2. An Approach to Measure and Model Heat Transfer in He II 21.04.2015 T. Winkler, T. Koettig, R. van Weelderen, J. Bremer, H.J.M ter Brake BLMTWG - Tiemo Winkler2

3. Content Measuring Heat Transfer Measurement Idea Example Measurements Modelling Heat Transfer Classical Heat Transfer He II Heat Transfer Defining Thermal Conductivity Simplified Model for a Superconducting Rutherford Cable 21.04.2015 BLMTWG - Tiemo Winkler3

4. Measurement Idea To measure the heat transfer between a superconducting cable and the surrounding helium bath, heat needs to be generated inside the cable. An AC magnetic field can be used to generate heat. So, the idea comes down to the following set-up: A cable sample with temperature sensors is placed inside another superconducting coil. This coil is ramped up and down very fast and the resulting magnetic field heats up the cable. 21.04.2015 BLMTWG - Tiemo Winkler4 superconducting cable He bath Kapton insulation AC magnetic field

5. Sample Measurement 21.04.2015 BLMTWG - Tiemo Winkler5

6. Classical Heat Transfer 21.04.2015 BLMTWG - Tiemo Winkler6

7. He II heat transfer 21.04.2015 BLMTWG - Tiemo Winkler7 Taken from vanSciver – Helium Cryogenics 2012 ΔT, Δ p Typical behaviour of temperature und pressure across a channel for different heat fluxes in counterflow He II. Note the change in slope when exceeding a certain heat flux. Zooming in on the changing point: The lower regime depends on the channel diameter. The transition point is not a constant value. The upper regime is not linear.

8. He II Heat Transfer 21.04.2015 BLMTWG - Tiemo Winkler8

9. He II – Thermal Conductivity 21.04.2015 Interpretation of the 1/3 Power Law in more than 1D: 1D 2D/3D y x BLMTWG - Tiemo Winkler9

10. What is missing now is the connection between the two regimes. There needs to be a point for which the regimes have the same thermal conductivity: At this point the transition from the laminar regime to the turbulent regime occurs. This point has the units of a temperature gradient. He II – Thermal Conductivity 2 21.04.2015 BLMTWG - Tiemo Winkler10 T(K )

11. Simplified Model for a SC Rutherford Cable The idea is to identify single cells in a Rutherford cable. Since the goal is to investigate global losses in the cable the single cell does not necessarily has to have the exact same geometry as the single cell in the real cable. The simple model consist of two layers of superconductor with a thin gap in- between the two. The ratios of the layers are chosen in such a way that the ratios in the cable are represented. The edge of the model is blocked by a porous Kapton insulation layer. 21.04.2015 BLMTWG - Tiemo Winkler11 Cable strands He channels cable helium Kapton insulation

12. Summary A proof of concept of this method to investigate heat transfer in He II has already been done. A model with the thermal conductivity as show has been implemented in COMSOL. Ongoing sample preparation. Heat Transfer measurements of a coil sample for known applied pressures. Measurements in saturated as well as in pressurized helium are planned. Ongoing work to improve the model as the transition between the two regimes is not that sharp. The defining parameter set for the simplified model still needs to be defined. 21.04.2015 BLMTWG - Tiemo Winkler12