Transverse orthotropic stress modelling in TF coil #992

Closed jonmaddock closed 1 year ago

In GitLab by @skahn on Jan 17, 2020, 16:11

Description

The current stress model calcualtions are made using a isotropic Hooke's law. This means that they are mostly addapted for layer of one material (steel case for example). However the Winding pack section has very different proporties in the vertical direction compared to the toroidal direction driving the necessity of considering the vertical stress a posteriori as a constant quantity.

However it is possible to extend the stress model calculations to take this anisotropy into account as discussed at the end of the following document :

Derivation_of_multilayer_generalized_plain_strain.docx

This allow to used the generalized plane strain formulation for superconducting TF coils.

Copper resistive coil implementation

Copper TF coils are actively cooled in PROCESS. For this reason a fraction of the coil is dedicated to cooling. Assuming constant strain properties (parallel case), the associated young modulus $E_z$ is given by :

$\bar{E_{z}} = (1-f_\mathrm{cool})E_\mathrm{Cu}$

with $f_{cool}$ the fraction of the toroidal mid-plane cross section allocated for cooling. After the stress calculations, the copper vertical stress $\sigma_{zz}^{copper}$ is obtained from the smeared one calculated by the stress model by :

\sigma_{zz}^\mathrm{Cu} = \frac{E_\mathrm{Cu}}{\bar{E_{z}}} \bar{\sigma_{zz}}

Show the comparison for the copper FNSF test case

Aluminium resistive coil implementation

The aluminium magnet design is thought to contain a CS in his middle and hence has to containt turn insulation. In addition to the cooling void fraction, the turn insulation must be considered for the smearing/un-smearing procedure, so the smeared young modulus becomes

$\bar{E_{z}} = f_\mathrm{ins}E_\mathrm{ins} + (1-f_\mathrm{ins})(1-f_\mathrm{cool})E_\mathrm{Cu}$

and the stress unsmearing

$\sigma_{zz}^\mathrm{Al} = \frac{E_\mathrm{Al}}{\bar{E_{z}}} \bar{\sigma_{zz}}$

Superconductor coil implementation

A little more consideration must be taken into account for SC coil (as usual ...). The external casing layer, facing the plasma takes a fraction of the vertical force, while not included in the stress modelling. Assuming the vertical average strain to be the same (parallel smearing), the fraction of vertical force to be considered in the stress calculation is :

f_{F_\mathrm{vert}} = \frac{ E_\mathrm{ins} A_\mathrm{ins} + E_\mathrm{steel} A_\mathrm{steel}^\mathrm{WP + case\ nose} }{ E_\mathrm{ins} A_\mathrm{ins} + E_\mathrm{steel} A_\mathrm{steel}^\mathrm{tot} }

with $A_\mathrm{ins}$ the insulation cross-section area (tunrn insulation + WP ground insulation), $A_\mathrm{steel}^\mathrm{tot}$ the total steel cross-section incluing the plasma facing side casing and $A_\mathrm{steel}^\mathrm{WP + case\ nose}$ the stell area of the WP (lateral casing included) + the nose casing excluding the plasma side casing.

The definition of these area are a bit tricky to as they must fit with the circular geometry used in the model and the rectangular geometry of the WP. A middle ground has been found defining the area

$A_\mathrm{steel}^\mathrm{WP + case\ nose} = A_\mathrm{steel}^\mathrm{tot} - \pi \left({R_\mathrm{TF}^\mathrm{out}}^2 - {R_\mathrm{WP}^\mathrm{out}}^2 \right)$

The WP vertical smeared young modulus is on the other hand defined as

$\bar{E_{z}} = \frac{A_\mathrm{ins}}{A_\mathrm{WP}^\mathrm{tot}}E_\mathrm{ins} + \frac{A_\mathrm{steel}^\mathrm{WP}}{A_\mathrm{WP}^\mathrm{tot}} E_\mathrm{steel}$

with

$A_\mathrm{steel}^\mathrm{WP} = A_\mathrm{steel}^\mathrm{tot} - \pi \left({R_\mathrm{TF}^\mathrm{out}}^2 - {R_\mathrm{WP}^\mathrm{out}}^2\right) - \pi \left({R_\mathrm{WP}^\mathrm{in}}^2 - {R_\mathrm{TF}^\mathrm{in}}^2\right)$

and

$A_\mathrm{WP}^\mathrm{tot} = \pi \left({R_\mathrm{WP}^\mathrm{out}}^2 - {R_\mathrm{WP}^\mathrm{in}}^2\right)$

Hopefully a simpler unsmearing is obtained by :

$\sigma_{zz}^\mathrm{steel} = \frac{E_\mathrm{steel}}{\bar{E_{z}}} \bar{\sigma_{zz}}$

Comparison with the PROCESS plane stress model

On DEMO 2018 : Steel structure stress with generalized plane strain (new)

Steel structure with plane stress (old)

The trend is to observe a larger vertical stress in the bucking region, while a smaller one in the winding pack region.

It is interresting to note that if the vertical and toroidal young modulus are the same, the case vertical stress gets larger.

For comparison the smeared stress of the generalized plane strain case is the following

Totally orthotropic model?

This task is much more difficult as the differencial equation changes its form, and J. Last has not found any analytical solutions. He derived however an approximation of the solution that he claimed to be valid up to 1%.

This is however not a priority for free standing TF coils as J. Last has proven that the value of the radial young modulus does not matter for ITER like magnets.

It could be however helpfull to have a fully orthotropic formulation to verify the assumption that the value of the radial young modulus does not matter in the stress calculation.

Two configurations where radial stress can be quite important might require more attention :

ST configuration
Bucked and wedged (the radial displacement is much lower reducing the toroidal stress)

Best regards
Seb

Ref : @mkovari @jmorris-uk @stuartmuldrew

In GitLab by @mkovari on May 21, 2020, 16:16

I don't understand why the copper and aluminium coils are treated differently.

In GitLab by @skahn on Jul 6, 2020, 12:29

closed

In GitLab by @skahn on Nov 3, 2020, 13:10

The current Cooper magnet desing we consider has no insulation layer.

There is indeed no genuine reason to forbid turn insulation in copper. I will change that. in the future develppement/other issue.

In GitLab by @pc6783 on Aug 2, 2021, 19:06

@skahn Would you please upload the derivation document in PDF format, also, for compatibility?

ukaea / PROCESS