quench simulation studies: program documentation of spqr...
TRANSCRIPT
LHC Project Note 265
July 20, 2001
Quench simulation studies:
Program documentation of SPQRSimulation Program for Quench Research
F. Sonnemann, M. Calvi
Keywords: quench simulation, quench propagation, superconducting magnets, FDM, SPQR
Summary
Quench experiments are being performed on short models and full-size prototypes of the supercon-ducting magnets and busbars to determine the adequate design and protection. Many tests can onlybe understood correctly with the help of quench simulations that model the thermo-hydraulic andelectrodynamic processes during a quench. In some cases simulations are the only method to scale theexperimental results of prototype measurements to match the situation of quenching superconductingelements in the LHC.
This note introduces the theoretical quench model and the use of the simulation program SPQR(Simulation Program for Quench Research), which has been developed to compute the quench processin superconducting magnets and busbars. The model approximates the heat balance equation withthe finite difference method including the temperature dependence of the material parameters. SPQRallows the simulation of longitudinal quench propagation along a superconducting cable, the transversepropagation between adjacent conductors, heat transfer into a helium bath through an insulation layer,forced quenching by heaters, and the impact of induced eddy currents due to a changing magneticfield. The simulation output includes quench data such as the longitudinal and transverse quenchpropagation velocity, the impact of cooling conditions on the hot spot temperature, the quench heaterdelays, and the quench back effect.
The numerical approach is adequate for a more precise modelling of the complex quench processeswith respect to analytical models. After the theoretical description of the model, the numericalrealization is presented. Following that, the program description and its documentation are given.Some examples how to use SPQR are shown.
Contents
1 Introduction 4
2 Heat balance equation 52.1 One dimensional heat balance equation . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Thermal impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1
2.1.2 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Internal heat generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Current decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.5 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.6 External heat pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Insulation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Constant gradient approximation . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Matrix evaluation of the heat flux . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Quench heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Two-dimensional heat balance equation . . . . . . . . . . . . . . . . . . . . . . . 142.5 Three-dimensional approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Discretisation 193.1 One dimensional FDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Heat transfer trough insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Quench heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Two dimensional FDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Three dimensional FDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.6 Matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7 Plug and λplate calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.8 Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Temperature dependence of the material parameters 28
5 Program documentation 305.1 Input variable names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Output variable names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.5 Program routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5.1 Read-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5.3 Interpolating routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.5.4 Time-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.5.5 Write results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.5.6 Abort and end of execution . . . . . . . . . . . . . . . . . . . . . . . . . 475.5.7 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Calibration and tests 476.1 Numerical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Comparison with analytical models . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Calibration of heat transfer through insulation . . . . . . . . . . . . . . . . . . . 49
2
7 Simlation examples 507.1 Bus-bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.1.1 Bus-bar cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.1.2 Plug calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.1.3 Matrix evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.1.4 Main bus-bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.2 Corrector magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2.1 Transverse propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2.2 Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.3 Main magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3.1 Quench heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3.2 Eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3.3 Transverse quench propagation . . . . . . . . . . . . . . . . . . . . . . . 52
8 Acknowledgements 53
A Analytical formulas 54
B Variable names 55
C Derivation of the discrete form of the heat balance equation 56
3
1 Introduction
In order to improve the understanding of quench propagation in superconducting busbars andmagnets it is essential to study the heating process and the heat transfer to liquid helium.Analytic models are always limited since they use simple geometries and assume simplifieddependencies of material properties on the temperature. As tests of the protection system arelimited by the available time and resources a FDM program has been developed to solve theheat balance equation and to help optimising the protection system for the series of different su-perconducting elements to be installed in the LHC. As most quench experiments are performedon short models of magnets and busbars smaller than those to be installed in the LHC, onlycomputations allow extrapolating to the case of a quench in the LHC. Sometimes experimentalresults that can even be performed on full-size prototype magnets require simulation studies tobe interpreted correctly.
This note is a description and documentation of the simulation tool Simulation P rogramfor Quench Research. SPQR has been used to better understand experimental quench data, todetermine the adequate protection and dimensioning of the LHC busbars, to help optimisingthe quench heater strip layout, and to study the impact of eddy currents for quenches in LHCmain and corrector magnets.
The theoretical model is introduced and the numerical approach is presented. Exampleshow to use SPQR to simulate a quench in the main magnets, the corrector magnets, mainbusbars and small busbar cables are given.
Program execution
At the program start the material properties are calculated according to the geometry andinput parameters (for instance the copper resistivity as a function of RRR and applied magneticfield). The coordinate system is defined in Fig. 1. The current flows in x direction. An initial
z
x
y
conductor
I
Figure 1: The coordinate system of SPQR
Gaussian temperature profile is computed to provoke a quench with peak temperature T initand sigma xT decay (also yT decay and zT decay for the three dimensional model). Thefurther temperature evaluation depends on the balance of the internal heat generation, thelongitudinal heat conductivity, the heat transfer from or into the surrounding environment and
4
the heat capacity of the structure.Various models for the heat transfer into helium are available including the modelling of
the heat transfer through the insulation layer. Quenching by heaters can also be studied. Theinternal heat generation is determined by the resistance development, the quench detectionvariables and the applied current.
The cross-section of the superconducting structure that is usually a single strand super-conducting cable or a Rutherford type cable can be any mixture of niobium-titanium, copper,and helium (if helium cross-section is not specified, it is assumed to be zero). The materialproperties of copper are calculated dependent on the entry parameters RRR and magnetic field.According to the fraction of Cu, NbTi, He, the averaged heat capacitance, conductivity andresistivity as a function of temperature are determined and stored in a lookup-table. Duringthe program execution the structure is treated as being homogeneous per unit length.
The simulation results are written into ASCII table files readable for all common dataanalysis tools.
Several type of quench calculations are possible with SPQR. All models have in commonthat they solve the heat balance equation using the finite difference method. The standardsetup assumes a symmetric temperature development around the hot spot which means thatcalculating half of the propagating normal zone is sufficient. However, the user can also choosewhere the quench starts and force the endpoints to be kept at a constant temperature thatallows estimating worst case scenarios.
The quench processes that can be simulated with the different models are summarised inTable 1, 2.
1D 1DI 1DH 1DHM 3D 3DC ED EDM Plongitudinal propagation x x x x x x x x xtransverse propagation x xheater delays x x x x x xheater performance x x x x x xquench back x xcooling impact x x x x x x x x xinsulation layer x x x x x x x xmatrix model x x x xplug calculations x
Table 1: Overview of the possible combinations of quench processes included in SPQR andtheir applications.
2 Heat balance equation
The heat balance equation is a second order differential equation in space and first order dif-ferential equation in time. It has to be solved in order to study the resistive transition along asuperconducting structure as a function of time. The propagation of normal conducting zonealong the conductor is known as the longitudinal quench propagation. The modelling of thisprocess requires the time transient solution of the one dimensional heat balance equation for
5
1D one dimension, Joule heating, conduction, direct heat transfer into helium bath,thermal impedance
1DI as 1D with modelling of the insulation layer with linear approximation1DH as 1DI with forced quenching by heaters through an insulation layer1DM as 1D with discretised modelling of the insulation layer1DHM as 1DM with forced quenching by heaters3D as 1DH for three dimensions for main magnets (with Rutherford type cables)3DC as 1DH for three dimensions for corrector magnets (fully impregnated coils,
helium only at the end elements)ED as 1DH with modelling of induced eddy currentsEDM as 1DHM with modelling of induced eddy currentsP as 1DM with two different materials of the insulation layer along the conductor
Table 2: Names for the quench processes.
the conductor including the transverse influence of the cooling mechanism. If the transversepropagation in between adjacent turns inside a magnet coil has to be simulated, the heat bal-ance equation becomes three dimensional (see Section 2.5). The one dimensional heat balanceequation along the conductor has to be solved in order to determine the stability of a super-conductor as a function of copper stabilisation, cooling conditions, applied magnetic field andtransport current.
Under the assumption of constant material properties and the temperature being either thecritical temperature Tc or the bath temperature Tb, this equation can be solved analytically [1,2, 3]. These analytic models offer a fast first approach but are strongly limited in precisionand cannot describe complex quenches. The material properties often range over several ordersof magnitude in the temperature range of 1.9–300K. In order to ensure a reliably-workingprotection system for the superconducting bus-bars and series of magnets to be installed in theLHC, it is essential to have a more precise picture of the quench process, i.e. to be able tounderstand the heat transfer to liquid helium rather than using adiabatic calculations only.
The program package iterates the material properties linearly using a lookup table (temper-ature steps of 0.1K). The values for the table are taken from different literature sources [4, 5, 6].The heat balance equation is solved time transient. The temperature is repeatedly calculatedfor all elements at discrete times (the time step can vary as a function of actual values of thematerial properties). The numerical realisation of the temperature evaluation is presented inSection 3.
2.1 One dimensional heat balance equation
The one dimensional heat balance equation to be solved is
d
dx
(k(T (x, t))
dT (x, t)
dx
)A(x)− hHe(T (x, t), t)P (x) + q +G(T (x, t), t)
= c(T (x, t))A(x)dT (x, t)
dt. (2.1)
Its unit is power per unit length [W/m] with x being the longitudinal coordinate, t being thetime, A(x) being the conductor cross-section as a function of its longitudinal position. Eq. 2.1
6
contains several terms to be explained as follows:
2.1.1 Thermal impedance
The term on the right side of Eq. 2.1 is called the thermal impedance that is the heat storedper volume and time. It is given by the heat capacitance c as a function of local temperaturemultiplied by the time derivative of the temperature.
2.1.2 Thermal conductivity
The first term on the left side of Eq. 2.1 describes the heat conduction along the conductor withconductivity k as a function of local temperature.
2.1.3 Internal heat generation
The internal heat generation G(T (x, t)) is given by
G(T (x, t), t) =
0 if T (x, t) ≤ Tcs(t)
ρcu(T (x, t))I(t)2
A(x)T (x,t)−Tcs(t)Tc(t)−Tcs(t)
if Tcs(t) < T (x, t) ≤ Tc(t)
ρcu(T (x, t))I(t)2
A(x)if T (x, t) > Tc(t)
(2.2)
The variables used in Eq. 2.2 stand for the conductor cross-section A; the specific resistivity ρas a function of the temperature at a certain place and time; and the current I as a function oftime. The heat generation is zero below the current sharing temperature when the conductoris still superconducting. A linear model of the current sharing is assumed so that the heatgeneration increases linearly beyond the current sharing temperature Tcs until Tc is reachedand the whole current is taken by the copper part of the conductor.
This model assumes a linear relation of the current transition from superconductor intocopper as a function of temperature. The values of the current sharing and critical temperatureand the critical current are calculated as a function of present temperature, magnetic field andcurrent density [7].
The critical temperature as a function of applied magnetic field is given by [7]
Tc(t) = 9.2 ·(1− | B(t) |
14.5
)0.59
. (2.3)
Despite of Eq. 2.3 the critical temperature can be set as an entry parameter.The magnetic field is assumed to change with current as follows [8]
| B(t) |= c1 + c2 · I(t). (2.4)
In standard mode the applied magnetic field is constant along the conductor. The user canalso set the magnetic field to be independent from the excitation current.
The current sharing temperature is calculated as a function of current and temperature [7]
Tcs(t) = Tb + (Tc(t)− Tb) ·(1− I(t)
Ic(t)
). (2.5)
The initial critical current is given as an entry parameter. The time behaviour follows
Ic(t) =
I0c ·(1− T (t)−Tb
Tc(t)−Tb
)if T (t)−Tb
Tc(t)−Tb≤ 1
0 else(2.6)
7
2.1.4 Current decay
The current decays according to the chosen program or decay option. It starts to decay whenthe resistive voltage Ures exceeds a threshold Uthres.
I(t) =
I0 if Ures(t) < UthresI0 exp
(− t−tdet
τ
)if Ures(t) ≥ Uthres∧ mode=exp
I0 exp(− (t−tdet)
2
τ2
)if Ures(t) ≥ Uthres∧ mode=gauss
I0 − k · (t− tdet) if Ures(t) ≥ Uthres ∧ I(t) > 0∧ mode=lin0 else
(2.7)
The detection and quench validation time is tdet. If the quench originates in the first element(standard mode) the program evaluates the propagation in one direction only, thus savingcomputation time. In this case the resistive voltage is doubled to take into account that thequench propagates in two directions along the conductor in reality.
The current decay inside magnets can be modelled with a Gaussian function [9]. A pureexponential decay of the current is appropriate for the quench simulation in a busbar. Thecurrent decay can also be evaluated using the resistance growth of the normal conducting zoneplus an optional external resistance
I(t) =
I0 if Ures(t) < Uthres
I0 · exp(−
t∫tdet
R(t′)+Rex
Ldt′ · (t− tdet)
)if Ures(t) ≥ Uthres∧ mode=R(t)
(2.8)
However, a more precise model has been developed that includes a bypass diode or a par-allel resistor. This model requires detailed knowledge of the time-scale in which the currentcommutates to the diode and is not described here.
2.1.5 Cooling
The cooling is given by P and h. P is called the wetted perimeter or effective wetted perimeter.For a circular conductor, it is P = f · 2πr, r being the radius and f being the effective fraction.P corresponds to the surface per unit length of the conductor that is surrounded by liquidhelium. This parameter is unknown. It can either be estimated from experience or be modifiedsuch that the simulation results reproduce measured results, i.e. quench velocity and hot spottemperature versus quench load. The heat transfer models available in the program assume aninfinite helium bath with constant temperature. The means that a possible quench accelerationdue to convection of forced or free-flowing helium is excluded.
The cooling conditions can be approximated using different models of heat transfer to heliumaccording to:
hHe1 (T (x, t), t) =
0 if T (x, t) ≤ Tb ∨ t− t(T > Tf.boil) > tf.boila1 · (T (x, t)4 − T 4
b ) if T (x, t) ≤ Tf.boila2 · (T (x, t)− Tb) if T (x, t) > Tf.boil ∧ t − t(T > Tf.boil) < tf.boil
(2.9)
This model of heat transfer is based on a fourth order Kapitza resistance approach with constanta1 if the local temperature to the power of four below the film boiling temperature and on afirst order Kapitza resistance approach if the local temperature is beyond the film boilingtemperature of liquid helium and the helium is not yet completely evaporated.
8
hHe2 (T (x, t), t) =
0 if T (x, t) ≤ Tb ∨ t− t(T > Tf.boil) > tf.boila1 · (T (x, t)4 − T 4
b ) if T (x, t) ≤ Tf.boila3 if T (x, t) > Tf.boil ∧ t − t(T > Tf.boil) < tf.boil
(2.10)
This heat transfer model is slightly modified such that a constant is used above film boilingtemperature instead of a constant times the temperature difference with respect to the bath.
hHe3 (T (x, t), t) =
0 if T (x, t) ≤ Tba1 · (T (x, t)n − T n
b ) if T (x, t) ≤ Tn.boila2 if Tf.boil > T (x, t) > Tn.boila3 · (T (x, t)m − Tm
b ) if T (x, t) > Tf.boil ∧ t− t(T > Tn.boil) < tf.boila4 if t − t(T > Tf.boil) > tf.boil
(2.11)
The modified model allows for fitting the heat transfer model according to the surface treatmentand differs between nucleate and film boiling conditions [10, 11]. The heat transfer is keptconstant during between the starting temperature of nucleate boiling and film boiling.
The simplest model for heat transfer to liquid helium is
hHe4 (T (x, t), t) = a · (T (x, t)− Tb) (2.12)
The constants for the different models of heat transfer into helium strongly depend on thesurface treatment. The used values can be found in [12, 13]. The model 3 for the heat transferrequires 9 parameters whereas the model 4 is set up with 2 parameters.
2.1.6 External heat pulse
Another term in Eq. 2.1 is the external heat pulse which provokes a quench when the minimumquench energy is exceeded. The term q stands for the initial energy which causes the quenchingprocess. When the program is executed the temperature profile is initially calculated as aGaussian like temperature profile obtained after a perturbation (except for a forced quench byheaters - see below). The difference of the enthalpy between the calculated temperature profileand the enthalpy of the helium bath temperature determines the initially deposited energy.This allows studying the stability and the minimum quench energy by changing the initialtemperature profile.
2.2 Insulation model
The low heat conductivity of the insulation material limits the amount of heat that is trans-ported into helium. An adequate modelling of the heat transfer through an insulation layer intohelium requires a two-dimensional model in case the insulation layer thickness is comparablewith that of the dimension of the conductor. This approach would require a significant amountof computational time. For that reason a different approach is followed. The heat balanceequation 2.1 can be generalised as
d
dx(A(x)jx) +G(x, t) + Ft(T,
dT
dt) = c(T )(A(x)
dT
dt(2.13)
9
Ft is the transverse heat flux from the surface of the superconducting cable, which is a functionof the temperature and its time derivative because of the thermal impedance of the insulationlayer. As the longitudinal heat propagation inside the insulation layer is negligible with respectto that in the conductor, the transverse heat flux Ft can be evaluated separately, which allowsthe reduction of the two-dimensional to two one-dimensional problems.
y or z
l
T(x,t)
stru
ctur
e
iso
supe
rcon
dutin
g
Cu,
SC
(,H
e)
Insu
latio
n
Hel
ium
bat
h
isoT (x,t)
T
Figure 2: Schematic temperature profile including heat transfer through insulation.
2.2.1 Steady state
In order to study the steady state solution, the function Ft(T, 0) has to be evaluated. Thesteady state means that the temperature inside the insulation remains constant with time andchanges only as a function of radial position. The heat balance equation is
∮P
"jt(r, ϕ) · d"l = const (2.14)
When the superconducting cable is radial symmetrical the heat balance equation reduces to
r · kiso · dTdr
= R1 · hHe(Tiso) (2.15)
Eq. 2.15 can be solved for particular points with the following numerical integral
∫ Tiso
Tcu
kiso(T )dT = R1 · hHe(Tiso)∫ R1
R0
1
rdr (2.16)
This solution allows investigating stability temperature points and a first estimation of Tmax.As the quench propagation velocity is related to the propagating front of the temperatureprofile, in which the temperature varies with time in the order of thousands K/s, the steadystate solution is insufficient to model the entire quench process.
10
2.2.2 Constant gradient approximation
A linear temperature profile inside the insulation material is assumed, which is an adequateapproach if the thickness of the insulation is small compared to the dimensioning of the super-conducting cable.
d
dx
(k(T (x, t))
dT (x, t)
dx
)A(x) + kiso
(T (x, t) + Tiso(x, t)
2
)P (x)
Tiso(x, t)− T (x, t)
liso+
q +G(T (x, t), t) = c(T (x, t))A(x)dT (x, t)
dt. (2.17)
The temperature Tiso on the surface of the insulation that is exposed to the helium bath isevaluated as follows:
kiso
(T (x, t) + Tiso(x, t)
2
)P (x)
T (x, t)− Tiso(x, t)
liso− hHe(Tiso(x, t), t)P (x)
= ciso(Tiso(x, t))A(x)dTiso(x, t)
dt. (2.18)
2.2.3 Matrix evaluation of the heat flux
A precise evaluation for the transverse heat transfer requires to disretise the radial heat balanceequation time transient. When neglecting the longitudinal heat propagation in the insulationlayer with respect to that of the superconducting structure, this equation is
−"∇ ·"j(r, ϕ) = ciso · dTisodt
(2.19)
Assuming radial symmetry Eq. 2.19 changes to
−1
r
d
dr(r · kiso · dT
dr) = ciso · dTiso
dt(2.20)
The temperature evaluation starts with a homogeneous temperature profile at Tb inside theinsulation layer. The temperature of the superconducting wire is increased with a constantrate dT
dtand the heat flux is evaluated for each temperature step. This procedure is repeated
for various rates of dTdt
and the results are stored in a two-dimensional matrix file as functionof T and dT
dt. The calculated values are linearly interpolated to get Ft(T, dT
dt) which is required
for the simulation of the quench propagation in Eq. 2.13.
2.3 Quench heaters
Quench heaters are used for magnet coils in case the natural quench propagation is not fastenough to avoid overheating and excessive voltages. They consist of stainless steel strips in-stalled along the magnet between the coil and the collars and are fired by a capacitor bankdischarge after quench detection. The current pulse heats the strips and the heat is transferredthrough the insulation layer into the coil, which provokes a quench. In order to reduce thenumber of required capacitor banks, heater strips can be plated partially with copper to reducetheir resistance. This means that the heaters provoke a quench in the cable only below a nonplated part using the natural quench propagation in between the various quench origins transfer
11
0.025 mm12 cm
width
15 mm
heater strip length 15 m
copper plating 40 cm thickness
Figure 3: Heater strip design.
the entire conductor into the normal conducting state [9]. The design of a quench heater stripis sketched in Fig. 3.
In order to study the quench process after firing quench heaters, an additional term has tobe added to Eq. 2.1 or 2.13.
The temperature in the heater strip is determined solving the one dimensional heat balanceequation. Heat conduction along the heater strip is neglected as the heat generation is the sameat every part of the heated zone. The heat transfer between heater strip and liquid helium isalso not taken into account. The resistance and temperature increase in the copper plated partsof the heater strip is neglected as the heat generation inside the superconducting cable once aquench starts is bigger than the heat generation in the copper plated part by several orders ofmagnitude (typical initial heater currents are about 50 to 60A [14]).
ρh(Th(t)) · I2h(t)
Ah−khiso
(T (x, t) + Th(t)
2
)wh
liso·fh · (Th(t)−T (x, t)) = ch(Th) ·Ah · dTh(t)
dt. (2.21)
In addition to the terms introduced in Eq. 2.1, Eq. 2.21 contains the heat conduction through theinsulation into the superconducting cable. In order to calculate the heat generation, a constantcapacitance C of the heater power supply is assumed for evaluating the heater current. Theinitial current is I0
h = U0h/Rh(T = Tb), with U0
h being the loading voltage and Rh(T = Tb) beingthe heater resistance at bath temperature.
The temperature dependent heater strip resistance is evaluated with the new temperatureafter each timestep as
τh(T (t)) = Rh(T (t))C. (2.22)
The heater resistance is calculated as
Rh(T (t)) =ρh(T (t))lh
Ah
lFehlFeh + lcuh
, (2.23)
with lh being the length, lFeh being the length of the non-plated parts, and lcuh being the length
of the plated parts of the heater strip. The heaters are fired when a quench is detected at timetdet. The detection requires that the resistive voltage exceeds a threshold for a certain timeinterval. The detection time can be set to zero if the quench should be provoked by heatersonly. This function is used to study experimental results of heater delay tests.
The time dependence of the heater current can be expressed as
Ih(t) =
0 if t < tdetI0h · exp(− t
Rh(T (t))Ch) if t ≥ tdet
(2.24)
12
The heat transfer through the insulation into the superconducting cable is given by thethermal conductivity of the insulation material khiso(T (t)), the heater strip width wh whichcorresponds to the surface per unit length and the insulation length liso. The factor fh hasbeen introduced to adjust the effective fraction of heat which is conducted from the heaterstrip into the superconducting cable. The simplest model sets fh equal to 0.5 (half of thegenerated heat is conducted into the cable the other half towards the collars). If the insulationthickness differs between the coil and the heater strip lhiso with respect to the thickness betweenthe collars and the heater strip lhcoll, it is more convenient to set
fh =lhiso + lhcoll
lhiso(2.25)
The heat capacitance for the insulation is not considered, as its volume per unit length is byfar smaller than that of the superconducting cable.
Alternatively, the temperature evaluation of the heater strip can include the heat transferfrom the surface of the heater strip into the magnet collars.
ρh(Th(t)) · I2h(t)
Ah− khiso
(T (x, t) + Th(t)
2
)wh
lhiso· fh · (Th(t)− T (x, t))
−khiso
(T (x, t) + Tb
2
)wh
liron· (Th(t)− Tb) = ch(Th) · Ah · dTh(t)
dt(2.26)
The new term takes into account the heat transfer through the insulation into the iron magneticyoke. For this simple model the length liron has to be chosen such that the temperaturedifference Th(t) − Tb can be used as an approximate approach. The heat balance equationEq. 2.1 including the impact of quench heaters heaters becomes
d
dx
(k(T (x, t))
dT (x, t)
dx
)A(x)− hHe(T (x, t), t)P (x) + q +G(T (x, t), t) +
g(x) · khiso(T (x, t) + Th(t)
2
)lwidthlhiso
(Tht− T (x, y, t)) = c(T (x, t))A(x)dT (x, t)
dt(2.27)
with lwidth being the width of the cable towards the heater strip. The function g(x) determineswhether the cable is below a heated part of the quench heater strip or not (see also Section 3.3).Eq. 2.27 and Eq. 2.26 have to be solved alternately.
Eq. 2.27 changes to Eq. 2.28 when the direct heat transfer into helium is exchanged by heattransfer through insulation
d
dx
(k(T (x, t))
dT (x, t)
dx
)A(x) + g(x)khiso
(T (x, t) + Th(t)
2
)lwidthlhiso
(Th(t)− T (x, t)) +
+q +G(T (x, t), t) + kiso
(T (x, t) + Tiso(x, t)
2
)P (x)
Tiso(x, t)− T (x, t)
liso+
= c(T (x, t))A(x)dT (x, t)
dt. (2.28)
13
2.4 Two-dimensional heat balance equation
The two dimensional heat balance equation for the homogeneously-treated superconductingcable includes the heat conduction from the heater through the insulation. It also containsheat conduction inside along the y direction inside the superconducting cable (the broad sideof the Rutherford cable).
k(T (x, y, t))
(d2T (x, y, t)
dx2+
d2T (x, y, t)
dy2
)+
G(T (x, y, t), t)
A(x, y)− hHe(T (x, y, t), t)
P (x, y)
A(x, y)+
g(x, y)khiso
(T (x, y, t) + Th(t)
2
)wt
lhisoA(x, y)(Tht− T (x, y, t)) +
q
A(x, y)=
c(T (x, y, t))dT (x, y, t)
dt, (2.29)
with ht being the height (layer) in y-direction and wt being the average width (turn) in z-direction), corresponding to surface exposed to the heater per unit length. The definitionsof the variables used in Eq. 2.29 are identical with those introduced in Eq. 2.1. The functiong(x, y) expresses that heat conduction between the cable and the heater strip only occurs onthe side of the cable that faces a heated part of the heater strip.
2.5 Three-dimensional approach
In order to study the quench propagation from one quenching turn to the next, a three dimen-sional model has to be used to solve the heat balance equation. The average turn width is forthe discretisation in z.
The turn thickness is averaged (∆z) which corresponds to wt in the two-dimensional model.The heat transfer between two turns is calculated by heat conduction through the insulationbetween two turns, using the thermal conductivity ktiso and the width htiso.
The finite volume element is dV = dx·dy·dz and the finite surface element is A(x, y) = dy·dz.The other terms are the equivalent of those in Eq. 2.29:
k(T (x, y, z, t))
(d2T (x, y, z, t)
dx2+
d2T (x, y, z, t)
dy2
)+
q
A(x, y, z)+
G(T (x, y, z, t), t)
A(x, y, z)+
ktiso(T (x, y, z, t))
ltiso
dT (x, y, z, t)
dz− hHe(T (x, y, z, t), t)
P (x, y)
A(x, y, z)+
g(x, y, z)khiso
(T (x, y, z, t) + Th(t)
2
)wt
lhisoA(x, y, z)(Th(t)− T (x, y, z, t))
= c(T (x, y, z, t))dT (x, y, z, t)
dt. (2.30)
Eq. 2.30 is formulated for the three-dimensional model for a Rutherford cable with discretisationalong the cable and of its broad side. The three-dimensional approach models the quenchpropagation from turn to turn. For a corrector magnet the quench propagation between turnsand layers is of general interest. Since both transverse dimensions of the conductor used towind corrector coils are of similar size the discretisation in y-direction is used to model the
14
quench propagation between layers.
k(T (x, y, z, t))d2T (x, y, z, t)
dx2+
G(T (x, y, z, t), t)
A(x, y, z)+
kliso(T (x, y, z, t))
lliso
dT (x, y, z, t)
dy+
q
A(x, y, z)+
ktiso(T (x, y, z, t))
ltiso
dT (x, y, z, t)
dz− hHe(T (x, y, z, t), t)
P (x, y)
A(x, y, z)+
g(x, y, z)khiso
(T (x, y, z, t) + Th(t)
2
)wt
lhisoA(x, y, z)(Th(t)− T (x, y, z, t))
= c(T (x, y, z, t))dT (x, y, z, t)
dt. (2.31)
2.6 Stability
One of the main questions related to superconducting cables is their stability. Cables can bedesigned such that the superconducting phase cannot disappear as long as the cooling remains.The domain is called cryo-stable. The simplest equation to distinguish between stable andnon-stable superconductors is the Stekly criterion [15]
α =G(T (x, t), t)
hHe(T (x, t), t)P (x). (2.32)
For most applications of superconductors in accelerators it is α >> 1 and therefore cryo-stability is not achievable at nominal operating current. However, at lower currents (beginningof the ramp) the conductor might become cryo-stable.
As the heat generation and cooling conditions are non linear function of time and temper-ature, any stability criterion changes with time during the quench process.
An approach that takes the temperature dependence into account is the equal-area theoremby Maddock et al. [16]. The steady state one dimensional heat balance equation can be rewrittento
d
dxk(T (x))
dT (x)
dx= h(T (x))
P (x)
A(x)−G(x). (2.33)
If k is assumed to be independent of T the normal conducting zone would not expand thefollowing equation is valid
T1∫T0
(h(T (x))− A(x)
P (x)G(T (x))
)dT = 0. (2.34)
The conditions requires that the difference between cooling and heat generation integratedover temperature must be zero. In other words: In a certain temperature interval the heatgeneration might be stronger than cooling. This regime has to be compensated by anothertemperature interval in which the cooling is stronger than the heat generation. If both regimehave the same area (in this sense area has the unit of power times temperature per surface[W·K/m2]) the normal conducting zone neither expands nor collapses. The inaccuracy of thisapproach is that the relation of the heat conductivity with temperature is neglected and timedependences are neglected.
15
Transient effects also play an important role as the heat transfer from the surface of theconductor into the helium bath changes dependent on the state of helium and the pressuredevelopment. Some transient aspects for stability that are based on simple relations betweenmaterial properties and temperatures are treated in Wilson [1]. The models for heat transferused in this model presume either constant pressure (for simulations of quenches in bus-bars)or an increase of pressure (for simulations of quenches in magnets). Effects of turbulent orforced flow helium are neglected.
As the heat generation is much larger than the cooling for cables used in the LHC magnets(not necessarily for busbars), more complicated time dependent stability criteria are not needed.Stability is the basic question to study whether a quench can occur in first place. As soon asthe current starts to bypass the superconductor and flows through the copper, a recovery isalmost excluded and the evaluation of the hot spot temperature and the required protectionare the most important issues.
Two parameters determine whether a normal conducting zone expands or collapses:
• the minimum quench energy (MQE) and
• the minimum propagation zone (MQZ).
Both parameters are strongly related. For the evaluation of MQZ, the impact of cooling isneglected. The MQZ is the mimimum normal conducting zone that is required to avoid thatheat conductance along the cable causes a reduction of the peak temperature in presence ofohmic heat generation and within a recovery of the superconducting state.
lMPZ =
√2kA(Tc − Tb)
G; (2.35)
with G being defined for the composite conductor as in Eq. 2.2.The energy to warm up enough material to the critical temperature for the length of the
MQZ is the MQE.EMQE = (H(Tc)−H(Tb)) · A · lMQZ ; (2.36)
with H being the enthalpy.The required normal conducting length for an expanding quench is reduced if the critical
temperature is exceeded at one point or if an additional heat source such as quench heaters isinvolved.
In reality the required normal conducting length has to be longer than the MPZ to developa natural quench due to the presence of helium cooling. The normal conducting zone requiredto provoke a quench by heaters can be less than the MPZ if the energy transfered from theheater into the superconducting cable is higher than the MQE.
SPQR allows studying the MPZ and the MQE by changing the temperature profile atprogram start (by changing the initial temperature T init and the sigmas xT decay, yT decay,zT decay).
2.7 Eddy currents
An electric field "E is induced in presence of a time dependent magnetic field "B due to Maxwell’slaw
d "B
dt= ∇× "E. (2.37)
16
The electric field is related to a current density in a conductor by Ohm’s law
"J = σ "E
σ =1
ρ(2.38)
This section treats the effect of dynamic losses due to eddy currents and magnetisation lossesthat are induced during the decay of the magnetic field after a quench and that limit the ramprate of magnets. The influence on the field quality is discussed for example by T. Ogitsu [17].
The numerical model is based on the work of Devred, Morgan, Niessen, Ogitsu, Verweij andothers [8, 17, 18, 19]. The results of the simulations are applied to the quench process and theconsequences are discussed.
The eddy currents that occur in superconducting magnets are either coupling currents overcontact resistances inside the superconducting cable or eddy currents in the copper wedges thatact like a secondary inductance. The copper wedges are installed to separate the different coilsof a magnet in order to approximate the current density which is required to obtain the desiredmagnetic field.
Different types of eddy currents occur in the magnet and in the cable:
• Eddy currents in copper spacers, collars and yokesThe changing magnetic field induces eddy currents not only in the cable but also in thecopper spacers in between conductor blocks, the magnet collars and the iron yoke. Thecollars and the iron yoke are made of laminations to reduce the strength of the inducededdy currents. The eddy currents in the copper spacers cause these parts behaving assecondary inductances that extract energy [20]. For the case of the LHC main magnets,the amount of energy extracted into the copper wedges was calculated to be less than afew percent of the stored magnetic energy. This effect is not included in SPQR.
• Interfilament coupling currentsEach strand is made of many NbTi filaments that are embedded in a copper matrix. Thefilaments are twisted over the length Lfp . The trajectory along the strand (in x direction)of two neighbouring filaments in the projection perpendicular to the applied externalmagnetic field are two sinusoid curves. At the crossing point the two filaments have theminimum distances dmin. An applied magnetic field change induces a coupling currentfrom one filament to the other. Due to the symmetric structure the strengths of thisinterfilament coupling current can be calculated independent from x as a function of thefilament diameter df , L
fp and the changing applied magnetic field dB/dt.
• Interstrand coupling currentsThe superconducting cable that is used for the main busbars and magnets for the LHCis a Rutherford type cable consisting of several strands (36 strands in the cable for theouter layer of the dipole magnets and 28 strands in the cable for the inner layer of thedipole magnets). The cable has a width w and a depth d. The strands are twisted withthe cable pitch Lsp. The strands cross in both, the x − y and the x − z plane with thecorresponding contact resistances Ry and Rz . The contact resistances are a result ofvarious reasons, which are not discussed here [8]. SPQR uses averaged values of Rc andRa. The averaged strength of the coupling currents is numerically calculated as a functionof applied magnetic field components parallel B‖ and perpendicular B⊥ to the broad side
17
of the cable, Ry and Rz, w and Lsp, and the number of parallel strands Np and layers ofstrands Ns.
If the magnetic field varies over the width of the cable the interstrand coupling currentscan only be numerically evaluated. Interstrand currents can close a loop that is by farlarger than the length of one cable pitch when the magnetic field varies along the cableor when the value of the contact resistances Rc and Ra are not constant. In this casethe induced eddy currents are known as boundary induced coupling currents that werepredicted and experimentally verified [21, 22].
A quench starts if the interfilament or the interstrand currents together with the trans-port current exceed the critical current of a filament or a strand for the actual tempera-ture, applied magnetic field and its value of the time derivative. The quench condition isI(t) > Ic(T (t), B(t), dB/dt(t)).
A quench can also occur due to the temperature that is increased by the power of in-terfilament and interstrand coupling currents. If the temperature exceeds the critical tem-perature that depends on the actual current density, applied magnetic field and its value ofthe time derivative the superconductivity disappears. This leads to the quench conditionT (t) > Tc(I(t), B(t), dB/dt(t)).
Both effects are known as quench back. In order to calculate the power of the inducedcoupling currents their built-up time that is characteristic for the dimensions of the filaments,the strands and the cable has to be considered. Another effect is due to possible magnetisationlosses: A high enough external magnetic fields can penetrate the filament and cause a mag-netisation that results in a hysteresis and an additional contribution to the loss power duringa changing field strength d/dt
∫M(B)dB.
A quantitative simulation study demonstrated that the quench back in the LHC dipolemagnets is due to the temperature increase caused by the heat generation at the contactresistance [23].
The formalism presented below for the calculation of the eddy currents was summarised byA. Vervweij [8]. The magnetisation loss per unit volume can be calculated according to
M = − 2
3πJcdf
dB/dt
|dB/dt| (2.39)
The induced interfilament coupling currents are given by
Iif(t) =dB
dt
L2pif
dif
4π2ρ(T (t))·(1− exp
(− t
τif
))(2.40)
τif =µ0
2ρ(T (t))
(Lpif
2π
)2
(2.41)
Both effects lead to the combined interfilament losses per unit volume of coupling currents andmagnetisation
P/Vif =1
ρ
(dB
dt
)2 (Lpif
2π
)2
·(1− exp
(− t
τif
))2
+2
3πJcdif
∣∣∣∣∣dBdt∣∣∣∣∣(1− exp
(− t
τif
))(2.42)
18
The interstrand currents have been derived for a Rutherford cable of the LHC type [8] as follows
Is =
(0.0415
LpswNs,l
Rc
dB
dtcos(φ) + 0.25
Lpsh
RaNs,l
dB
dtsin(φ)
)(1− exp
(− t
τis
)); (2.43)
τis = 1.68 · 10−8Lps(N2s,l − 4Ns,l)
Rc(2.44)
The angle φ determines the parallel and rectangular component of the magnetic field withrespect to the cable axis. The interstrand losses per unit volume are
P/Vis =
(0.170
Lpsw2(1− 1/Ns,l)
Ra(cos(φ))2 + 0.125
Lpsh2
Ra(sin(φ))2+
8.49 · 10−3Lpsw2(N2
s,l −Ns,l)
Rc(cos(φ))2
)(dB
dt
)2 (1− exp
(− t
τis
))2
(2.45)
3 Discretisation
The discretisation used in the program is shown in Fig. 4. The current flows in the x-direction,in which the discretisation element is Dx. When modelling corrector magnets discretisationin the y-direction Dy corresponds to the different layers, whereas Dy for main magnets is thediscretisation of the Rutherford cable. The discretisation in z-direction Dz refers to variousturns of the coil. The quench heater strips lie in the x-z plane. The discretisation width in spaceand time has to be carefully adjusted in order to avoid numerical instabilities as oscillationsand diverging temperatures [24, 25]. For that reason the chosen step-width and initial timestephave to be studied for every new simulation. For a fast execution the number of elements shouldbe matched to the required value. In addition the initial timestep and the variables responsiblefor the timestep ∆t adjustment (tcontrol and α) can be increased but compromising precisionand reliability.
3.1 One dimensional FDM model
In the one dimensional approach the cable is discretised in Nelem nodes of Dx length. Thetimestep ∆t is dynamically adjusted by the program depending on the material properties(∆tact). The user sets the initial and minimal timestep with the variable timestep. The actualtime is updated according to tnew = told +∆tact. In discrete space the continuous temperatureT (x) becomes T (ix) meaning T at ix. The same notation is used for P and A as a function ofx. The time is noted as t although it is discrete tn with a changing time interval between twopoints. The algorithm for the evaluation of the temperature development contains two nestedloops
• Program initialisation
• Time loop with dynamically calculated timestep = ∆tact
• Nested loop for temperature evaluation for Nelem elements with length Dx
19
y
discretisation in x,ytaking an averagevalue for z
turns
x
z
turn
turn
turn
Dy
Dx
dV=DxDyDz
Figure 4: Discretisation of the superconducting structure.
20
The quench energy at program start is calculated from the enthalpy difference between thecomputed temperature profile and helium bath temperature.
q = Dx · (∑ix
A(ix)(H(T (ix))−H(Tb)). (3.1)
After initialisation the temperature is evaluated with the discretised one dimensional heatbalance equation (see also Eq. 2.1). The new temperature T new(ix, t
new) is given by
T new(ix, tnew) = T (ix, t) +
∆tactc(T (ix, t))
·(k(T (ix, t))
T (ix+1, t) + T (ix−1, t)− 2T (ix, t)
Dx2
− hHe(T (ix, t))P (ix)
A(ix)+
G(T (ix, t), t)
A(ix)
). (3.2)
Eq. 3.2 is valid only for constant cross-section A along the conductor, otherwise the derivativedA/dx has to be included.
The terms of Eq. 2.1 are expressed in Eq. 3.2 as follows:
• The time derivative of the temperature becomes (T new − T )/∆tact.
• The first time derivative of the temperature in x is (T (ix+1 −T (ix−1)/(2Dx). The secondtime derivative of the temperature in x is given by (T (ix+1, t)+T (ix−1, t)−2T (ix, t))/Dx.
• The heat generation is given by G(T (ix, t), t).
• The values of the the material properties ρ, c, k and the heat transfer to liquid heliumhHe are linearly interpolated from a lookup table that is calculated at the start of thesimulation (according to input parameters like magnetic field (b − field), RRR, rSC/CU ,etc). The common material parameters are calculated as a function of temperature ac-cording to proportions of the materials Asc/Atot, Acu/Atot, and AHe/Atot; and are storedin a lookup table using a temperature step width of 0.1K (see Section 4).
The temperature evaluation procedure ends when an abort criterion is fulfilled (usually whenthe actual time exceeds the end-time fintime). The abort criteria can also be a very high tem-perature (greater than 1000K) in the cable; a temperature or voltage threshold; zero current;length of normal conducting zone; or a collapsing quench.
The timestep is calculated according to
∆tnewact =
α · c(min(T (ix, t))) Dx2
k(min(T (ix,t)))if ∆tact < ∆tnewact < ∆tcontrol · α
∆t0act if ∆tnewact ≤ ∆t0act∆tcontrol · α if ∆tnewact < ∆tcontrol · α
(3.3)
In order to reduce the required computation time, the number of terms in the heat balanceequation can be reduced when no heat generation is present. If the current decayed to zero,the temperature is evaluated with a equation that does not include the ohmic heat generationterm.
21
3.2 Heat transfer trough insulation
Eq. 3.2 changes to a system of two differential equation when the cooling model takes heattransfer through the insulation of the material into account
T new(ix, tnew) = T (ix, t) +
∆tactc(T (ix, t))
·(k(T (ix, t))
T (ix+1, t) + T (ix−1, t)− 2T (ix, t)
Dx2+
kiso
(T (ix, t) + Tiso(ix, t)
2
)P (ix)
A(ix)
Tiso(ix, t)− T (ix, t)
liso+
G(T (ix, t), t)
A(ix)
). (3.4)
Eq. 3.4 contains the heat conductance through the insulation material, which requires the eval-uation of the temperature on the surface between insulation and helium bath Tiso(ix, t) asfollows
T newiso (ix, t
new) = Tiso(ix, t) +∆tact
ciso(Tiso(ix, t))· (3.5)
(kiso
(T (ix, t) + Tiso(ix, t)
2
)P (ix)
A(ix)
T (ix, t)− Tiso(ix, t)
liso− hHe(T (ix, t))
P (ix)
A(ix)
).
Due to the alternating evaluation of temperature in the conductor a numerical problem canoccur. The temperature dependent heat transfer coefficient lead to more heat transferred intohelium than from the conductor through the insulation layer. This would cause the temperaturein the insulation layer being below Tb. This unphysical result is prohibited by resetting T new
iso =Tb.
It is possible that the cooling is strong enough to transfer more heat into the helium baththan the amount of heat that is transferred through the insulation. In this case Eq. 3.5 wouldevaluate a temperature T new
iso being below Tb. If a temperature T newiso or T new is calculated to be
below Tb it is set to be equal to Tb in order to avoid the instability.The temperature evaluation process is similar to that described in Section 3.1.
• Program initialisation
• Time loop with dynamically calculated timestep = ∆tact
• Inside two nested loops:
– Temperature T (ix, t) evaluation for Nelem elements with length Dx
– Temperature Tiso(ix, t) evaluation for Nelem elements with length Dx
In standard mode (AHe=0; neglecting the heat transfer through insulation), the wettedperimeter is the parameter to calibrate the simulation model to fit measured values. Whena helium fraction inside the superconducting structure is specified and the insulation is notneglected, the wetted perimeter is not a fit parameter anymore. The only parameter that isnot precisely known is the amount of helium inside the superconducting structure and theheat transfer into helium on the surface of the insulation material. The effective amount ofhelium can be fit to reproduce the measured quench propagation velocity whereas the heattransfer from the insulation into the helium bath plays only a role in the longterm temperatureevaluation.
22
Dy
Cu
Cu
Cu
Cu
Cu
Dx
heater strip
Dz
Figure 5: Discretisation of the coil geometry including quench heater strips.
3.3 Quench heaters
In order to simulate quenches provoked by quench heaters, the heat balance equation for thequench heater has to be approximated alternately with the one for the conductor.
T newh (tnew) = Th(ix, t) +
∆tactch(Th(ix, t))
·(ρh(Th(ix, t))
Ih(t)2
Ah
−fh · khiso(Th(ix, t) + T (ix, t)
2
)Th(ix, t)− T (ix, t)
lhisoth
). (3.6)
The heater cross-section Ah is wh · th (heater strip dimensions are width wh in z direction andthickness th in y direction). The time behaviour of the current flowing through the heater isdescribed in Eq. 2.24.
The heat conductance inside the heater strip is neglected. Heat transfer from the heaterstrip into the superconducting structure occurs only at the heated parts of the strip. For thatreason it is sufficient to calculate a single temperature of the heater strip that changes withtime, which reduces the number of elements to be evaluated per timestep and accelerates thesimulation run. This approximation simplifies Eq. 3.6 to
T newh (tnew) = Th(t) +
∆tactch(Th(t))
·(ρh(Th(t))
Ih(t)2
Ah
−fh · khiso(Th(t) + Tu.h.(t)
2
)Th(t)− T (t)
lhisoth
). (3.7)
23
The average temperature in the cable under the heater strip Tu.h. is given by
Tu.h.(t) =1
Ncov
Nelem∑ix=i0
T (ix, t) · cix (3.8)
Ncov =Nelem∑ix=i0
cix
cix =
1 if element ix is under a heated part of the heater strip0 if element ix is under a heated part of the heater strip
The integer cix determines whether an element ix is located under a heated part of the quenchheater strip or not. The first i0 elements are not used for the evaluation of Tu.h.(t) in order toavoid an error due to the expanding normal conducting zone. Eq. 3.7 changes to Eq. 3.9 whenheat transfer into the magnetic collars is included.
T newh (tnew) = Th(t) +
∆tactch(Th(t))
·(ρh(Th(t))
Ih(t)2
Ahkhiso
(Th(t) + Tu.h.(t)
2
)Th(t)− T (t)
lhisoth
−khiso
(Th(t) + Tb
2
)Th(t)− Tb
lironth
). (3.9)
The temperature in the the superconducting cable including heat transfer from heaters isevaluated as follows
T new(ix, tnew) = T (ix, t) +
∆tactc(T (ix, t))
·(G(T (ix, t), t)
A(ix)− hHe(T (ix, t))
P (ix)
A(ix)
+
(k(T (ix, t))
T (ix+1, t) + T (ix−1, t)− 2T (ix, t)
Dx2
)+
g(ix)khiso
(T (ix, t) + Th(t)
2
)Th(t)− T (ix, t)
lhisoDy
). (3.10)
Eq. 3.10 changes to Eq. 3.11 if the cooling model includes heat transfer through the insulation
T new(ix, tnew) = T (ix, t) +
∆tactc(T (ix, t))
·(G(T (ix, t), t)
A(ix)
+kiso
(T (ix, t) + Tiso(ix, t)
2
)Tiso(ix, t)− T (ix, t)
lisoDy
+
(k(T (ix, t))
T (ix+1, t) + T (ix−1, t)− 2T (ix, t)
Dx2
)+
g(ix)khiso
(T (ix, t) + Th(t)
2
)Th(t)− T (ix, t)
lhisoDy
). (3.11)
The routine for temperature evaluation of the superconducting structure including heattransfer from quench heater strips is summarised as follows:
• Program initialisation
• Time loop with dynamically calculated timestep = ∆tact
24
• Nested loops:
– Temperature evaluation of Th(t) or Th(ix, t) for Nelem elements
– Temperature evaluation of T (ix, t) for Nelem elements using
∗ Direct heat transfer into helium or
∗ Heat transfer through the insulation material including the temperature evalu-ation of Tiso(ix, t) for Nelem elements
The possibility of studying the effect of quench heaters requires to set the logical variableheater=1.
3.4 Two dimensional FDM model
The temperature Th(ix, t) in the heater at ix is evaluated including the heat transfer to the firstelement in the superconducting cable iy = 0 with temperature T (ix, 0, t) (see Eq. 2.23)
T newh (ix, t
new) = Th(ix, t) +∆tact
ch(Th(ix, t))·(ρh(Th(ix, t))
Ih(t)2
Ah
−fh · khiso(Th(ix, t) + T (ix, 0, t)
2
)Th(ix, t)− T (ix, 0, t)
lhisoth
). (3.12)
The discretised two dimensional heat balance equation including heat transfer from quenchheaters is
T new(ix, iy, tnew) = T (ix, iy, t) +
∆tactc(T (ix, iy, t))
·(G(T (ix, iy, t), t)
A(ix)− hHe(T (ix, iy, t))
P (ix)
A(ix)
+
(k(T (ix, iy, t))
T (ix+1, iy, t) + T (ix−1, iy, t)− 2T (ix, iy, t)
Dx2+
T (ix, iy+1, t) + T (ix, iy−1, t)− 2T (ix, iy, t)
Dy2
)+
g(ix, iy)khiso
(Th(ix, t) + T (ix, 0, t)
2
)Th(ix, t)− T (ix, 0, t)
lhisoDy
). (3.13)
Eq. 3.13 includes two terms for the heat conduction inside the superconducting cable along xand y (setup for a Rutherford cable). The other terms are identical to those of Eq. 3.2. Theg(ix, iy) is the discrete version of g(x, y) defined in Section 2.4.
3.5 Three dimensional FDM model
The complete solution of the heat balance equation requires a three dimensional model in spacein order to be able to simulation the quench process in a coil. The model is similar to the twodimensional approach. It includes an additional term for the heat transfer between two adjacentturns (via the insulation thickness ltiso with heat conductivity ktiso). The heat conduction in zdirection inside the superconducting cable is neglected because the width of a turn is an orderof magnitude less than its height (typically 1–2mm with respect to about 15mm for the LHC
25
dipole magnet cables). The discrete-sized three dimensional heat balance equation for the mainmagnets is
T new(ix, iy, iz, tnew) = T (ix, iy, iz, t) +
∆tactc(T (ix, iy, iz, t))
·(G(T (ix, iy, iz, t), t)
A(ix)+
(k(T (ix, iy, iz, t))
T (ix+1, iy, iz, t) + T (ix−1, iy, iz, t)− 2T (ix, iy, iz, t)
Dx2
+T (ix, iy+1, iz, t) + T (ix, iy−1, iz, t)− 2T (ix, iy, iz, t)
Dy2
)+
ktiso(T (ix, iy, iz, t))T (ix, iy, iz+1, t) + T (ix, iy, iz−1, t)− 2T (ix, iy, iz, t)
ltisoDz−
g(ix, iy, iz)khiso
(T (ix, iy, iz, t) + Th(ix, t)
2
)Th(ix, t)− T (ix, iy, iz, t)
lhisoDy+
hHe(T (ix, iy, iz, t))P (ix)
A(ix)
). (3.14)
In Eq. 3.14 it is A(ix) ≡ A = Dy · Dz. The other terms are equivalent to those introduced forthe one and two-dimensional approach. The cooling model that takes heat transfer throughinsulation into account can also be chosen. The function g(ix, iy, iz) is the three dimensionalversion of g(ix, iy) in the two dimensional model. If the effect of quench heaters is studied, theheat balance equation of the heater strip Eq. 3.12 has to be solved in addition to Eq. 3.14.
The three dimensional model for the corrector magnets takes into account that heat trans-fer into the helium bath takes only place at the boundary elements since the coils are fullyimpregnated. The material for the insulation layer can either be a polyimide or epoxy. Theheat balance equation becomes
djxdx
+j−y − j+
y
dy+
j−z − j+z
dz+
ρ · I2
A2=
du
dt
j−y =K(Ti−1)−K(Ti)
dy
j+y =
K(Ti)−K(Ti+1)
dy
j−z =K(Ti−1)−K(Ti)
dz
j+z =
K(Ti)−K(Ti+1)
dz
K(T ) =∫
kins(T′) dT ′ (3.15)
j− is the inner flux and j+ the output flux in a turn. The evaluation is made using a steadystate approximation:
j = k(T ) · dTdx
= constx∫ x2
x1
jdx =∫ x2
x1
k(T (x)) · dTdx
dx
26
For elements at the boundary of the conductor, the heat transfer into helium through theinsulation layer is evaluated using the matrix model (see Section 3.6).
Figure 6: Three dimensional model for corrector magnets.
3.6 Matrix model
As pointed out in Section 2.2.3, the correct modelling of the heat flux through the insula-tion layer requires a radial discretisation. This discretisation is needed if the insulation layerthickness is of the same order of magnitude as the diameter of the conductor. Assuming ra-dial symmetry, the discretisation of the insulation layer adds another dimension to evaluate thetemperature (N longitudinal elements for the conductor and M radial insulation layer elementsper longitudinal element of the conductor). In order to reduce the number of heat balance equa-tions to be solved every timestep, the heat flux as a function of radial position and the timederivative of the temperature dT/dt is evaluated before the quench computation for a givenwidth and material of the insulation layer. The first radial element is heated with different ratesof dT/dt. The heat flux through the radial elements is evaluated with the boundary conditionof the heat transfer into helium at the last radial element. The results are stored in a matrixF (T, dT/dt) in the file matrix.dat which is read at the beginning of the quench computationand linearly interpolated to find the actual heat flux out of the conductor element. The matrixfile is computed when the logical input parameter make matrix=1 is given. The use of thematrix model for the tranverse cooling during the quench computation is switched on with thelogical input parameter use matrix=1. Both variables cannot be used at the same time.
TNewi = Ti +
∆t
Cins ·∆2·Ti+1 − Ti−1
4(2kins(Tii+ i0
+
kins(Ti)− kins(Ti−1)) + kins(Ti)(Ti+1 + Ti−1 − 2Ti)i0 =
R(0)
∆r(3.16)
27
3.7 Plug and λplate calculations
In order to study the effect of quenching running into or starting in an interconnect (plugs andλplates) which are insulated usually with epoxy, the heat transfer through different materialsof the insulation layer along the conductor has to be considered. Plugs are used to avoid heliumflow along the conductor. λplates separate a superfluid from a normal fluid helium bath. Forthat reason two matrices are calculated first (polyimide as standard insulation material andepoxy for the insulation of a plug or λplate). As explained in Section 3.6 a matrix is evaluatedfor each insulation material and its layer thickness.
The two matrix files are read in at the start of the quench computation. The logical variableplug=1 determines to start the evaluation of a quench occurring in a plug or in a λplate. Inorder to provide the SPQR with the required matrix files for the two different insulation layers,the program has to be executed twice beforehand to compute the matrix files (using the inputvariable kindiso 1.0 for epoxy; 0.0 for polyimide).
The matrix file matrix plug.dat is used for evaluating the heat transfer through the insu-lation layer in the plug elements for the first Nplug elements (Nplug is the number of plugelements). The file matrix.dat is used for evaluating the heat transfer through the insulationlayer in the other Nelem-Nplug elements.
3.8 Eddy currents
Following the description of induced eddy currents in Section 2.7, the numerical realizationrequires to evaluate the external magnetic field as a function of current.
B(I(t)) = a + b · I(t); dB
dt=
B(ti−1)−B(ti)
timestep(3.17)
The resistivity is evaluated as ρ ≡ ρ(B(I(t)), RRR, T (t)) for each element and timestep.The eddy current effects are calculated when the logical variable eddy is set 1. The geometry
parameters are specified at program start e.q., the filament diameter d fil, filament and strandtwist pitch Lp fil and Lp strand, the number of cable layers and strands N strand, the angle inbetween the cable axis (width) and the magnetic field, and the constants a and b for evaluatingthe field as function of current.
Functions are called that compute the strength of the induced eddy currents and their timeconstants. The contribution of the dynamic losses is added to the heat generation to evaluate thetemperature increase in a generalised functionG(B(x, I(t)), I(t), T (x, t), ρ(B(x, I(t)), RRR, T (x, t))).The modelling of interstrand losses requires the input parameter rutherford=1.
For superconducting cables of Rutherford type, the logical variable rutherford must be setequals one.
4 Temperature dependence of the material parameters
The lookup table has been produced using the entry parameters of various reported measure-ments [4, 26, 12, 27, 6, 5]. The material properties for the homogeneous cable are given by
c =rcu/sc
1 + rcu/scccu +
rcu/sc1 + rcu/sc
csc (4.1)
28
k =rcu/sc
1 + rcu/sckcu +
rcu/sc1 + rcu/sc
ksc (4.2)
ρ =1 + rcu/sc
rcu/scρcu (4.3)
The resistivity of copper as a function of the applied magnetic field, temperature and RRRvalue is calculated as follows
ρcu(RRR, T,B) = (1 + r)ρcu(RRR, T,B = 0)
ρcu(RRR, T,B = 0) = ρ0 + ρi + ρi0
ρ0 =15.53 · 10−9
RRR
ρi =1.171 · 10−17 · T 4.49
1 + 4.498 · 10−7 · T 3.35 exp(−(50/T )6.428)ρi0 = 0.4531
ρ0ρiρ0 + ρi
log r = −2.662 + 0.3168 log s+ 0.6229(log s)2 − 0.1839(log s)3 + 0.01827(log s)4
s =15.53 · 10−9 · B
ρcu(RRR, T,B = 0)(4.4)
The copper thermal conductivity is computed according to the Wiedemann-Franz law, as-suming that the Lorentz constant L is independent of the magnetic field
kcu(RRR, T,B) =2.44 · 10−8 · Tρcu(RRR, T,B)
(4.5)
The niobium-titanium thermal conductivity is assumed to be independent of the magneticfield and is computed as a function of temperature as
kNbTi(T ) = 7.5 · 10−3 · T 1.85 (4.6)
As for the cable parameters, the heat capacitance and resistivity of iron (304) and thethermal conductivity of Kapton is read from a lookup table and interpolated. The heat transferto liquid helium is calculated with various heat transfer models in steps of 0.1K presented below
hHe1 (T (x, t), t) =
0 if T (x, t) ≤ Tb ∨ t− t(T > Tf.boil) > tf.boil180
[W
m2K4
]· (T (x, t)4 − T 4
b ) if T (x, t) ≤ Tf.boil
100[Wm2K
]· (T (x, t)− Tb) if T (x, t) > Tf.boil ∧ t− t(T > Tf.boil) < tf.boil
hHe2 (T (x, t), t) =
0 if T (x, t) ≤ Tb ∨ t− t(T > Tf.boil) > tf.boil180
[W
m2K4
]· (T (x, t)4 − T 4
b ) if T (x, t) ≤ Tf.boil
10000[Wm2
]if T (x, t) > Tf.boil ∧ t− t(T > Tf.boil) < tf.boil
hHe3 (T (x, t), t) = 670[
W
m2K
]· (T (x, t)− Tb) (4.7)
Most parameters are interpolated from lookup tables that are included are read in during theprogram start. The data is stored in temperatures steps of 0.1K.
29
5 Program documentation
This section describes the program routines, the input and output files of the program and theinput parameters.
5.1 Input variable names
The variable names are presented in alphabetic containing all input variables. Please note thatsome variables required for the conductor description can be redundant (total area and ratiocopper to superconductor or copper and superconductor areas). In these cases the user hasto provide a logical input file since SPQR does not check the input file for unphysical inputparameters for the time being.
• ACU :Copper cross-section [m2].
• anormal:
• AHE:Helium cross-section [m2].
• alpha:Help variable for timestep regulation function. The likely-hood of an increased timestepreduces with smaller values of alpha.
• Area:Cross-section of superconducting cable (y-z plane) [m2]
• ASC:Superconductor cross-section [m2].
• b0:Iron yoke contribution to the magnetic field [Tesla]; used for evaluating B(I(t)).
• b1:Coefficient for evaluating B(I(t)) [Tesla/A].
• b angle:Angle in between the magnetic field and the cable axis (width) [rad], used to calculatethe parallel and perpendicular field component.
• b perp:If not calculated with Bfield and b angle, the perpendicular magnetic field component.
• b par: If not calculated with Bfield and b angle, the parallel magnetic field component.
• Bfield:Average magnetic field applied to the superconducting cable [tesla].
30
• calcul:Integer variable to be given as float. If zero the old material properties are read. If thevalue is one the parameters are displayed on the screen. If the value is bigger than onethe new interpolated parameters are saved. If the value is equal or bigger than three theparameters evaluation is not shown on screen.
• control:All stored variables are written into heater.dat and time.dat after every time interval oflength control.
• covered:Number of finite difference elements in x direction that are not exposed to helium coolingstarting from zero. This variable can be used to study whether a quench can be stoppeddue to worse cooling conditions at certain parts of the conductor.
• cu plated:Length of one copper-plated part of the heater strip (only for quench sim) [m].
• d fil:Interfilament effective diameter [m].
• d strand:Interstrand effective diameter [m].
• decay mode:Variable to be given as float that determines the current decay mode in quench sim (0exponential decay; 1 Gaussian decay; 2 linear decay; 3 R(t)/L decay, this option requiresthe setting of r extract, L series; dynamic decay: this option requires the setting ofr extract, ex diode, t transfer, L magnet, L series. The current decays with a timedependent time constant of R(t)/L series first and R(t)/L magnet after the diode isturned on).
• deltat I read:Time interval with constant current before the linear interpolation to the current decayvalues that are read with the logical variable read I decay [s].
• dk iso:Length of the radial elements of the insulation layer thickness [m].
• dt iso:The timestep for the matrix generation function [s].
• Dx:Length of each cable finite difference element in x direction [m].
• Dy:Length of each cable finite difference element in y direction [m].
• Dz:Length of each cable finite difference element in z direction [m].
31
• eddy :Logical variable to calculate the effects of a changing magnetic field with time.
• ex diode:Turn-on voltage threshold of the parallel protection diode [V]; if set to zero, the calculationof the protection with a parallel resistance is possible (using r help = value of parallelresistor).
• ex res:Threshold of the detection voltage [V]. The current decay starts when ex res is exceeded.The quench detection has to be validated by heat det time if the heater mode is switchedon.
• fe np:Length of one heating part of the heater strip [m].
• fend:End time for quench propagation velocity calculation [s].
• fintime:End time of the simulation [s]. If no abort criterion is fulfilled the simulation will endafter exceeding this time limit.
• fstart:Start time for quench propagation velocity calculation [s].
• h cable:Rutherford cable height (thickness) [m].
• h wedge:Copper wedge (spacer) height (thickness) [m].
• heat det time:Quench validation time [s]. The detected quench must exceed the threshold for the timeperiod of heat det time before the current decay starts and heaters are fired.
• heater:Integer variable given as float to allow for quenching due to quench heaters [1 = quenchheater variables have to be set, 0 = no quench heaters].
• heat const0:Heat constant for model 1 of heat transfer into helium.
• heat const1, 2, 3:Heat constants for model 2,3,4 of heat transfer into helium.
• heater iso:Same as l iso; the insulation layer thickness between the conductor and the heater strip[m].
32
• heat pot:Power for Kapitza regime of heat transfer.
• heat pom:Power for model 3 of heat transfer after film boiling.
• heat transfer:Integer variable to be given as float. This variable determines which model for heattransfer to liquid helium shall be used. If the value is zero, it is h(T ) ≡ 0. Otherwise thevalue i of heat transfer corresponds to the heat transfer model for hHei in Eq. 4.7.
• heat cap:Capacitance of the heater power supply [F].
• heat fraction:Fraction of heat that is conducted from the heater strip into the superconducting cable.If this value is set with a negative value the variable is understood as the insulationthickness between the heater strip and the collars. The heat fraction used to quench thecable is calculated with Eq. 2.25.
• heat length:Total length in x direction of the heater strips connected in series with a power supply ofinitial voltage heat U0 and capacitance heat cap [m].
• heat stop:Integer variable to be given as float that stops the program execution if the entire cableis quenched.
• heat thick:Thickness of the heater strip (in y direction) [m].
• heat U0:Initial voltage of the heater power supply [V].
• heat width:Width of the heater strip (in z direction) [m].
• helium channel:Helium channel diameter for helium phase modelling [m].
• helium phase:Logical variable to use helium phase modelling (set to 1, standard value 0).
• Icrit:Critical current [A]. If it is set to zero or left out the value is calculated as a function oftemperature and magnetic field.
• Iinit:Initial current [A].
33
• insul:Integer variable given as float to allow for heat transfer through insulation material [1 =heat transfer through insulation, 0 = direct heat transfer into helium].
• keep ends:Integer variable given as float to keep the end elements at helium bath temperature [1 =keep temperature, 0 = allow for increase].
• kindiso:Logical variable that decides which insulation material is used for the matrix evaluationand the linear approximation of the insulation layer temperature (1=epoxy, 0=polymidefilm, 0 is standard value).
• liso:Insulation layer thickness between the cable and the helium bath [m].
• l iron:Radius between heater strip and helium bath outside the iron yoke (radial) [m].
• l iso:Insulation layer thickness between heater strip and superconducting cable (in y direction)[m].
• L magnet:Inductance of the quenching magnet [Henry=Vs/A].
• L series:Inductance of the entire magnet chain [Henry=Vs/A].
• l turn:Insulation layer thickness in between two adjacent turns of the superconducting cable (inz direction) [m].
• lp fil:Filament twist pitch length [m].
• lp cable:Rutherford cable twist pitch length [m].
• make matrix:Logical variable to start matrix evaluation (if set to 1, standard set to 0).
• matrix cool:Variable used for steering the wetted perimeter in the matrix model (multiplying factor).
• N filaments:Number of filaments per strand.
• N layer:Number of strand layers (2 for LHC Rutherford cables).
34
• N strands:Number of strands per Rutherford cable (36 for LHC outer layer cable).
• Niter:Number of finite difference elements for the radial discretisation of the insulation layer.
• Nelem:Number of finite difference elements in x direction.
• Nplug:The firstNplug elements ofNelem elements are covered with the plug insulation material,the others with the standard insulation material.
• Nyelem:Number of finite difference elements in y direction.
• Nzelem:Number of finite difference elements in z direction.
• Pheat:Wetted perimeter [m] after tboil is exceeded.
• plug:Logical variable to calculate plugs and λ-plates (set to 1, standard set to 0).
• Pwet:Effective wetted perimeter [m]. Pwet is the parameter that has to be adjusted in orderto fit the simulation results to measured data. For a circular cable it is Pwet = f · 2πr,r being the radius of the cable, f being the effective fraction.When heat transfer through the insulation is included, Pwet is the entire circumference.
• P C:Wetted perimeter for averaged insulation layer [m]. P C is set to Pwet if not specified.
• P K:Wetted perimeter at the surface of the conductor [m]. P K is set to Pwet if not specified.
• quench v:Integer variable to be given as float that triggers the calculation of the quench velocityfor different analytical models (see SectionA).
• qvel:Not used in current version.
• ratio:Ratio between copper and superconductor in the cable rcu/sc.
• read a:Integer given as float. If set as 0.0, no profile is read. If set as 1.0, the file dat/a profile.datis read to allow for simulating the effect of reduced cross-sections along short stretches.If set as 2.0, the file dat/p profile.dat is read in addition to allow for changing coolingconditions Pwet(x).
35
• read I decay:Logical variable. If set to one, the current decay file (filename dat/i decay.dat) is read.
• read in:Integer value to be given as float in order to read an initial temperature profile. If thevalue is one the initial temperature profile is read from the current directory and filedat/read in.dat. The number of finite difference elements must be in agreement with theinput parameters Dx, Dy, Dz, Nelem, Nyelem, Nzelem.
• rest ext:Value of the extraction resistor. Not used in current version. Current decay time constantis given with tI decay.
• R a:Interstrand contact resistance in between parallel strands in the same layer [Ohm].
• R c:Interstrand contact resistance in between strands of different layers [Ohm].
• R extract:Extraction resistor used for the complex current decay calculation [Ohm].
• r help:Equivalent resistance of the diode at the time when the turn-on voltage is reached [Ohm].
• r multiply:Factor to multiply the developing resistance used for the complex current decay calcula-tion (NOTE: It’s a factor, not a unit).
• RRR:Value of the RRR of copper.
• rutherford:Logical value to define Rutherford type cable (set to 1 as standard).
• start elem:Number of element (given as float) that has temperature T init at t=0s [default=0.0].
• t transfer:Time in between turn-on voltage of the parallel diode is reached and its equivalent resis-tance is negligible [s].
• timestep:Minimal timestep ∆t [s]. The timestep is recalculated at the end of each evaluation ofthe temperature array. The user has to set timestep small enough to avoid unwantedfailures like oscillations of temperatures during the simulation. Typical values are a fewmicroseconds.
36
• tI decay:Current decay time constant [s]. The current decay starts after quench detection andvalidation. The user can set whether a Gaussian, an exponential or a linear current decayis wanted (see decay mode).
• tboil:Time interval for helium film boiling [s].
• Tbath:Helium bath temperature [K].
• Tboil:Temperature when helium film boiling starts [K].
• Tcrit:Critical temperature [K]. If it is set to zero or left out the value is calculated as a functionof current density and magnetic field.
• Tfilm:Temperature when helium film boiling starts [K] for model 3 and 4. For this models Tboilis the temperature when helium nucleate boiling starts. The heat transfer coefficient iskept constant between Tboil and Tfilm.
• T init:The initial temperature profile is calculated as Gaussian like distribution using T init asthe initial peak temperature (see also xT decay, yT decay, zT decay) [K].
• Tmax:If this maximum allowed temperature [K] is reached in any element during the tem-perature evaluation the program stops to write the achieved results in order to avoid aexecution failure.
• t out:The temperature profile is written into temp.dat every time interval of length t out [s]starting with t=0.0 s. The number of time stamps has to be less than 200.
• t switch:Switch delay before the current decay starts after detection of the quench [s].
• use variable:Logical variable to use the matrix model (set to 1, standard set to 0).
• w cable:Width of the Rutherford cable (defines cable axis) [m].
• w wedge:Width of the copper wedge (spacer) [m].
37
• write out:Integer value to be given as float to write out the temperature profile at the end of theexecution. If the value is set to one the profile is written to current directory into the filedat/profile.dat.
• xT decay:Decay length in x direction of the initially calculated temperature profile [m].
• yT decay:Decay length in y direction of the initially calculated temperature profile [m].
• zT decay:Decay length in z direction of the initially calculated temperature profile [m].
• Option -d:Three dimensional model of the coil for main magnets.
• Option -c:Three dimensional model of the coil for corrector magnets.
• Option -o:Optimisation routine starts (to fit wetted perimeter to obtain the measured quench prop-agation velocity).
5.2 Output variable names
• all quenched:Time at which all elements are quenched [s].
• ex.time:Execution time of the simulation run [s].
• fstart:Starting time for the calculation of the quench propagation velocity [s].
• fend:Ending time for the calculation of the quench propagation velocity [s].
• miits:Value of quench load [A2s].
• MQE:Calculated quench energy (from enthalpy difference of starting temperature profile andbath temperature) [J].
• pf m:Maximum strength of interfilament coupling current losses [W/m].
• ps m:Maximum strength of interstrand coupling losses [W/m].
38
• quench prop:Calculated quench propagation velocity [m/s].
• rho(T adia):Specific resistivity at adiabatic temperature T adia [Ωm].
• Temp abs max:Maximum temperature during the calculation [K].
• time abs max:Time of maximum temperature Temp abs max [s].
• tpf m:Time of maximum interfilament coupling losses [s].
• tps m:Time of maximum interstrand coupling losses [s].
• T adia:Calculated adiabatic temperature for the quench load value miits [K].
• t detect:Time at which the detection threshold ex rex is reached by the expanding normal con-ducting zone (in two directions) [s].
• xmin:Minimum position of the conductor for temperature profile plot [K].
• xmax:Maximum position of the conductor for temperature profile plot [K].
• xstart:Starting point for calculation of the quench propagation velocity [m].
• xend:Ending position for the calculation of the quench propagation velocity [m].
• ymin:Minimum temperature during the calculation minus 1K for temperature profile plot [K].
• ymax:Maximum temperature during the calculation plus 1K for temperature profile plot [K].
5.3 Input
The input parameters are stored in todo.dat. This file is read in at the beginning of programexecution.
Depending on the input parameter, an initial temperature profile can be read in fromdat/read in.dat. Otherwise the initial temperature profile is calculated. Some examples of
39
todo.dat are given in 7. If the optimisation routine is used the file todo.opt must be edited (seeSection 5.5.7.)
When the copper cross-section or the cooling conditions are varied along the superconduct-ing structure, the input files dat/aprofile.dat and/or dat/pprofile.dat are read. The commonmaterial parameters are then calculated for the average cross-section.
When the matrix model is used for the heat transfer through the insulation layer or theplug model, the file dat/matrix.dat must be readable.
5.4 Output
Some examples of the output of the program are shown in Fig. 7 to Fig. 10.
x (m)
T (
K)
0.00E+0s1.20E+1s2.40E+1s3.60E+1s4.80E+1s6.00E+1s7.20E+1s8.40E+1s9.60E+1s1.08E+2s1.20E+2s1.32E+2s1.44E+2s1.56E+2s1.68E+2s1.80E+2s1.92E+2s2.04E+2s2.16E+2s2.28E+2s2.40E+2s2.52E+2s2.64E+2s2.76E+2s2.88E+2s3.00E+2s3.12E+2s3.24E+2s3.36E+2s3.48E+2s3.60E+2s3.72E+2s3.84E+2s3.96E+2s4.08E+2s4.20E+2s4.32E+2s4.44E+2s4.56E+2s4.68E+2s4.80E+2s4.92E+2s5.04E+2s5.16E+2s5.28E+2s5.40E+2s5.52E+2s5.64E+2s5.76E+2s5.88E+2s
50
100
150
200
250
300
350
400
0 5 10 15 20 25 30 35 40
Figure 7: Example of the temperature profile as a function of x and t as 2D plot. The resultwas obtained with quench propagation for the study of the quench propagation in the mainbusbar.
Fig. 7 shows the temperature evaluation as a function of x in a 2-dimensional plot. Thetime dependence is included with temperature profiles in different colours for various times.
40
0
50
100
150
200
250
300
350
400
xpos (m)
time (s)
T (
K)
05
1015
2025
30
0100
200300
400500
0
50
100
150
200
250
300
350
400
Figure 8: Example of the temperature profile as function of x and t as 3D plot. The result wasobtained with quench propagation.
41
Fig. 8 shows the same simulation result in a 3-dimensional representation. The differentcolours represent the temperature for a given place and time.
The examples are obtained with SPQR for the quench protection study of the main busbarsincluding heat transfer to liquid helium.
time (s)
curr
ent (
A)
time (s)
volta
ge (
V)
time (s)
resi
stan
ce (
Ohm
)
time (s)
ener
gy (
J)
time (s)
quen
ch r
egio
n (m
)
time (s)
quen
ch T
(K
)0
2500
5000
7500
10000
0 200 400 6000
2.5
5
7.5
10
0 200 400 600
0
0.001
0.002
0 200 400 6000
2500
5000
7500
10000
x 10 3
0 200 400 600
0
10
20
0 200 400 6000
5
10
15
20
0 200 400 600
time (s)
quen
ch r
egio
n (o
ne s
ide)
(m
)
v(12000A)=0.306303+/-0.0107472m/s
2
4
6
8
10
12
14
16
10 20 30 40 50
Figure 9: Example of the recorded global variables during the quench propagation (on the left)and the fitted quench propagation velocity (on the right). These plots allow the calibration ofthe simulation model with measured data.
The recorded global variables like current, voltage, development of the dissipated energy,resistance and normal conducting zone are shown in Fig. 9 (on the left). The fit of the expandingnormal conducting zone versus time derives the quench propagation velocity shown on the right.
An example of a collapsing quench is shown in Fig. 10. The program package also allowsstudying minimum quench energy in presence of liquid helium after the cooling parameters arecalibrated to measured data.
Two groups of output files are written into the path currentdirectory/dat/. The parameterfiles are the following:
• mat spec:According to input parameters, the material properties as a function of temperature arerecalculated for the applied B-field, values of RRR and rcu/sc. The temperature step-widthis 1K.
• heater iso.dat:The parameters of the heater strip and the insulation material as a function of temperatureare stored here in steps of 1K.
• hel heat.dat:The coefficient hHe for the heat transfer to helium is stored as a function of temperaturein steps of 0.1K.
The second group of output files are the results of the program execution:
• temp.dat:This file includes the entire temperature profile at times intervals of the variable t out.
42
x (m)
T (
K)
0.00E+00s3.00E-02s6.00E-02s9.00E-02s1.20E-01s1.50E-01s1.80E-01s2.10E-01s2.40E-01s2.70E-01s3.00E-01s3.30E-01s3.60E-01s3.90E-01s4.20E-01s4.50E-01s4.80E-01s5.10E-01s5.40E-01s5.70E-01s6.00E-01s6.30E-01s6.60E-01s6.90E-01s7.20E-01s7.50E-01s7.80E-01s8.10E-01s8.40E-01s8.70E-01s9.00E-01s9.30E-01s9.60E-01s9.90E-01s1.02E+00s1.05E+00s1.08E+00s1.11E+00s1.14E+00s5
10
15
20
25
30
35
40
0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 10: Example of a collapsing quench when the initial energy is not sufficient. The resultis shown in the 2 dimensional presentation as described in Fig. 7.
43
• time.dat:The global variables are stored at the rate of the time intervals given by the variablecontrol.
• heater.dat:The heater related variables are written into this file at the rate of the time intervalsgiven by the variable control. .
• param.dat:All input parameters are characteristic simulation results like quench detection time, valueof the quench load, etc. are stored.
• profile.dat:If specified in the input file the final temperature profile is written into this file.
• If the user starts the simulation with an option -iX, X being an integer number, theoutput files are copied to outfile.X.date.time.dat.If the option -jX is used, the output files are copied to outfile.X.dat.In both cases the simulation run is stored in the file control.out with its characteristicparameters for documentation.
• eddy.dat:If the user specified eddy 1.0, the eddy currents are calculated. This file contains thestrengths of the interfilament and interstrand coupling currents and losses at the timeintervals defined by the control variable.
• insul.dat:The temperature of the insulation layer is stored in this file.
5.5 Program routines
5.5.1 Read-in
The program start searching for todo.dat in the current directory. All further action depend onthe input parameters in that file (see Sections 5.3, 5.1).
5.5.2 Setup
The program routines to prepare the simulation run are the following:
• make properties:This routine calculates the copper resistivity and prepares the lookup tables. The resultsare written into the parameter output files.
• read properties and write them:If the parameters remain unchanged this routines reads the parameter files.
• make average properties:The routines calculates the common material properties for the actual fractions of copperand niobium titanium in the cable to receive a homogeneous material.
44
• read profile:The initial temperature profile can be read with this routine.
• room init:This routine calculates the initial temperature profile according to the input parametersand evaluates which elements are covered by heated parts of the heater strip.
• write out parameters in param.dat:The read input parameters are stored in param.dat. This file is appended at with simu-lation results at the end of the simulation run.
5.5.3 Interpolating routines
The interpolating routines return the interpolated material property for a given temperature.The different routines are the following:
• rho Treturns interpolated copper resistivity ρ ≡ ρcu for a given T
• c Treturns interpolated heat capacitance of the cable c ≡ ccommon for a given T
• ent Treturns interpolated enthalpy of the cable ent ≡ entcommon for a given T
• k Treturns interpolated thermal conductivity of the cable k ≡ kcommon for a given T
• h Treturns interpolated heat transfer coefficient to helium h for a given T
• G Treturns heat generation as a function of I(t), T (t) and ρ(T ) and sums up resistance
• R Tsums up resistance if the current has already decayed
• hqvel Tnot used in present version
• c heat Treturns interpolated heat capacitance of the heater cheat for a given T
• rho heat Treturns interpolated heater resistivity ρheat for a given T
• k iso Treturns interpolated thermal conductivity of the heater insulation kiso for a given T
• k turn Treturns interpolated thermal conductivity of the turn to turn cable insulation kturn for agiven T
45
• T cscalculates the current sharing temperature (linear model) as function of current and B-field
• T ccalculates critical temperature as function of current and B-field
• Ic Tcalculates critical current as function of T (t), B-field and current density
• analyitcal qvelcalculates the quench velocity for different analytical models for a given environment
5.5.4 Time-loop
After initialisation the time-loop is started for the temperature evaluation (the routine is calledtimeintegration). During the heating process the program calls the interpolating routines toobtain new temperature values for the next timestep using Eq. 3.2 to Eq. 3.14. Depending onactual values further changes are made (abort criteria or storage). An abort routine is includedwhich allows storing the data obtained so far; continuing with the temperature evaluation; orquitting immediately without saving the data (the command CTRL+C calls the interruptingabort routine). The program routine checks the following points:
• The program calculates the initial output parameters for t=0.0s.
• Check output parameters and start time-loop
– Check resistance. If the voltage over the normal conducting zone exceeds the detec-tion level, start current decay, declare detection time, calculate new current
– For heater models:
∗ Calculate heater current
∗ Evaluate new temperature in heater
– Evaluate new temperatures in 1D (3D respectively) space (loop dependence on cur-rent)
– Adjust critical temperature and critical current
– Check for maximum temperature and increase of normal conducting zone
– Check step number for output to screen and files
– Continuously record H, quench load, cooling and heat generation power
– Adjust duration of timestep
• For eddy currents
– Evaluate B(I(t)) and ρ(RRR,B(I(t)), T (x, t))
– Calculate strength of eddy currents and their losses considering their time constant
– Take the impact of the dynamic losses into account for the temperature increase
46
– Let the eddy currents decay if that element quenched either due to exceeding Tc orIc
• If end time is exceeded leave routine and go on
• If the temperature of any element exceeds abort criterion stop and leave routine
• If abort tried by user call interrupting abort routine
5.5.5 Write results
After the temperature evaluation process is finished, the output routines are called. Their tasksare as follows:
• Write temperature map as a function of space and time
• Write macro variables as a function of time
• Write temperature profile for further simulation if specified.
5.5.6 Abort and end of execution
As mentioned the program can be interrupted asking for an end with save option, abort orcontinue. The program run ends if the maximum temperature is exceeded, a unexpected erroroccurred, the calculation is finished or an abort is induced by the user interface. It announcesthe total execution time.
5.5.7 Optimisation
If the program is started with the option -o, the optimisation routine is started. It allows tochange the range of the wetted perimeter to match the measured quench propagation velocityor to fit the measured heater delays.
The optimisation routine is not usable for the more complex cooling models (insulation layermodelling), eddy current calculations, or three dimensional approaches.
6 Calibration and tests
The program has been checked with experimental results obtained from quench tests on shortmodels and full-size prototype dipole magnets, corrector magnets and bus bars. Examples touse quench heaters and the main busbar are presented in Section 7. This section deals withnumerical issues and debugging tests.
6.1 Numerical issues
The precision of the simulation result depends not only on the model that is used but also onthe dimensioning of the finite difference elements. The temperature map might be wrong if thelength of Dx, Dy, Dz is set too long (problem of setting the mesh) due to the linear evaluationand interpolation of the temperature and material properties. A diverging temperature or
47
temperature oscillations are possible if the timestep is to big. This occurs for example if thetemperature jumps over several Kelvin in an element that becomes normal conducting within∆t. The timestep is recalculated according to the material parameters.
∆t = αc(min(T (x, y, z)))min(Dx,Dy,Dz)2
k(min(T (x, y, z))); (6.1)
with α being an input parameter to steer the timestep evaluation (normally 0.5). ∆t is propor-tional to the length of the finite difference element squared. As the precision of the temperatureevaluation relies on how fine the finite difference element mesh is set up, the timestep must bereduced accordingly. For example, if the simulation includes heat transfer to helium for thecable of the LHC dipole magnets, the Dx may not be set higher than 1 cm. This requires aninitial timestep ∆t of about 0.1–1µs. The other limiting factor is the helium model on a longtime basis (simulation of temperature evaluation after several seconds).
The program does not offer the ability to adjust the initial timestep according to values usedfor the finite difference element dimensions which means that the user has to be aware of thisnumerical instability. Therefore the dependence of the temperature on the input parametersDx, Dy, Dz and timestep for a given cable and its environment should be investigated first.
6.2 Comparison with analytical models
In order to test if the numerical algorithm is calculating the heat conduction along the cablein the correct way, the code has been compared with an analytical model.
The temperature profile T (x, t) for a thin one dimensional conductor that receives an initialheat Q at x=0 for t=0 can be analytically calculated as follows
T (x, t) = T0 +Q
c√(π4Dwt)
exp
(− x2
4Dwt
)(6.2)
with t ≥0 and Dw = k/c.Averaged material constants are used in Eq. 6.2 and the simulation run. The error of the
numerical results depends on the mesh dimensions (see Section 6.1).The energy dissipated into the normal conducting magnet can be calculated via the enthalpy
and via the current and the resistance development. Without heat transfer to liquid heliumis allowed both values must be identical. This can be used as a check of the program code.When heat transfer to liquid helium is included the difference of both values corresponds tothe energy which is taken by the helium bath.
The energy obtained via the current is as follows:
E(t) =
t∫0
R(t′)I(t′)2dt′
E(tn) =n∑i=0
R(ti)I(ti)2∆ti. (6.3)
The resistance R(ti) is given by
R(ti) = 2Nelem∑j=0
ρ(Tj(ti))Dx
Aor 3-dimensionally
48
R(ti) = 2Nelem∑j=0
Nyelem∑k=0
Nzelem∑l=0
ρ(T (j, k, l)(ti))Dx
Dy ·Nyelem ·Dz. (6.4)
The energy obtained via the enthalpy is:
E(t) = (H(t)−H(t0))V
E(tn) =Nelem∑j=0
(H(Tj(tn))−H(Tj(t0)))Vj or for the three-dimensional model
E(tn) = 2Nelem∑j=0
Nyelem∑k=0
Nzelem∑l=0
(H(T (j, k, l, tn))−H(T (j, k, l, t0)))DxDyDz; (6.5)
with V being the total volume, Vj=Dx · A.This test also allows the test of the routine at the endpoints which are assumed to be the
extrema of the temperature profile.
6.3 Calibration of heat transfer through insulation
When the material properties of the insulation material are used as the thermal conductivityand heat capacitance of the busbar, it is possible to verify that the temperature profile insidethe insulation layer can be assumed to be a linear gradient (see Section 2.2.2).
x (cm)
T (
K)
0.00E+00s
4.00E-04s
8.00E-04s
1.20E-03s
1.60E-03s
2.00E-03s
2.40E-03s
2.80E-03s
3.20E-03s
3.60E-03s
4.00E-03s
4.40E-03s
4.80E-03s
5.20E-03s
5.60E-03s
6.00E-03s
6.40E-03s
6.80E-03s
7.20E-03s
7.60E-03s
8.00E-03s
8.40E-03s
8.80E-03s
9.20E-03s
9.60E-03s
10
20
30
40
50
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Figure 11: Test of the heat transfer model through the insulation.
Even if the temperature difference between the surfaces of a 200µm thick insulation layeris 50K, the temperature gradient reaches a linear profile in less than 50ms.
This test showed that the discretisation of the insulation layer is only required if its thicknessis not small with respect to the diameter of the conductor (e.g. for the 600A busbar cable).
49
x (mm)
T (
K)
0.00E+00s
2.00E-03s
4.00E-03s
6.00E-03s
8.00E-03s
1.00E-02s
1.20E-02s
1.40E-02s
1.60E-02s
1.80E-02s
2.00E-02s
2.20E-02s
2.40E-02s
2.60E-02s
2.80E-02s
3.00E-02s
3.20E-02s
3.40E-02s
3.60E-02s
3.80E-02s
4.00E-02s
4.20E-02s
4.40E-02s
4.60E-02s
10
20
30
40
50
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225
Figure 12: Test of the heat transfer model through the insulation.
7 Simlation examples
Please note that after each number a new line has to be started for the program execution.
7.1 Bus-bars
The busbar calculations are performed with a longitudinal discretisation of the conductor.
7.1.1 Bus-bar cable
Example for todo.dat to calculate a quench profile in a 600A bus-bar cable using the matrixcooling model:
Nelem 500.0 start elem 00.0 keep ends 0.0 timestep 0.0000005 fintime 1.5 al-pha 0.01 Tbath 1.9 ACU 0.000001809504 ASC 0.000000201056 AHE 0.0 Tinit 10.0xT decay 0.1 t out 0.001 t switch 0.1 control 0.0001 tI decay 1.0 ex res 1000 Tmax1000.0 Icrit 1500.0 calcul 3.0 Bfield 0.0 Iinit 600.0 RRR 120.0 heat transfer 3.0 tboil1000.0 Tboil 5.0 Tfilm 7.0 P K 0.005024 P C 0.005338 Pwet 0.005652 Pheat 0.001covered 0.0 write out 1.0 read in 0.0 Dx 0.002 fstart 0.1 fend 0.5 insul 1.0 heater0.0 liso 0.0002 use matrix 1.0 make matrix 0.0 Niter 35 dt iso 0.000005
7.1.2 Plug calculations
A plug calculation example is shown below:
50
Nelem 100.0 Nplug 40.0 start elem 00.0 keep ends 0.0 timestep 0.000001 fintime5.0 alpha 0.01 Tbath 1.9 ACU 0.000001809557 ASC 0.000000201062 AHE 0.0 Tinit10.0 xT decay 0.1 t out 0.01 t switch 0.1 control 0.001 tI decay 1.0 ex res 1000 Tmax1000.0 Icrit 1500.0 calcul 3.0 Bfield 0.0 Iinit 600.0 RRR 120.0 heat transfer 3.0 tboil1000.0 Tboil 5.0 Tfilm 7.0 P K 0.005024 P C 0.005338 Pwet 0.005652 Pheat 0.001covered 0.0 write out 1.0 read in 0.0 Dx 0.001 fstart 0.1 fend 0.5 insul 0.0 heater 0.0liso 0.009 plug 0.0 use matrix 1.0 make matrix 0.0 Niter 35 dt iso 0.000005 kind iso1.0
7.1.3 Matrix evaluation
See example under plug calculations and set use matrix 0.0, make matrix 1.0.
7.1.4 Main bus-bar
Nelem 600.0 start elem 00.0 keep ends 0.0 timestep 0.00001 fintime 20.0 alpha0.05 Tbath 1.9 ASC 0.0000065 ACU 0.0002965 AHE 0.0000029 Tinit 30.0 xT decay1.0 t out 25.0 control 1.0 tI decay 104.0 ex res 1.0 Tmax 1000.0 Icrit 15000.0 calcul3.0 Bfield 0.1 Iinit 12000.0 RRR 70.0 heat pom 1.0 heat transfer 4.0 tboil 0.02 Tboil5.0 heat pot 2.4 heat pom 0.0 Pwet 0.072 Pheat 0.0072 covered 0.0 write out 1.0read in 0.0 Tfilm 10.0 Dx 0.05 fstart 0.5 fend 15.0 insul 1.0 heater 0.0 liso 0.0002heat const3 100.0 heat pot 3.5
7.2 Corrector magnets
7.2.1 Transverse propagation
This example has to be started with option -c (3d corrector model).
Nelem 400.0 timestep 0.0000005 fintime 1.1 alpha 0.01 ACU 0.0000004242 ASC0.0000002651 AHE 0.0000000 Tbath 1.9 Tinit 10.0 xT decay 0.05 yT decay 0.0001zT decay 0.0001 t out 0.001 control 0.0001 heater 1.0 tI decay 0.10 ex res 0.2 Tmax1000.0 Icrit 1400.0 calcul 3.0 Bfield 0.0 Iinit 600.0 RRR 120.0 liso 0.000120 in-sul 1.0 covered 0.0 write out 1.0 read in 0.0 tboil 0.03 Dz 0.00113 Dy 0.00061 Dx0.002 Nyelem 5. Nzelem 5. Pwet 0.001 Pheat 0.0001 tboil 0.03 Tboil 5.0 Tfilm10.0 heat U0 0.000001 heat width 0.0151 l iso 0.000120 l iron 0.1 l turn 0.000120heat thick 0.000025 heat length 30.0 cu plated 0.40 fe np 0.12 decay mode 1.0 heat cap0.00705 heat det time 1.0 heat stop 0.0 heat fraction 1.0 t switch 0.02 fstart 0.001fend 0.02
7.2.2 Eddy currents
Nelem 300.0 timestep 0.0000001 fintime 0.05 alpha 0.1 ACU 0.0000004242 ASC0.0000002651 AHE 0.0000000 Tbath 1.9 Tinit 15.0 xT decay 0.01 yT decay 0.001zT decay 0.001 t out 0.001 control 0.0001 heater 1.0 tI decay 0.10 ex res 0.2 Tmax1000.0 Icrit 1400.0 calcul 3.0 Bfield 0.0 Iinit 600.0 RRR 100 liso 0.000240 insul 1.0heat transfer 4.0 heat const1 180. heat const2 1000. heat const3 1000. covered0.0 write out 1.0 read in 0.0 tboil 0.03 Dz 0.00113 Dy 0.00061 Dx 0.005 Nyelem 5.
51
Nzelem 5. Pwet 0.000 Pheat 0.000 tboil 0.03 Tboil 5.0 Tfilm 10.0 heat U0 0.000001heat width 0.0151 l iso 0.000240 l iron 0.1 l turn 0.000240 heat thick 0.000025 heat length30.0 cu plated 0.40 fe np 0.12 decay mode 1.0 heat cap 0.00705 heat det time 1.0heat stop 0.0 heat fraction 1.0 t switch 0.02 fstart 0.001 fend 0.01
7.3 Main magnets
7.3.1 Quench heaters
Nelem 200.0 start elem 00.0 keep ends 0.0 timestep 0.0000001 fintime 1.0 alpha0.0 Tbath 1.9 ASC 0.0000065 ACU 0.0000125 AHE 0.00000080 Tinit 30.0 xT decay0.3 t out 0.05 control 0.001 tI decay 0.2 ex res 0.150 Tmax 1000.0 Icrit 15000.0 calcul3.0 Bfield 5.0 Iinit 11800.0 RRR 100.0 heat pom 1.0 heat transfer 4.0 tboil 0.02 Tboil5.0 heat pot 2.4 heat pom 0.0 Pwet 0.032 Pheat 0.0032 covered 0.0 write out 1.0read in 0.0 Tfilm 10.0 Dx 0.01 fstart 0.001 fend 0.01 insul 0.0 heater 0.0 liso 0.0001l iso 0.00015 heat U0 800.0 heat cap 0.0075 heat width 0.015 heat thick 0.000025heat length 30.0 fe np 0.12 cu plated 0.4 l iron 100.0 heat fraction 1.0 heat const3100.0
7.3.2 Eddy currents
Nelem 300.0 timestep 0.000001 fintime 0.3 alpha 0.01 ACU 0.000012601 ASC0.000006635 AHE 0.0000005 Tbath 1.9 Tinit 10.0 xT decay 0.1 t out 0.005 control0.0001 heater 1.0 tI decay 0.30 ex res 0.2 Tmax 1000.0 Icrit 14200.0 calcul 3.0 Bfield5.0 Iinit 11800.0 RRR 100 liso 0.00015 insul 1.0 heat transfer 4.0 heat const1 180.heat const2 1000. heat const3 1000. covered 0.0 write out 1.0 read in 0.0 tboil 0.03Dz 0.00129 Dy 0.015 Dx 0.01 Pwet 0.001 Pheat 0.0001 tboil 0.03 Tboil 5.0 Tfilm10.0 heat U0 800.0 heat width 0.0151 l iso 0.0005 l iron 0.1 l turn 10.0 heat thick0.000025 heat length 30.0 cu plated 0.40 fe np 0.12 decay mode 1.0 heat cap 0.00705heat det time 0.015 heat stop 0.0 heat fraction 1.0 t switch 0.02 fstart 0.001 fend0.02 rutherford 1.0 R a 0.00002 R c 0.00002 eddy 1.0 b0 0.3 b1 0.00055 fintime 0.2t switch 0.001 use matrix 0.0 heat det time 0.025
7.3.3 Transverse quench propagation
This example has to be started with option -d (3d main magnets model).
Nelem 300.0 timestep 0.000001 fintime 0.3 alpha 0.01 ACU 0.000012601 ASC0.000006635 AHE 0.0000005 Tbath 1.9 Tinit 10.0 xT decay 0.1 t out 0.005 control0.0001 heater 1.0 tI decay 0.30 ex res 0.2 Tmax 1000.0 Icrit 14200.0 calcul 3.0 Bfield5.0 Iinit 11800.0 RRR 100 liso 0.00015 insul 1.0 heat transfer 4.0 heat const1 180.heat const2 1000. heat const3 1000. covered 0.0 write out 1.0 read in 0.0 tboil0.03 Dz 0.00129 Dy 0.015 Dx 0.01 Nyelem 3. Nzelem 3. Pwet 0.001 Pheat 0.0001tboil 0.03 Tboil 5.0 Tfilm 10.0 heat U0 800.0 heat width 0.0151 l iso 0.0005 l iron0.1 l turn 0.0005 heat thick 0.000025 heat length 30.0 cu plated 0.40 fe np 0.12decay mode 1.0 heat cap 0.00705 heat det time 0.015 heat stop 0.0 heat fraction1.0 t switch 0.02 fstart 0.001 fend 0.02
52
8 Acknowledgements
Many thanks go to Robert Herzog who initially worked on the quench propagation in mainbusbars and Rudidger Schmidt who contributed with many fruitful discussions and interestingapplications and who helped reviewing this documentation. The author also wishes to thankD. Hagedorn, G. Riddone, F. Rodriguez-Mateos, S. Russenschuck, G. Vandoni and A. Verweijfor providing required material properties and their support.
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53
[17] T. Ogitsu. Influence of cable eddy currents on the magnetic field of superconductingparticle accelerator magnets. SSCL-N848, 1994.
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[25] A. Bejan. Heat Transfer. John Wiley & Sons, Inc., 1993.
[26] A. Devred et al. Quench analysis of ssc prototype magnets. SSC note, 1994.
[27] D. Hagedorn and F. Rodriguez-Mateos. Material properties of polyimide films. CERN -AT-Note, 1992.
A Analytical formulas
Analytical expressions of the quench propagation velocity are limited in precision as they usuallybased on constant material properties and simple temperature dependencies or geometries. Thequality of the quench propagation velocity calculation strongly depends on the value of the heattransfer into the helium bath h. Nevertheless, the section summarises some formulas to evaluatethe quench propagation velocity analytically for a first approach.
Dresner’s task formulas:
v =J
cb
(kbρb
Tc − Tb
)1/2
(A.1)
v =J
cb
(L0Tb
Tc − Tb
)1/2
· TcTb
(A.2)
v =J
cb
(L0Tb
Tc − Tb
)1/2
·(
2TbTb + Tc
)1/2
(A.3)
54
v =J(L0Tc)
1/2
(cc∫ TcTb
c(T ′)dT ′)1/2(A.4)
Cryo-stability criterion:
α =ρcuI
2c
fcuAPh(Tc − Tb)(A.5)
Wilson’s formula including cooling:
v =J
C
(ρk
Tc − Tb
)1/2(1− 2y)
(yz2 + z + 1− y)1/2(A.6)
y =hP (Tc − Tb)
I2/Aρ(A.7)
z =η2P 2
J2ρA2C(Tc − Tb)(A.8)
Transverse propagation velocity:
αtr =vtransversevlongitudinal
=Ccond
Call
[ktransklong
]1/2
(A.9)
B Variable names
symbol units meaningI [A] currentA [m2] cross-sectionJ [A/m2] current densityk [W/mK] thermal conductivityc [J/m3] specific heat per volumeρ [Ωm] specific resistivityR [Ω] resistanceB [T ] magnetic fieldL [H] self inductanceM [H] mutual inductance
RRR ratio of ρcu(T=300K)ρcu(T=10K)
rcu/sc ratio of copper to superconductorf fractionα Steekly parameter (ratio of heat generation and coolingL0 Lorenz constant, 2.45·10−8V2K−2
Table 3: Variable names (nomenclature).
55
C Derivation of the discrete form of the heat balance
equation
The value of a function close to a given value at point x0 can be approximated using Taylor’srow development up to the second order:
f(x0 + h) = f(x0) + h · df
dx
∣∣∣∣∣x0
+h2
2· d2f
dx2
∣∣∣∣∣x0
+ (
d3f
dx3
∣∣∣∣∣x0
)(C.1)
f(x0 − h) = f(x0)− h · df
dx
∣∣∣∣∣x0
+h2
2· d2f
dx2
∣∣∣∣∣x0
+ (
d3f
dx3
∣∣∣∣∣x0
). (C.2)
Subtracting Eq.C.2 from Eq.C.2 the first derivative of function f at x0 becomes
df
dx
∣∣∣∣∣x0
=f(x0 + h)− f(x0 − h)
2h± h2
6· d3f
dx3
∣∣∣∣∣x0
(C.3)
The second derivative is obtained by adding Eq.C.1 and Eq.C.2
d2f
dx2
∣∣∣∣∣x0
=f(x0 + h) + f(x0 − h)− 2 · f(x0)
h2± h2
12· d4f
dx4
∣∣∣∣∣x0
. (C.4)
For the discrete space the definition
x0 ≡ xi
x0 − h ≡ xi−1
x0 + h ≡ xi+1 (C.5)
is used. If h ≡ ∆x is small compared to the change of the function f within the interval h thefirst and second derivatives of function f at xi can be written as
df
dx
∣∣∣∣∣xi
=f(xi+1)− f(xi−1)
2 ·∆x(C.6)
d2f
dx2
∣∣∣∣∣xi
=f(xi+1) + f(xi−1)− 2 · f(xi)
(∆x)2(C.7)
The general term of the heat conduction in the heat balance equation is
∂
∂xk(T (x))
∂T (x)
∂x. (C.8)
symbol meaningcu coppersc superconductor (NbTi if not specified otherwise)b material property at T = Tb, cooling temperaturecs material property at T = Tcs, current sharing temperaturec material property at T = Tc, critical temperature
Table 4: Index names (nomenclature).
56