Detailed model: 3Df

A detailed 3D model, for referral called ’3Df’, was created including all of the oven parts that also are included in the 2D model and using the same boundary conditions. The basis for the geometry is the CAD model that was used to produce the oven and can be found in the CERN database EDMS[60].

Several adaptions had to be made to allow a proper simulation of the thermal radiation.

This mainly consisted of avoiding overlapping geometries, as some bodies in the original CAD model share geometry at the same place in space. The tip of the filament was reshaped to not stick into the surrounding alumina cylinder. Also at the connection of the outer oven cover and the end of the cane some parts of the geometry needed to be cut away to allow heat exchange by radiation.

The mesh which is then created from the geometry is a compromise to the computing time. It was adjusted to be fine enough at certain places to represent the complex geom-etry from e.g. the filament and the filament holder and is coarser at places with simple geometries like the oven cane. Figure 3.17 depicts the mesh for some of the oven parts.

The ANSYS mesh metric can be used to asses the quality. It is a number between 0 (lowest quality) and 1 (highest quality) that calculates the ratio of the volume of an element and the sum of its edge lengths. Here on average the mesh quality is 0.73. Even with this rather coarse mesh the model is computational heavy and the iterative process can take a long time (For 10 W approximately 13 d of CPU-time on the CERN High Performance Computing (HPC) cluster lxplus and using 48 cores[61]).

The loads

The loads used in the model are the same as in the 2D model. This mainly means that all surfaces are radiating the heat with the respective temperature dependent emission coefficient, for their material. Figure 3.18 shows the material assignments of the 3D model which also determines the assignment of the thermal emissivity that is being used.

Figure 3.19 shows the temperature dependent emissivity values that are assigned to each material and that are also used in the 2D model[55].

Figure 3.20 shows the other loads of the 3D model besides the radiation loads, which are assigned to every surface of the model. In the 2D model the connection between the oven cane and the piece connecting the oven to the cane was evaluated have a thermal conduc-tance coefficient of 14 500 W m⁻²K⁻¹([55]). In the 3D model two additional connections with conduction are introduced which are also shown in the figure 3.20. As here the oven

Oven cane Outer enclosure

Crucible

Heating filament

Filament holder

Figure 3.17.:Appearance of the mesh for the three dimensional oven model in ANSYS. The mesh is a compromise between representing the geometry correctly and not introducing too many nodes.

3.7. 3D model in ANSYS

Stainless steel Alumina Tantalum

Copper Kapton

Figure 3.18.:The material assignments at the 3D model of the oven.

is welded the connections are assumed to be perfectly conducting, i.e. the temperature of both bodies at the connection needs to be the same.

While the radiation boundaries can be directly transferred from the 2D model, the internal heat generation has to be adapted. For a scheme of the current carrying parts of the oven see figure 2.8.

In the 2D model the bodies are represented by planes. But if these planes are revolved around the symmetry axis the resulting shape is not the same as in the 3D model. For example the 3D model features the real shape of the filament, which is a single long body and the volume of this body can be a different one than the theoretical ring shaped filaments from the symmetrical 2D model.

When the heat generation is given as a density, e.g. in[W m⁻³]the total heat generated depends on the volume of the body. Therefore the heat density input in the 3D model needs to be adapted. To convert the internal heat generation from the 2D model (h_2D/[W m⁻³]) to a corresponding heat generation for the 3D body (h_3D/[W m⁻³]), the volumeV_cond.2Dof the theoretical symmetrical conductors from the 2D simulation is needed. This can be calculated from the respective surface areaa_cond.2Dof the conductors in 2D:

V_cond.2D=2πr_rot.a_cond.2D , (3.19)

wherer_rot.is the distance of the surface centre to the symmetry axis. Using this volume the internal heat generation can be adapted:

−200 0 200 400 600 800 1,000 1,200 1,400 0

0.2 0.4 0.6 0.8

T [°C]

ε[dim.less] Stainless steel

Tantalum Alumina

Copper Kapton

Figure 3.19.:Thermal emissivities of the materials that are used in the 3D model. The values at temperatures below 20 °C are used for ANSYS to make the correct interpolation.

Internal heat gen. SS

Internal heat gen. Ta Internal heat gen. Copper

FixedT at cane end

Connection with conduction

Figure 3.20.:Loads and contacts of the 3D oven model, apart from the radiation. The heat gen-eration happens in all parts that transport the heating current, weighted with their respective resistances.

h_3D=h_2DVfil.2D

Vfil.3D

. (3.20)

Table 3.3 shows the heat generation values of the 2D conductors and the derived heat generation for the 3D model.

The difference in volume for the copper conductor in 2D and 3D stems from the different representation of this body in 3D. In the 2D model the cane is simplified and does not feature the vacuum feedthrough of the inner conductors. As described in appendix B, simulations containing temperature dependent values like conduction or emissivity (and generally radiation) are not linear and need to be solved by using an iterative process. It helps to apply the loads like heat generation stepwise. For the heating power of only 10 W the simulation of the detailed 3D model was run with 100 sub steps.

3.7. 3D model in ANSYS

Material Power fraction h_2D[W mm⁻³] V_cond.2D/V_cond.3D h_3D[W mm⁻³]

Tantalum 0.95 8.47 10⁻³ 1.04 8.81 10⁻³

Stainless steel 0.027 2.00 10⁻⁶ 0.9957 1.99 10⁻⁶

Copper 0.021 9.9 10⁻⁵ 1.621 1.6 10⁻⁴

Table 3.3.:Heat generation values of the 2D conductors (h_2D), the conversion factor ^V_V^fil.2D

fil.3D and the derived heat generation for the 3D modelh_3Dfor the different conductors.