[^1]

*Abstract*---A luminosity upgrade of the CERN Large Hadron Collider
(LHC) is planned to coincide with the expected end of life of the
existing inner triplet quadrupole magnets. The upgraded inner triplet
will have much larger heat loads to be removed from the magnets by the
cryogenics system. As part of the LHC Accelerator Research Program
(LARP), a design study has been completed to investigate the required
characteristics of the cold mass cooling system within the framework of
a design temperature profile. These characteristics are the beam pipe
annulus, collar radial cooling channels, yoke radial cooling channels,
yoke longitudinal cooling channels, and heat exchanger connecting pipe.
Using these parameters in conjunction with energy deposition
calculations, longitudinal and radial temperature profiles for an entire
inner triplet are calculated and presented.

*Index Terms*---accelerator magnets, cooling, cryogenics, helium

# Introduction

[The]{.smallcaps} inner triplets of the CERN Large Hadron Collider (LHC)
are expected to reach their end of life after several years of operation
due to radiation damage. Subsequent replacement of the inner triplets it
expected to coincide with a luminosity upgrade of the accelerator. Such
an upgrade would result in increased heat loads to the inner triplet
magnets, requiring a new design instead of simply replacing them with
another identical triplet. A number of possible upgrade scenarios are
being considered: Nb~3~Sn coil magnets \[1\], NbTi coil magnets \[2\],
and 'slim' quadrupole magnets placed very close to the interaction point
(IP).

Cold mass cooling studies are required to ensure the heat load deposited
by the beams can be removed from the cold mass within acceptable
temperature limits. Inadequate cooling of the magnet coil would result
in quenching when operating at high luminosity.

One such cold mass cooling study is presented here for a luminosity of
10^35^/cm^2^-s, a factor of 10 higher than the nominal luminosity of the
LHC. This paper details the cooling study of an inner triplet consisting
for four main quadrupoles and four correctors. The four main quadrupoles
are Q1 (closest to the IP), Q2a, Q2b, and Q3 (farthest from the IP). The
four corrector magnets are MCBX-1 (a nested horizontal and vertical
dipole orbit corrector at the non-IP end of Q1), MCBX-2 (a nested
horizontal and vertical dipole orbit corrector between Q2a and Q2b),
MQSXA (a skew quadrupole in line with nested skew octupole, octupole,
and skew sextupole coils at the Q3 IP end), and MCBXA (an MCBX package
with nested sextupole and dodecapole coils at the Q3 non-IP end). This
inner triplet would be placed on either side of the IP at the position
of the existing inner triplet. The cold mass magnetic lengths are 6.3 m
for Q1 and Q3, and 5.5 m for Q2a and Q2b. The coil aperture is 90 mm.
The beam pipe is made of the tungsten-rhenium alloy W-25 Re with a
thickness of 11.5 mm in the Q1 and 2.5 mm elsewhere. The inner triplet
operates in a stagnant bath of He II that is cooled by a He II heat
exchanger located outside the cold mass. The external heat exchanger
currently used in the inner triplet is described elsewhere \[3\]-\[4\],
as are initial heat exchanger studies for the upgraded inner triplet
\[5\].

This paper presents the methodology used to calculate the values for the
critical cooling parameters for this particular configuration. Radial
and longitudinal temperature profiles are then calculated for single
magnets and the entire inner triplet from the beam pipe annulus to the
magnet heat exchanger.

# Design Temperature Profile

A design temperature profile was established as a starting point for the
cold mass cooling studies. The design temperature profile takes the
total temperature drop in the cryogenic system, from the magnet beam
pipe to the cryoplant cold compressors, and partitions it into segments
with the ultimate goal of ensuring the magnet coil remains in He II.

There were several considerations in arriving at the design temperature
profile. An initial study examined the thermal limitations of the
existing inner triplet and identified thermal bottlenecks as the heat
load is increased \[6\]. These bottlenecks were then allocated a
relatively larger temperature drop.

Another consideration is possible upgrades to the LHC cryogenics system.
For example, upgrading the cold compressors and moving them closer to
the inner triplets would increase the available temperature drop and
reduce the pressure drop in the piping between the inner triplets and
cold compressors.

Finally, CERN work in the Next European Dipole (NED) program and
interaction region upgrade studies also provided input for the design
temperature profile \[7\]-\[8\].

The design temperature profile used for this design study allocates a
150 mK temperature drop (2.15-2.00 K) from the beam pipe annulus to the
heat exchanger connecting pipe, and a 50 mK temperature drop (2.00-1.95
K) within the heat exchanger connecting pipe. The envisioned heat
exchanger connecting pipes are illustrated in Fig. 1.

![](media/image1.png){width="3.0305555555555554in"
height="0.9784722222222222in"}

Fig. 1. Illustration of heat exchanger connecting pipes, which transfer
heat from the cold mass He II volume to an annular He II volume around
the heat exchanger pipe. Results presented here assume a connecting pipe
at each end of each cold mass, as shown.

The existing inner triplet has one connecting pipe per interconnect,
whereas it is assumed the upgraded inner triplet will have a connecting
pipe at each end of each main cold mass.

# Heat Deposition

A required input to calculate temperature profiles is the longitudinal
heat deposition. Fig. 2 shows the calculated dynamic heat load to the He
II as a function of distance from the IP \[9\]. The locations of the
four main quadrupole cold masses are indicated. The four corrector
magnets are also shown, each indicated by a single vertical bar outside
the brackets of the main quadrupoles. The heat is assumed to be
deposited equally in the horizontal and vertical planes. Radial
variations in heat deposition are also not taken into account. Whether
heat is deposited in the beam pipe, the inner coil, or the outer coil,
it is assumed that all heat reaches the He II of the beam pipe annulus.

![](media/image2.png){width="3.48125in" height="2.0840277777777776in"}

Fig. 2. Dynamic heat loads to the He II system of the upgraded inner
triplet. Increasing the LHC luminosity would substantially increase the
He II heat loads, requiring careful design of the cold mass cooling
system.

# Thermal Center And Heat Removal Distribution

The thermal center is the longitudinal position where there is a change
in the direction of heat flow. On the IP side of the thermal center,
heat flows toward the IP. On the non-IP side of the thermal center, heat
flows away from the IP. The thermal center provides a zero heat flux
boundary condition for subsequent calculations.

A first estimate of the longitudinal position of the thermal center of a
cold mass is calculated by (1), where *q(x)* is the heat flow at a given
position, *L* is the length of the cold mass, and *x~TC~* is the
longitudinal position of the thermal center. The cubic exponent is
characteristic of He II heat transfer. Equation (1) assumes the
longitudinal temperature drops from the thermal center to each end of
the cold mass are equal.

$$$$

$\int_{0}^{x_{\text{TC}}}{q^{3}(x)\text{ dx}} = \int_{x_{\text{TC}}}^{L}{q^{3}(x)\text{ dx}}$
(1)

Based on the calculated location of the thermal center of each main
quadrupole cold mass, Fig. 3 shows the amount of heat to be removed from
each end of each cold mass. The numbers in parentheses are the heat
loads deposited in the corrector magnets. The numbers without
parentheses are the heat loads deposited in the main quadrupoles.

![](media/image3.png){width="3.35in" height="2.5722222222222224in"}

Fig. 3. Heat loads to be removed from each end of each inner triplet
quadrupole. These values result from estimating the locations of the
cold mass thermal centers using (1).

# Design Parameters

There are a number of critical design features that must be specified in
order to adequately remove heat from the cold mass coils: beam pipe
annulus, collar radial cooling channels, yoke radial cooling channels,
yoke longitudinal cooling channels, and heat exchanger connecting pipe.
The sensitivity and effect of each individual parameter can be studied
using (2), where *q''* is the heat flux, *L* is the cooling channel
length, *f* ^-1^*(T,P)* is the He II thermal conductivity function, and
*T* is temperature.

$q\text{''} = \left( \frac{1}{L}\int_{T_{1}}^{T_{2}}{f^{- 1}(T,P)\text{dT}} \right)^{\frac{1}{3}}$
(2)

Heat exchanger connecting pipes transfer heat from the ends of the cold
masses to the external heat exchanger. Each inner triplet now has five
connecting pipes: one at the IP end of Q1, one between each pair of cold
masses, and one at the non-IP end of Q3. Those at the Q1 IP end and at
the Q3 non-IP end have an inner diameter of 82.8 mm. Those in the
interconnects have an inner diameter of 95.2 mm. The maximum size of
this connecting pipe is limited by the geometries of the cold mass and
cryostat piping. The cooling scheme presented here uses pipes having a
168.3 mm (6.625 in) outer diameter and a 3.4 mm (0.134 in) wall
thickness.

For the Nb~3~Sn magnets being studied here, \[1\] has specified 400
cm^2^ of yoke longitudinal cooling channels. The existing inner triplet
magnets have 113 cm^2^.

Yoke radial cooling channels are required to transfer heat from the yoke
lamination inner diameter to the yoke longitudinal cooling holes. These
cooling channels may be created by removing a small portion of the full
thickness of the yoke lamination at the parting plane or by machining
cooling channels into the surface of the laminations. Whichever method
is used, it must be ensured that the heat load is able to be distributed
among the yoke longitudinal cooling holes. The cooling scheme described
here considers 2.4% of the yoke inner diameter open for cooling
channels.

Collar radial cooling channels are required to transfer heat from the
beam pipe annulus to the iron yoke. Collar cooling channel size is
expressed as a percentage of the collar pole tip open for He II heat
transfer. The current LHC Q2 quadrupole magnets have a cooling channel
equivalent to 4% of the inner pole tip area, or one missing 1.5 mm thick
collar lamination in each quadrant per 38 mm of length. If the second
generation magnets are constructed similarly to the first generation
magnets, it is recommended to have a collar cooling channel area
equivalent to 7% of the pole tip area, or one missing 1.5 mm thick
lamination in each quadrant per 21 mm of length.

A heat load due to secondary particles produced by the proton-proton
collisions at each IP is deposited in the beam pipe and the coils,
concentrated in the horizontal and vertical planes. This heat is
transferred to the He II and is conducted 45^o^ circumferentially
through the beam pipe annulus to the collar pole tips. The existing
inner triplet has a 1 mm annular gap between the beam pipe and the
coils. The same 1 mm annular gap is used here.

This is one possible combination of parameters and is summarized in
Table I. The value of each parameter is noted along with the calculated
radial temperature drop near the non-IP end of Q1, where the heat loads
are largest. The maximum calculated temperature is 2.089 K, leaving a
significant temperature drop available from the coil to the He II bath.

TABLE I Cold mass cooling system parameters

  --------------------- --------------------- ---------------------------
  Design parameter      Value                 Calculated temperature
                                              range \[K\]

  Beam pipe annular gap 1 mm                  2.089 - 2.061

  Collar radial cooling 7% of collar pole tip 2.061 - 2.008
  channels              open                  

  Yoke radial cooling   2.4% of yoke inner    2.008 -- 2.000
  channels              diameter open         

  Yoke longitudinal     400 cm^2^             2.000
  cooling channels                            

  Heat exchanger        161.5 mm (6.357 in)   2.000 - 1.971
  connecting pipe       inner diameter        
  --------------------- --------------------- ---------------------------

# Magnet Temperature Profile

Once these parameters have been specified, the longitudinal and radial
temperature distribution of one half of the cold mass (from the thermal
center to the cold mass end) can be calculated. A finite difference
model using temperature-dependent properties of He II was constructed to
calculate the temperature profiles. Fig. 4 shows the resulting
temperature map of the non-IP half of Q1 where the heat loads are the
highest, resulting in substantial temperature gradients.

It was assumed that all heat transfer is radial, except in the yoke
longitudinal cooling holes. The beam pipe annulus allows a slight
redistribution of the heat load, permitting it to find a path of least
resistance through the cold mass cooling channels. The result would be a
slight smoothing of the calculated magnet temperature profiles.
Neglecting this effect is therefore conservative in that a larger
temperature range is calculated. This redistribution of the heat load
has been calculated to be 3 W/m or less, which is small compared to the
dynamic heat loads presented in Fig. 2.

![](media/image4.png){width="3.4895833333333335in"
height="2.459722222222222in"}

Fig. 4. Temperature profile of the non-IP half of Q1. All locations
within the magnet remain in He II, below the lambda point of 2.17 K. The
yoke longitudinal cooling channels are 2.00 K at the non-IP end of the
cold mass as specified by the design temperature profile. At the thermal
center, the temperature gradient in the yoke longitudinal cooling
channels is zero. Positions where temperatures are calculated are
specified in the magnet cross-section inset.

# Triplet Temperature Profile

These calculations can be extended to calculate the radial and
longitudinal temperature profiles of the entire inner triplet. The
finite difference model was applied to each half of each cold mass to
calculate the temperature profiles after additional iterations to more
precisely locate the cold mass thermal centers and generate better curve
fits of the longitudinal heat load distribution. This and the
temperature-dependent properties of He II account for the difference at
the non-IP end of Q1 between Fig. 4 and Fig. 5, which presents the
calculated temperature profile of the specific inner triplet
configuration considered here. Discontinuities in the temperature curves
are the result of small errors in determining the exact thermal centers
of the magnets.

![](media/image5.png){width="6.775in" height="2.40625in"}

Fig. 5. Calculated longitudinal and radial temperature profiles of the
inner triplet. The cooling system presented here would result in sizable
temperature drops in the Q1 cold mass but very small temperature drops
in the other cold masses. The positions where temperatures are
calculated are the same as those identified in the Fig. 4 inset.

# Conclusion

A methodology for calculating longitudinal and radial temperature
profiles of an inner triplet has been presented. A complete thermal
analysis also requires consideration of the heat exchanger and heat
transfer between the coil and the He II.

While results for one specific inner triplet configuration and cold mass
geometry are detailed here, this approach can be used to investigate
other inner triplet configurations. One such alternate configuration is
the use of heat exchanger tubes internal to the cold mass \[10\].

Convergence on a magnet design will ultimately require iteration on both
thermal and mechanical analyses.

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[^1]: ^^Manuscript received August 24, 2007. This work was supported by
    the U.S. Depart­ment of Energy under contract DE-AC02-76CH03000.

    R. J. Rabehl is with the Fermi National Accelerator Laboratory,
    Batavia, IL 60510 USA (phone: 630-840-8855; fax: 630-840-6724;
    e-mail: rabehl@fnal.gov).