a temperature dependency the conductivity values are given in tabular form for several temperatures and the program will do a linear interpolation between the inserted values (see figure 3.19). As heat conduction might also occur between bodies the connection between bodies needs to be defined. In the case of the oven between most bodies it is assumed that no heat conduction occurs as the touching pressure between the bodies is low due to the construction of the oven. Only where the cane is connected to the oven a conducting boundary is assumed. After the definition of the geometry, the assignment of the materials and the definition of connections, the mesh is created.
3.6.2. Boundary conditions
The next step is to define the boundary conditions of the model, the loads. Several loads are possible:
• Temperature: An edge (1D), a plane/surface (2D), or a body (3D) can have a prede-fined temperature.
• Convection: A surface or an edge might be exposed to a heat transport due to con-vection.
• Radiation: Edges or surfaces can emit thermal radiation that will transport heat between the body and other bodies that have a defined radiation boundary.
• Heat flow and heat flux: A defined heat flow or heat flux through an edge or a surface.
• Internal heat generation: A plane or a body generates a defined amount of heat that will be evenly distributed over the body.
Temperature Two regions in the 2D model have a fixed temperature, the end of the cane on the opposing side to the oven and the outside of the outer enclosure. In both cases the temperature here is 22 °C. For the end of the cane this simulates the cooling at the feedthrough, where the cane is inserted into the plasma chamber. The outside of the outer enclosure is the outer boundary of the simulation and in case of the test stand it resembles the test stand walls that are at room temperature and are cooled by the surrounding air.
Convection It is assumed that convection does not play a role for the thermal profile of the oven, as the plasma chamber as well as the test stand are evacuated during its operation.
It is however possible that the lead gas jet introduces heat transport by convection that is not modeled in the simulations.
3.6. 2D model of the micro oven in ANSYS
Radiation Radiation is the dominating heat transport mechanism between bodies in the oven assembly, as the setup of the oven insulation is designed to prevent conduction.
In the 2D model the radiation happens between the planes representing a body and the radiating parts are the edges of these planes. In ANSYS this means that radiation heat loads are defined on the edges of the planes. Each radiation heat load requires a given emissivity.
When the emissivity of the material depends on the temperature it can be given in tabular form, with pairs of an emissivity and a temperature value. The used values are presented in[55]. As they where also used in the 3D model, figure 3.19 in section 3.7.1 shows the emissivity values for the different oven materials.
Internal heat generation The oven is operated at fixed power levels, that are realized by a loop controlling the power supply providing the heating current. A certain power level equals a certain amount of heat being transported into the oven setup by the current. This can be realized as an internal heat generation within the planes or bodies that represent the filament. For the internal heat generation the input value is a heat density in W m−2 in 2D or W m−3in 3D. In the 2D model, the resistances of the different conductors that transport the heating current was taken into account, to calculate the fractions of the total power that are lost in each individual conductor. For a given power level the result of this calculation is a total amount of heat that needs to be generated in the planes that represent these conductors. With the area of the plane this can then be evaluated to an internal heat generation value.
3.6.3. Solution and adaption to measurements
The result of a the simulation is a temperature field for every part of the model. Additionally it is possible to derive a field of the total heat that leaves or enters a body at its boundaries.
As the dominating heat transfer is radiation, the models behaviour is depending on the input of the emissivity values. It was shown by C. Fichera et al. that the emissivity of tantalum affects the model the most, and the emissivity values of copper, stainless steel and tantalum where adapted to resemble temperature measurements at the test stand[55].
Figure 3.14 shows the resulting temperature for two parts of the oven, a copper pin inside of the crucible and the oven cover, for heating powers between 1 W and 20 W. The copper pin was used to fixate a thermocouple inside of the crucible. As each body in the simulation has a distribution of temperatures, the plot shows the temperature range that is present in the respective distribution at a certain heating power.
0 5 10 15 20 400
600 800 1,000
P [W]
T[°C]
Copper pin within crucible
Meas 2D model
0 5 10 15 20
100 200 300 400 500
P [W]
T[°C]
Oven cover
Meas 2D model
Figure 3.14.:Temperature of the oven cover and a copper pin within the crucible, against heating power. The simulation and the measurement were done before this work and are presented in[55].
The resulting 2D model was used during this work to simulate different aspects of the oven. The following section presents results that where obtained using the 2D model.
3.6.4. Oven tuning needed due to the temperature profile
During the operation of the GTS-LHC ion source the heating power of the oven is one of the frequently adapted tuning parameters. It needs to be raised throughout the run to maintain stable ion beam production. Normally the oven power was raised starting from around 7 W to a final value of 20 W within two weeks of operation at the source.
However when the emitting lead surface within the crucible does not change in size or position, a constant temperature of this surface would lead to a constant evaporation rate (see section 3.4.2). Thus one would not expect that it is necessary to adapt the heating power.
It was studied whether the simulated temperature profile can partly explain the need for the raise in heating power. For this purpose two distributions of lead within the crucible where simulated using the 2D model.
The geometry
In the first case the complete crucible is filled with liquid lead, while in the second case the lead is only occupying a small fraction of the crucibles volume. In both cases it was
3.6. 2D model of the micro oven in ANSYS
assumed that the lead sample, when molten, stays in a cylindrical shape due to surface tension.
Crucible full Crucible empty
Surface representing the lead sample
Figure 3.15.:Geometry of the surface representing the cylindrical lead sample in the 2D simulation.
On the left: The crucible is newly filled. On the right: The crucible is almost empty with the remaining lead gathered at the tip of the crucible.
The real distribution is not known. At the oven test stand the remaining lead within the crucible, after a run where not all lead was consumed, was always in a cylindrical shape and in most cases in the front of the crucible.
Parameters of the molten lead
For the liquid lead the thermal conductivity and the emissivity needed to be defined. The conductivity was defined with the two values given in table 3.2[56] [57].
Temperature[°C] Conductivity[W m−1K−1]
327.5 15.5
1200 23
Table 3.2.:Conductivity values used in the 2D simulation for the molten lead from[56]and[57]. ANSYS interpolates and extrapolates the conductivity from these values, which is appro-priate as the temperature dependence of the conductivity of lead is close to linear.
No published data on the thermal emissivity of molten lead could be found but the simulation showed a low sensitivity on the emission coefficient. Two cases where tested, a constant emissivity value of 0.43 which is published for solid rough or oxidized lead and a value of 0.057 which is published for solid polished lead[58](Similar in[59]).
Results
Figure 3.16 shows the resulting temperature span of the surface, that represents the lead in the two simulations.
It can be seen that there is a temperature difference between the lead that fully fills the crucible and the lead that is only taking up the tip of the crucible. So, while the oven is
0 5 10 15 20 300
400 500 600 700 800 900 1,000
P [W]
T[°C]
ε=0.43
Full Empty
0 5 10 15 20
300 400 500 600 700 800 900 1,000
P [W]
T[°C]
ε=0.057
Full Empty
Figure 3.16.:Lead temperature from the thermal 2D simulation for two bodies that resemble a full and an empty crucible. Two different emissivities,ε, of the lead where assumed, 0.43 on the left and 0.057 on the right.
operated and the lead sample is consumed it is possible that the overall temperature of the lead changes at a stable power due to redistribution within the crucible.
However it is also evident that the difference in temperature would only need a correction of some watt and does not explain the necessity of raising the oven power e.g. from 8 W to 20 W. This hints that also other mechanisms are involved in reducing the lead gas flow from the crucible like another lead surface position. Other mechanisms that can affect the total evaporation, like blockages and temperature instabilities where found experimentally as presented in chapter 5.