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Heat Conduction in a Spherical Multidomain Geometry with Nonuniform Heat Flux

This example shows how to perform a 3-D transient heat conduction analysis of a hollow sphere made of three different layers of material, subject to a nonuniform external heat flux.

Problem Description

The physical properties and geometry of this problem are described in Singh, Jain, and Rizwan-uddin [1], which also has an analytical solution for this problem. The inner face of the sphere has a temperature of zero at all times. The outer hemisphere with positive value has a nonuniform heat flux defined by

and are azimuthal and elevation angles of points in the sphere. Initially, the temperature at all points in the sphere is zero.

Create Thermal Model

Create a thermal model for transient thermal analysis.

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

Define Geometry

Create a multilayered sphere using the multisphere function. Assign the resulting geometry to the thermal model. The sphere has three layers of material with a hollow inner core.

gm = multisphere([1,2,4,6],'Void',[true,false,false,false]);
thermalmodel.Geometry = gm;

Plot Geometry with Face and Cell Labels

Plot the geometry and show the cell labels and face labels. Use a FaceAlpha of 0.25 so that labels of the interior layers are visible.

figure('Position',[10,10,800,400]);
subplot(1,2,1)
pdegplot(thermalmodel,'FaceAlpha',0.25,'CellLabel','on')
title('Geometry with Cell Labels')
subplot(1,2,2)
pdegplot(thermalmodel,'FaceAlpha',0.25,'FaceLabel','on')
title('Geometry with Face Labels')

Generate Mesh

Generate a mesh for the geometry. Choose a mesh size that is coarse enough to speed the solution, but fine enough to represent the geometry reasonably accurately.

generateMesh(thermalmodel,'Hmax',1);

Specify Thermal Properties

Specify thermal conductivity, mass density, and specific heat for each layer of the sphere. The material properties are dimensionless values, not given by realistic material properties.

thermalProperties(thermalmodel,'Cell',1,'ThermalConductivity',1,'MassDensity',1,'SpecificHeat',1);
thermalProperties(thermalmodel,'Cell',2,'ThermalConductivity',2,'MassDensity',1,'SpecificHeat',0.5);
thermalProperties(thermalmodel,'Cell',3,'ThermalConductivity',4,'MassDensity',1,'SpecificHeat',4/9);

Boundary Conditions

The innermost face has a temperature of zero at all times.

thermalBC(thermalmodel,'Face',1,'Temperature',0);

The outer surface of the sphere has an external heat flux. Use the functional form of thermal boundary conditions to define the heat flux.

function Qflux = externalHeatFlux(region,~)
[phi,theta,~] = cart2sph(region.x,region.y,region.z);
theta = pi/2 - theta; % transform to 0 <= theta <= pi
ids = phi > 0;
Qflux = zeros(size(region.x));
Qflux(ids) = theta(ids).^2.*(pi - theta(ids)).^2.*phi(ids).^2.*(pi - phi(ids)).^2;
end

Plot the flux on the surface.

[phi,theta,r] = meshgrid(linspace(0,2*pi),linspace(-pi/2,pi/2),6);
[x,y,z] = sph2cart(phi,theta,r);
region.x = x;
region.y = y;
region.z = z;
flux = externalHeatFlux(region,[]);
figure
surf(x,y,z,flux,'LineStyle','none')
axis equal
view(130,10)
colorbar
xlabel 'x'
ylabel 'y'
zlabel 'z'
title('External Flux')

Include this boundary condition in the model.

thermalBC(thermalmodel,'Face',4,'HeatFlux',@externalHeatFlux,'Vectorized','on');

Initial Conditions

Define the initial temperature to be zero at all points.

thermalIC(thermalmodel,0);

Solve Problem

Define a time-step vector and solve the transient thermal problem.

tlist = [0,2,5:5:50];
R = solve(thermalmodel,tlist);

Animate Temperature Over Time

To plot contours at several times, with the contour levels being the same for all plots, determine the range of temperatures in the solution. The minimum temperature is zero because it is the boundary condition on the inner face of the sphere.

Tmin = 0;

Find the maximum temperature from the final time-step solution.

Tmax = max(R.Temperature(:,end));

Plot contours in the range Tmin to Tmax at the times in tlist.

h = figure;
for i = 1:numel(tlist)
    pdeplot3D(thermalmodel,'ColorMapData',R.Temperature(:,i))
    caxis([Tmin,Tmax])
    view(130,10)
    title(['Temperature at Time ' num2str(tlist(i))]);
    M(i) = getframe;

end

To see a movie of the contours when running this example on your computer, execute the following line:

movie(M,2)

Visualize Temperature Contours on Cross-section

Define a rectangular grid of points on the plane where .

[YG,ZG] = meshgrid(linspace(-6,6,100),linspace(-6,6,100));
XG = zeros(size(YG));

Interpolate the temperature at the grid points. Perform interpolation for several time-steps to observe the evolution of the temperature contours.

tIndex = [2,3,5,7,9,11];
TG = interpolateTemperature(R,XG,YG,ZG,tIndex);

Define the geometric spherical layers on the cross-section.

t = linspace(0,2*pi);
ylayer1 = cos(t); zlayer1 = sin(t);
ylayer2 = 2*cos(t); zlayer2 = 2*sin(t);
ylayer3 = 4*cos(t); zlayer3 = 4*sin(t);
ylayer4 = 6*cos(t); zlayer4 = 6*sin(t);

Plot the contours in the range Tmin to Tmax for the times in tIndex.

figure('Position',[10,10,1000,550]);
for i = 1:numel(tIndex)
    subplot(2,3,i)
    contour(YG,ZG,reshape(TG(:,i),size(YG)),'ShowText','on')
    colorbar
    title(['Temperature at Time ' num2str(tlist(tIndex(i)))]);
    hold on
    caxis([Tmin,Tmax])
    axis equal
    % Plot boundaries of spherical layers for reference.
    plot(ylayer1,zlayer1,'k','LineWidth',1.5)
    plot(ylayer2,zlayer2,'k','LineWidth',1.5)
    plot(ylayer3,zlayer3,'k','LineWidth',1.5)
    plot(ylayer4,zlayer4,'k','LineWidth',1.5)
end

Reference

[1] Singh, Suneet, P. K. Jain, and Rizwan-uddin. "Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere." ASME. J. Heat Transfer. 138(10), 2016, pp. 101301-101301-11.

See Also

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