BLMTWG - Tiemo Winkler 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 Winkler 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 Winkler 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

superconducting cableHe bath

Kapton insulation

AC magnetic field

BLMTWG - Tiemo Winkler 5

Sample Measurement

21.04.2015

𝜏=𝜌∗𝑐𝑝

𝑘∗𝐴∗

BLMTWG - Tiemo Winkler 6

Classical Heat Transfer

Classical Heat Transfer can be described by Fourier’s Law:

In the case of a solid the thermal conductivity is a function of temperature. For small temperature differences the heat flux and the temperature gradient show a linear dependency on each other.

21.04.2015

BLMTWG - Tiemo Winkler 7

He II heat transfer

21.04.2015

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.

BLMTWG - Tiemo Winkler 8

He II Heat Transfer

The two regimes are:

Laminar Regime or Landau Regime

Turbulent Regime or Gorter-Mellink Regime

For the laminar regime the thermal conductivity has besides the temperature dependency also a diameter dependency.

In the turbulent regime the thermal conductivity is temperature and gradient dependent as can be seen clearer on the next slide.

21.04.2015

BLMTWG - Tiemo Winkler 9

He II – Thermal Conductivity

21.04.2015

• Interpretation of the 1/3 Power Law in more than 1D:

1D 2D/3D

�̂�𝑥( 𝑑𝑇𝑑𝑥 )

13 �̂�

𝑥

𝛻𝑇|𝛻𝑇|

13𝛻𝑇

y

x

BLMTWG - Tiemo Winkler 10

What is missing now is the connection between the two regimes. There needs to be apoint for which the regimes have the samethermal conductivity:

At this point the transition from the laminar regimeto the turbulent regime occurs.This point has the units of a temperature gradient.

He II – Thermal Conductivity 2

21.04.2015

|𝛻𝑇|(K𝑚− 1)

T(K)

K (Wm−1K −1)

𝐾 𝐿𝛾

∝ 𝑓 −1/3

BLMTWG - Tiemo Winkler 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

Cable strands He channels

cable

cablehelium Kapton insulation

BLMTWG - Tiemo Winkler 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