PDE Toolbox: Constant external heating on 4 sides of a Cylinder

Hi,

i am trying to simulate the heating of a hollow carbon shaft through 4 Infrared radiators which are placed alongside the shaft. I already constructed the model in 3D, but i don't know how to simulate the external heatflow with the 4 differently positioned radiators. Which conditions do i have to consider in order to succesfully simulate the heatflux in Matlab? I guess i need to do it similar to this https://de.mathworks.com/help/pde/ug/heat-conduction-in-multidomain-geometry-with-nonuniform-heat-flux.html with a function handle, but now i have 4 heat sources which emit heat on different locations.

The radiatiors are placed like this:

Many thanks in advance,

Pascal

Answer 1

[Ravi Kumar]

Hi Pascal,

If there is no axial variation, then I suggest using a 2-D crosssecion to solve the problem. In 2-D you can easily draw the large cylinder and smaller heating elements as different circles. You then model the heat generation using internalHeatSource function.

If the problem is indeed 3-D, then you might make a couple of modelling approximation:

1. On the blue solid parts create either fully burried heating element as holes. Apply an equivalent heat flux on the inner wall surfaces.
2. If heating element is not fully embedded, they also radiate heat to atmosphere, then you can make slots/channels at the location of heating element and apply heat flux BC.

Regards

Ravi

[Pascal Kobuß]

Hi Ravi,

thank for your answer. So basically my modell needs to be 3d and the elements are not embeddes but they do have a small distance to the model. How is it possible to construct these slots or channels?

Thanks,

Pascal

[Ravi Kumar]

Hi Pascal,

Are planning to solve a stead-state problem or a transient problem?

Regards,

Ravi

[Pascal Kobuß]

Hi Ravi,

the problem is transient. Shall i show my code to understand my model better?

Regards,

Pascal

[Ravi Kumar]

Hi Pascal,

It looks like PDE Toolbox is lacking two features to support this kind of modeling.

1. Multidomain geometry.
2. Surface-to-surface radiation modeling.

It requires multidomain geometry capabality to model heating elements, ambient air, and cylinder. Given that cylinder and elements are only connected through radiation heat exchange you need to model surface-to-surface radiation. The radition BC in PDE Toolbox supports only radiation of heat from a boundary to ambient.

You can share the code, but I am less hopeful of setting up this problem in PDE Toolbox.

Regards,

Ravi

[Pascal Kobuß]

Hi Ravi,

okay this is unfortunate.

To simplify the problem, i could assume, that the heatflux is produced directly on the surface of the cylindric model. So, heat would be generated on 4 lines of the surface. Is there a possible solution for this scenario?

Thanks,

Pascal

[Ravi Kumar]

Hi Pascal,

I would be suggesting using 'HeatFlux' to functionally distribute the heat around circumference. That is, if you plot heat flux vs theta, theta from 0 to 360, then your heat flux function could contain four peaks centered at for location where you want to locate the heating elements.

Regards,

Ravi

[Pascal Kobuß]

Hi Ravi,

so do i need to make something like a coordinate transformation? Is there a good example for that?

Thanks,

Pascal

[Ravi Kumar]

Hi Pascal,

You get spatial coordinates in Cartesian coordinates system as input to the function you define. You use those coordinates to construct whatever heat flux distribution at those coordinates. Sometimes, like in your case, the function definition becomes easier if you convert the input Cartesian coordinates to polar coordinates. You can refer to the externalHeatFlux function in the example you cited in your original post. Here, cart2sph has been used to convert the input coordinates (region.x, region,y, region.z) int spherical coordinates. In your case, you can use cart2pol to convert the input coordinates to polar coordinates. And use the theta output to decide the value of heat flux.

Regards,

Ravi

[Pascal Kobuß]

Hi Ravi,

thank you, this definitly works for me! I definded my heatflux regionally now. So, if theta is between a value, then there will be generated a heatflux. I defined 4 regions for that. Mysteriously, my rod heat distribution now looks like that:

So you can see, heat only will be generated on two regions. What did i do wrong? Here is my code of the function for better understanding.

function Qflux = externalHeatFlux(region,~)
    [theta,rho,~] = cart2pol(region.x,region.y,region.z);
    r_Kern = 0.02; %Alukernradius
    d_Lage_CFK = 0.0005; %Dicke einer CFK-Lage in m
    n_CFK = 4; %Anzahl CFK-Lagen
    hl_Kern = 0.25; %Halbe Länge Kern in m
    V_cyl =pi *(r_Kern+d_Lage_CFK*n_CFK)*(r_Kern+d_Lage_CFK*n_CFK)*2*(hl_Kern); %Volumen des bestrahlten Zylinders
    y_dach = 1500; %maximal ankommender Wärmestrom
    heatflux_percentage = 0.35; %Wieviel Prozent des Strahlers werden benötigt
    heatflux = y_dach/ V_cyl * heatflux_percentage; %W/m^3, um DGL-Einheit zu erfüllen
    Qflux = zeros(size(region.z));
    if theta > 17/9*pi && theta < pi/9 %area between 340° and 20°
        Qflux = heatflux;
    end
    if theta > 7/18*pi && theta < 11*pi/18 %area between 70° and 110°
        Qflux = heatflux;
    end
    if theta > 8*pi/9 && theta < 10*pi/9
        Qflux = heatflux;
    end
    if theta > 25/18*pi && theta < 29*pi/18
        Qflux = heatflux;
    end  
end

[Ravi Kumar]

I think you are overwriting a Qflux with a subsequent assignment as your theta ranges could fall into more than one if branches. You should be fine if you insert return after each assignment of Qflux.

Regards,

Ravi

[Pascal Kobuß]

Thanks Ravi, i have solved the problem by myself. I didnt know that theta is defined in the area [-pi, pi]. This was the problem.

Regards,

Pascal