('thermal', 'steadystate-axisymmetric');
geometryFromEdges(model, g);
generateMesh(model);


% Define the Gaussian heat flux function----------------------------------
x0 = 0; %change these to apply the laser spot to different places on sample
y0 = 0;
%gaussianHeatFlux = @(region,state) B * exp(-((region.x -x0).^2) / (R^2)); %center at r=0
gaussianHeatFlux = @(region,state) B * exp( -(((region.x -x0).^2) / (R^2))+ ( ( (region.y -y0).^2) / (R^2))); %center at r=0



%Set BCs------------------------------------------------------------------

thermalBC(model,"Edge",[1,2,4,5],"Temperature",T0);  % ambient temp (K)
thermalBC(model, 'Edge',2, 'HeatFlux',gaussianHeatFlux);


%Set material properties---------------------------------------------------

thermalProperties(model, 'ThermalConductivity', sam_k,'Face',1); %sample properties
thermalProperties(model, 'ThermalConductivity', ins_k,'Face',2); %insulation properties


%solve model--------------------------------------------------------
results = solve(model);
T = results.Temperature;
msh = results.Mesh;


%plot figures------------------------------------------------------
figure
pdegplot(g,EdgeLabels="on",Facelabels="on"); 
xticks = get(gca, 'XTick');
yticks = get(gca, 'YTick');
set(gca, 'XTickLabel', xticks * 1e6);  % Convert X-axis labels to microns
set(gca, 'YTickLabel', yticks * 1e6);  % Convert Y-axis labels to microns
%axis([0 30 -10 10]);
title("Sample Geometry")
xlabel('r [\mum]');  % \mum for the micron symbol in LaTeX formatting
ylabel('z [\mum]');  % \mum for the micron symbol in LaTeX formatting


figure
axis equal
pdeplot(msh,XYData=T(:,end),Contour="on",ColorMap="jet"); 
axis equal
title("Temperature, Steady-State Solution")
% Scaling the axis labels to microns
xticks = get(gca, 'XTick');
yticks = get(gca, 'YTick');
set(gca, 'XTickLabel', xticks * 1e6);  % Convert X-axis labels to microns
set(gca, 'YTickLabel', yticks * 1e6);  % Convert Y-axis labels to microns
%clim([min(T_K(:)) max(T_K(:))]); % Set color axis to match the range of temperature in Kelvin
xlabel('r [\mum]');  % \mum for the micron symbol in LaTeX formatting
ylabel('z [\mum]');  % \mum for the micron symbol in LaTeX formatting


% Define a line along the top sample surface being heated
xline = linspace(0, samp_radius, 5000);  
yline = ones(size(xline))*samp_thick/2;        
y2 = ones(size(xline))*1e-6;
y3 = ones(size(xline))*-1e-6;
y4 = zeros(size(xline));
y5 = ones(size(xline))*-1*samp_thick/2;  



% Extract temperature at the specified points and time
T_x = interpolateTemperature(results, xline, yline);
T_x2 = interpolateTemperature(results, xline, y2);
T_x3 = interpolateTemperature(results, xline, y3);
T_x4 = interpolateTemperature(results, xline, y4);
T_x4 = interpolateTemperature(results, xline, y5);



% Plot the temperature as a function of x
figure
hold on
plot(xline, T_x, 'LineWidth', 2)
plot(xline, T_x2, 'LineWidth', 2)
plot(xline, T_x4, 'LineWidth', 2)
plot(xline, T_x3, 'LineWidth', 2)
plot(xline, T_x4, 'LineWidth', 2)

xticks = get(gca, 'XTick');
%yticks = get(gca, 'YTick');
set(gca, 'XTickLabel', xticks * 1e6);  % Convert X-axis labels to microns
xlim([0 samp_radius/2])
legend('sample surface','z = 1 um','z = 0', 'z = -1 um','bottom sample surface')
xlabel('r [\mum]');  % \mum for the micron symbol in LaTeX formatting
ylabel('Temperature (K)')
title('Temperature across sample heated surface')
grid off
hold off


%-------------------------------------------------------------------

Answer 1

[Ashok]

Hey @Kirtana,

Heat transfer through a transparent material predominantly involves radiative heat transfer.

The thermal model created using MATLAB function ‘createpde’ can be used to ‘solve conduction-dominant heat transfer problems with convection and radiation occurring at boundaries’ as stated in the following documentation page:

https://www.mathworks.com/help/pde/heat-transfer-and-diffusion-equations.html

I would suggest using the Unified Model approach and more specifically the ‘setupRadiation’ function should serve your need. Here are the relevant links.

• Unified Model - https://www.mathworks.com/help/pde/setup-and-solve.html
• setupRadiation - https://www.mathworks.com/help/pde/ug/femodel.setupradiation.html

I believe this will assist you!