Equation for plasma area/volume #513

Closed jonmaddock closed 1 year ago

In GitLab by @mkovari on May 5, 2017, 08:34

From: Coleman Matti
The conclusion in the end was:

From: Kemp Richard
The PROCESS code is below.

      zn = kap * a
      rci = ( (r + a)**2 - (r - tri*a)**2 - zn**2) / &
           (2.0D0 * (1.0D0+tri) * a)
      c1 = r + a - rci
      rc2 = (-(r - a)**2 + (r - tri*a)**2 + zn**2) / &
           (2.0D0 * (1.0D0-tri) * a)
      c2 = rc2 - r + a

The error in the paper is linked to an error in the code, which is not mathematical but is typographical: rc_x and c_x are the wrong way round: rc1 is the centre of the first arc and c1 is its radius, not the other way round (as you might actually expect...).

Note also that the inputs are for the separatrix rather than the 95% flux surface.

So for R = 9m, a = 3m, kappa_x = 1.78, delta_x = 0.5 I get:

rc1 = 6.58m
c1 = 5.42m
rc2 = 16.18m
c2 = 10.1m

and the separatrix is in the right position.

(The formula for volume is right too, it just also has the mislabelled variables.)

In GitLab by @ajpearcey on Mar 27, 2019, 15:00

The notation used in the fvol function in plasma_geometry.f90 has been updated to reflect above.

In GitLab by @ajpearcey on Mar 27, 2019, 15:00

closed

In GitLab by @mkovari on Mar 28, 2019, 08:15

@ajpearcey - Can you confirm that you have tested it and it produces exactly the same results as before, both for area (cross-section or surface area?) and volume, and that these results have been checked against an external reference or hand calculation?

Also - would you say these formulae are appropriate for a single null, symmetric double null, or (approximately) for both?

In GitLab by @mkovari on Mar 28, 2019, 08:15

reopened

In GitLab by @ajpearcey on Apr 25, 2019, 09:49

@mkovari I can confirm that these functions to work and give consistent plasma geometry, but only as long as trianguarity satisfies $\delta !=\pm1$ and $\kappa^2 > (1 + \delta)^2$ at which point the volume calculated will unphysical results and PROCESS will immediately stop the calculation.

The functions in plasma_geometry.f90 assume the triangular plasma geometry is given by considering two circular segments. These have been compared to the hand calculation following the approach using the information found here and applying Pappus's Theorem. I find different expressions to those given in PROCESS, but I find identical answers for all plasma geometry variables, these seem to be explained by some inscrutable use of some trig identities.

The limits on the validity of the trianguarity are due to variables denomo and denomi becoming either zero or negative. These issues would arise in either the currently used expressions or the ones I found.

I have scanned trigularity to check the behaviour of the plasma geometry variables and they are well behaved with the limits described above, although outside $\delta = 0.4-06$ we get an error message in the validity of the pedestal scaling.

Also, the formulae assume a symmetric double null shape.

In GitLab by @mkovari on Apr 25, 2019, 10:03

only as long as triangularity satisfies $-1 < \delta < 1$ and $\kappa > 1 + \delta$

In GitLab by @mkovari on Apr 25, 2019, 10:04

Is the develop branch now merged and up to date?

In GitLab by @ajpearcey on Apr 25, 2019, 10:17

The geometry calculations with fail if either $\delta$ equals precisely 1 or -1, or $\kappa < 1 +\delta$. The inputted triangulaity is also hard coded allow only values in between -1 and 1. It will run fine if the triangularity is 0.999... as long as the elongation is large enough. Sorry, that was a bit unclear.

The branch which includes the changes above is currently waiting to be merged into the develop branch.

In GitLab by @ajpearcey on Apr 25, 2019, 10:35

closed

In GitLab by @mkovari on Nov 20, 2019, 11:40

Still slightly confused about the correct formulae. It would be nice to rewrite the parts of section 9 of the document below that relate to plasma volume and area to reflect (a) the code as it stands now, (b) the correct mathematics and (c) formulae that have been checked numerically or using calculations external to PROCESS.

K:\Power Plant Physics and Technology\PROCESS\PROCESS documentation papers\PROCESS full documentation build, coils and heating.docx

Also, I understand that this calculation is done more than once. It would be good to have a reference to the relevant sections of code.

Also, note that in Plasmod the volume is calculated by numerical integration, giving a slightly different answer, which is output separately

@ajpearcey ?

In GitLab by @ajpearcey on Jan 14, 2020, 14:28

As there is no complete and clear derivation of plasma geometry, I have tried to create one I include here. One of the main issues is that within the module plasma_geometry.f90 there are two versions of every subroutine, older and newer versions, which are selected with a switch igeom. Neither of the expressions used in the two plasma volume calculation is at all clear, straight forward or documented. Therefore, I have derived the plasma volume again and find an expression with a different form to the default igeom=1 calculation but gives the same answer for the volume.

The word document detailing the calucation is called plasma_geometry.docx and is found at K:\Power Plant Physics and Technology\PROCESS\PROCESS documentation papers\plasma_geometry.docx

In GitLab by @mkovari on Jan 14, 2020, 14:45

A few suggestions.

@stuartmuldrew to check whether using igeom=1 for FNSF makes a difference.
If it does, check using Alex's formulae to see which is correct.
If all is OK, remove the switch igeom.
Add some text about the range of parameters for which the code is thought to give the correct answer.
Put something into the user manual to explain what is going on (!).
Also update the main text for this issue so that people can see the current situation at a glance.
Also write something for Laurens as he was asking about this.

In GitLab by @ajpearcey on Apr 11, 2023, 16:00

The word doc can also be found here plasma_geometry.docx

ukaea / PROCESS

Equation for plasma area/volume #513