Open jonmaddock opened 4 years ago
In GitLab by @jlion on May 26, 2020, 17:31
@jmorris-uk, @mkovari please have a look.
In GitLab by @mkovari on May 27, 2020, 16:44
RRR is an issue. Transmutation will gradually introduce impurities in the copper, and radiation damage will gradually cause lattice defects. Both factors will reduce the RRR during the lifetime of the machine.
For normal superconducting wire, 100 seems common. This should certainly be our highest assumption, but maybe we need to lower it for the reasons above.
Magnetoresistance will also be significant, increasing the resistance of the copper further.
Properties_of_Copper_and_Copper_Alloys_nistmonograph177.pdf
I have added a few queries in-line - in this format:
In GitLab by @jlion on May 28, 2020, 09:04
The integral is exact when taken the form as given and integrated to "infinity"
\tau_{dump}
$ the same as $\tau
$?yes
1.687\cdot 10^{-8}-2.213\cdot10^{-15} RRR
$ doesn't look right. The second term is always negligible.No, its the term that takes the RRR terms into account at low temperatures, I added a plot.
In GitLab by @jlion on May 28, 2020, 09:16
Is there any documentation for this voltage formula by the way? I am not so sure how to arrive at this form.
In GitLab by @jlion on May 26, 2020, 17:28
Quench Protection
protect
As the current routine
protect
(quench protection) is missing a documentation (?) I would like to propose a new more transparent implementation of basically the same formalism as before, but with clearer documentation and clearer assumptions.Assumptions
J(t)
$ with a decay time $\tau
$ and a delay time $t_d
$ (this is strictly speaking inconsistent with the rest)Equations
In thermal equilibrium and without losses the heat produced by the copper resistivity during a quench is equal to the heat needed to rise the temperature in the material by $
dT
$:Assuming the materials in the winding pack are equilibrated this is
where $
P
$ is the power deposited by the copper current in time $t
$, $i
$ runs over all winding pack materials and $V_i
$ stands for the volume of those materials. With $p=J^2 \eta
$, where $\eta
$ is the resistivity of copper and $J
$ the current in the copper, it is:Now, the quench restriction is to impose
The integral on the left hand side runs over the whole quench time while the integral on the right hand side goes from the operation temperature $
T_{op}
$ to a maximal $T_{max}
$. The difference $T_{max}-T_{op}
$ is usually chosen in the order of 150 K. The sum on the right hand side goes over all materials with fractions $f_i
$.The current density decay shall be written in the form:
(Note again that this is inconsistent and an assumption.)
Given this form the integral is $
\int J(t)^2 dt = J^2 (\frac{1}{2} \tau_{dump} + t_d)
$.Then:
with
and using
one ends up with:
To rewrite this as the winding pack current density and with $
1-f_{He}=f_{cond}
$ and $J_{WP} = J_{Cu} f_{Cu} f_{cond} (1-f_{case})
$:This is also the form of the previous implementation, apart from the added quench detection term $
t_d
$.Prerequisites
Needed are:
\eta_{Cu}(T)
$c_{Cu}(T)
$\rho_{Cu}
$c_{He}(T)
$\rho_{He}(T)
$c_{scu}(T)
$\rho_{scu}(T)
$I suggest using a Bloch-Grüneisen parametrization for the copper resistivity, along the lines of:
References, e.g.
https://www.copper.org/resources/properties/cryogenic/, https://en.wikipedia.org/wiki/Bloch%E2%80%93Gr%C3%BCneisen_temperature
This produces:
data from: https://nvlpubs.nist.gov/nistpubs/Legacy/MONO/nistmonograph177.pdf
And store it as a list for a certain RRR value.
I give some values for different temperatures below (helium at 6 bar):
T [K]
$\eta_{Cu}^{RRR=100} [n\Omega\,m]
$\eta_{Cu}^{RRR=1000} [n\Omega\,m]
$\rho_{He}[kg/m^3]
$c_{He}[J/(K kg)]
$c_{Cu}[J/(K kg)]
$Required data (q Integrals)
Only the q integrals will be useful in the function so they can be precalculated in advance and then included in the function later. Assume a maximal temperature rise of 150 K, no matter from which operating temperature. The $
q
$ integrals are then only dependent on the operating temperature and can be calculated by the above given formula to (RRR=100):T_{op}[K]
$q_{He} [s A^2/m^4]
$q_{Cu} [s A^2 / m^4]
$These values will enter the function.
Implementation
I suggest implementing these two functions in the module superconductors:
The maximum voltage is independent of this and should be treated in a seperate function, e.g.:
Todo
I have only tested this for a few values but it seems to be a bit more optimistic than the previous implementation which might be due to the used copper RRR value.
Todo: