How to apply a time decaying internal heat source?

Question

James Pollington 2021-1-3

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/707878-how-to-apply-a-time-decaying-internal-heat-source

编辑： Ayush 2024-5-16

Hi

I'm trying to model the decay of nuclear fuel following an accident, I need to be able to apply an equation for the decay heat produced by the fuel to the internal heat source of the model.

This is my current code with P being the equation that should represent the decay heat.

thermalmodel= createpde('thermal',"transient");

gm = multicylinder([4.655e-3 4.865e-3 5.59e-3],12.6e-3);

thermalmodel.Geometry = gm;

pdegplot(thermalmodel, "CellLabels",'on','FaceLabels','on',"FaceAlpha",0.2);

thermalProperties(thermalmodel,'Cell',1,'ThermalConductivity',7,"MassDensity",10.97, "SpecificHeat",370); % Uranium

thermalProperties(thermalmodel,'Cell',2,'ThermalConductivity',252.48e-3 ,"MassDensity",0.178e-3,"SpecificHeat",5.193); % Helium

thermalProperties(thermalmodel,'Cell',3,'ThermalConductivity', 21.5,"MassDensity",6.55,"SpecificHeat",0.328); % Zircalloy

tlist = (10:1:100);

for a = 1 : length(tlist)

P(a) = 100e6*0.066*((tlist(a).^(-0.2))- ((10*24*60*60)+tlist(a)).^(-0.2));

end

internalHeatSource(thermalmodel,P,"Cell",1);

thermalIC(thermalmodel,873);

generateMesh(thermalmodel);

pdemesh(thermalmodel);

tic

thermalresults = solve(thermalmodel,tlist);

toc

Tcenter = interpolateTemperature(thermalresults,0,0,0,1:numel(tlist))

plot(tlist,Tcenter)

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Answer 1

Ayush 2024-5-16

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/707878-how-to-apply-a-time-decaying-internal-heat-source#answer_1458596

编辑：Ayush 2024-5-16

在 MATLAB Online 中打开

Hi James,

To apply a time decaying internal heat source to your provided code, you would need to implement a decay heat function which would then be referenced in the “internalHeatSource” function. Please refer to the below documentation to know more about “Specifying Nonconstant Parameters of a Thermal Model”:

https://www.mathworks.com/help/pde/ug/pde.thermalmodel.internalheatsource.html

From the above reference, here is a possible implementation of the mentioned decay heat function which will be based on the sum of exponentials or a single exponential term that represents the decay of fission products. Here is an example code for your reference:

function P = decayHeatFunction(location, state)
    % Constants
    P0 = 100e6; % Initial decay heat in Watts
    alpha = 0.2; % Decay exponent
    % Check if state.time is NaN (solver's check for time dependency)
    if isnan(state.time)
        P = nan; % Return NaN to indicate time-dependent problem
    else
        % Calculate decay heat for each time point
        P = P0 * (state.time.^(-alpha));
    end
    % Ensure output is a row vector with the number of columns equal to the number of evaluation points
    P = repmat(P, 1, length(location.x));
end

After implementing the decay heat function, you would have to pass it as an anonymous function that only takes location and state as inputs into the “internalHeatSource” function as follows:

internalHeatSource(model, "Face", 2, @decayHeatFunction);

Hope this helps!