LARP IR Cryogenics: Quench Pressure Calculations

Roger Rabehl

January 9, 2007

# Introduction

Quench pressure calculations are required in order to design the magnet
cold mass for the pressures it must withstand during operation. The
calculations presented here for a LARP quadrupole include both the
initial pressure spike following a quench and the pressure in the magnet
cold mass during heat transfer to the bulk helium.

**Initial Pressure Spike**

The initial pressure spike is calculated using the two-volume model
\[1\]. The first volume of helium is that in close contact with the
magnet coils, the second volume is the bulk helium contained in the cold
mass structure. Immediately following a quench, the rapid expansion of
the helium in close contact with the coils causes an isentropic
compression of the bulk helium. Energy balances, as given by Equations
1-2, are written for both helium volumes. A mass balance, as given by
Equation 3, is written for the system with the assumption of constant
system volume.

Coil He energy balance:
$\text{M x }\left( u\left( T_{\text{x,2}},P_{2} \right)\text{ - u}\left( T_{1},P_{1} \right) \right) = \text{ Q - W}$
(1)

Bulk He energy balance:
$\text{W } = \text{ M }\left( \text{1 - x} \right)\left( u\left( s_{1},P_{2} \right)\text{ - u}\left( T_{1},P_{1} \right) \right)$
(2)

System mass balance (constant volume):
$\frac{\text{M x}}{\rho\left( T_{1},P_{1} \right)} + \frac{\text{M }\left( \text{1 - x} \right)}{\rho\left( T_{1},P_{1} \right)} = \frac{\text{M x}}{\rho\left( T_{\text{x,2}},P_{2} \right)} + \frac{\text{M }\left( \text{1 - x} \right)}{\rho\left( s_{1},P_{2} \right)}$
(3)

Two parameters are required in order to solve these equations: the
fraction x of the total helium in close contact with the coils, and the
temperature of this helium T~x,2~ immediately after its initial
expansion.

The estimated maximum coil temperature underneath the quench protection
heaters of a LARP Nb~3~Sn quadrupole is 127-180 K for heater coverages
of 25-50% \[2\]. If a 150 K temperature is assumed, and a ∆T of 110 K is
assumed as per the CERN calculations \[1\], the final temperature of the
expanded helium in close contact with the coils is 40 K.

The CERN calculations use 2% as the fraction of helium in close contact
with the coils \[1\]. This value is based on micrographs of NbTi coil
cross-sections. Micrographs of potted Nb~3~Sn coils show complete
impregnation \[3\]. The volume of helium in contact with the coils is
therefore assumed to be equal to the volume of the Kapton and ceramic
insulating wraps. Based on a 6 m long cold mass with the cross-section
given in \[2\] with the addition of two end domes, the estimated
fraction of helium in close contact with the coils is 0.15%.

Figure 1 shows the calculated quench pressure spikes as a function of
expanded helium temperature and fraction of helium x in contact with the
coils. The results agree with those in \[1\]. For a Nb~3~Sn magnet with
x=0.15% and a 40 K expanded helium temperature, the calculated pressure
spike is only 4.7 bar.

![](media/image1.png){width="3.9034722222222222in" height="2.9375in"}

**[Figure 1]{.underline}** Final temperature of expanded helium vs.
initial quench pressure spike for a range of helium fraction x in
contact with the magnet coils.

**Mixing**

Following the initial pressure spike, the two helium volumes (the
expanded helium near the coils, and the bulk helium) will mix. This
results in a rise in temperature of the now-single volume, and a drop in
the pressure. As the two helium volumes are mixing, heat is being
transferred from the magnet coils to the cold mass structure. The warmed
cold mass in turn heats the helium, causing the pressure to rise. This
pressure is eventually vented when the setpoint of the quench relief
valves is reached.

If this mixing was complete, without the addition of any more heat to
the helium, the pressure would become subatmospheric. For a given helium
fraction x, this mixed pressure is minimized when the initial heat input
results in the mixed volume reaching the lambda point. This is
illustrated in Figure 2.

There is good agreement with the CERN data, however. With 2% of the
helium in close contact with the coils, the after-mixing temperature is
calculated to be 3 K, as shown by Figure 2b. This agrees with Figure 14
in \[1\], where the measured temperature is 3 K a few seconds after the
quench.

![](media/image2.png){width="2.995138888888889in"
height="2.254861111111111in"}
![](media/image3.png){width="2.954861111111111in"
height="2.209722222222222in"}

\(a\) (b)

**[Figure 2]{.underline}** (a) Pressure and (b) temperature after mixing
as a function of the initial temperature of the expanded helium.

Similar to the method in \[1\], the process of heat transfer from the
cold mass to the bulk helium can be modeled using the lumped approach
where the magnet cold mass is treated as an isothermal mass and the cold
mass helium is treated as an isothermal fluid. The heat transfer
coefficient can be approximated by extrapolating from the data presented
in \[1\]. The LHC prototype string had 850 l of LHe and a stored energy
of 15.3 MJ \[1\]. The total cold mass is 10^5^ kg \[4\]. Collars are
austenitic stainless steel, and yokes are low carbon mild steel, with
the collars accounting for approximately 25% of the cold mass
cross-section. Treating the cold masses as lumped, the temperature
following a quench would reach approximately 25 K. A simple model can be
written to estimate the heat transfer coefficient hA from the cold mass
to the helium. With a value of hA=3000 W/K, after 60 s the helium
remaining in the cold mass has a temperature of a little over 8 K after
an initial temperature of 3 K. This is in good agreement with Figure 14
in \[1\].

This value of hA can then be corrected for the LARP quadrupole,
correcting for LHe-cold mass interface surface area. It is estimated
that one LARP 6-m quadrupole has approximately one-quarter the LHe-cold
mass interface surface area as the LHC test string. This then yields a
value of hA=750 W/K for the LARP quadrupole. The estimated LHe volume
for a single LARP quadrupole is 275 liters.

The maximum stored energy of a TQC magnet is 224 kJ/m \[5\], so a 6 m
cold mass will have a stored energy of 1.34 MJ. For the LARP magnet, the
ratio of the mass to the stored energy is less than that of the LHC test
string. The cold mass will thus reach a higher temperature, 32 K, but
the LARP quadrupole has a relatively large quantity of helium.

Figures 3 (a)-(b) show the calculated cold mass pressure and the cold
mass and helium temperatures for a closed system (i.e., no venting of
mass through a relief valve).

![](media/image4.png){width="2.7555555555555555in"
height="2.1493055555555554in"}
![](media/image5.png){width="2.8444444444444446in"
height="2.1840277777777777in"}

**[Figure 3]{.underline}** Closed system (a) pressure and (b) cold mass
and helium temperatures during heat transfer from the cold mass to the
bulk helium.

### Conclusions

The sequence of events can be summarized as follows. The initial
pressure spike reaches approximately 5 bar. As the stored energy
diffuses as heat through the cold mass, the mixed helium volume remains
as subcooled liquid at approximately 1.97 K/0.94 bar. The heat transfer
rate between the 32 K cold mass and the 1.97 K subcooled liquid is high,
quickly raising the pressure beyond the critical pressure of 0.2275 MPa
at a temperature of 2.86 K. Only 10% of the magnet stored energy is
required to do this. The cold mass temperature is reduced by 1 K, to 31
K. Heat transfer between the cold mass and the supercritical helium is
then greatly reduced to 750 W/K, and the cold mass pressure rises at a
rate of approximately 1 bar/s until the quench valve setpoint is
reached.

The accuracy of this document's calculation of the initial pressure
spike for a Nb~3~Sn coil is considered marginal, at best. One critical
parameter for this calculation, the fraction of helium in close contact
with the coil, does not seem applicable for a potted Nb~3~Sn coil.
Spaces between the cable strands are filled with epoxy rather than
helium. A helium layer surrounding the coil with a thickness equal to
the insulation thickness was assumed instead. Another important
parameter is the temperature of this expanded helium. A ∆T of 110 K
between the quenched coil and the surrounding helium is used, which for
lack of better information is the same as that for the LHC test string.
Prototype magnet tests are required to better understand these
parameters for the LARP quadrupoles.

#### References

\[1\] M. Chorowski, et al., "Thermohydraulics of Quenches and Helium
Recovery in the LHC Magnet Strings," LHC Project Report 154 (1997).

\[2\] A. Zlobin, et al., "Conceptual Design Study of Nb3Sn Low-beta
Quadrupoles for 2^nd^ Generation LHC IRs," Applied Superconductivity
Conference 2002 (Houston, TX).

\[3\] R. Bossert, "TQ Collar Design -- Details and Plans," U.S.
Department of Energy Review of the Fermilab High-Field Magnet Program,
November 1-4, 2005.

\[4\] J. Casas-Cubillos, et al., "Operation, Testing and Long Term
Behaviour of the LHC Test String Cryogenic System," LHC Project Report
210.

\[5\] R. Rabehl, "Verification of VMTF Relief Sizing for LARP High-Field
Magnet Testing," Fermilab Technical Division Note TD-06-042, August
2006.