LARP IR Cryogenics: Scaling of LHC I IR Cryogenics Model

Roger Rabehl

June 27, 2005

Rev. October 7, 2005

Version 2: March 24, 2006

# Introduction

A spreadsheet-based model was written during the design phase of the LHC
I interaction region (IR) cryogenics system \[1\]. This model calculates
temperature drops from the magnet center at the cold mass cooling holes
out to the cold compressors. This paper documents the equations used in
this model and the predicted temperature profiles resulting from scaling
up the IR dynamic heat load.

Temperature drops within the pressurized He II system are based on
static and dynamic heat loads of the inner triplet. The dynamic heat
load (heat deposition rate) is greatest at the Q1/Q2a end of the Q1
magnet, and therefore this interconnect is the basis for the model.
Figure 1 is a cutaway of the Q1 and Q2a magnets, showing the cold
masses, interconnect piping, TAS absorber, and a corrector magnet. The
locations of the calculated temperatures and temperature drops are
indicated in Figures 1 and 2, respectively. Each temperature drop is
described in Table 1.

![](media/image1.png)

**[Figure 1]{.underline}** Temperatures in the Q1 and Q2a magnets and
the associated interconnect.

![](media/image2.png)

**[Figure 2]{.underline}** Temperature drops in the Q1 and Q2a magnets
and the associated interconnect.

## Table 1 Temperature drops within the pressurized He II system.

  ------------------ ----------------------------------------------------
  Temperature drop   Description

  ∆T~1~              From the cold mass thermal center to the cold mass
                     end via the cooling holes

  ∆T~2~              Through the interconnect piping to the crossover
                     pipe

  ∆T~3~              Through the crossover pipe

  ∆T~4~              Through the pressurized He II in the annular volume
                     outside the corrugated magnet heat exchanger tube

  ∆T~5~              Across the corrugated magnet heat exchanger tube
                     wall
  ------------------ ----------------------------------------------------

There are additional temperature drops to consider in the subatmospheric
He system. These are equivalent temperature drops resulting from
pressure drops as the saturated He II vapor generated in the magnet heat
exchanger corrugated tube returns to the cold compressors. Figure 3
shows the locations of the equivalent temperatures and temperature drops
in the subatmospheric He system, and Table 2 describes these equivalent
temperature drops.

![](media/image3.png)

**[Figure 3]{.underline}** Equivalent temperature and temperature drops
in the subatmospheric He system.

**[Table 2]{.underline}** Equivalent temperature drops within the
subatmospheric He system.

  ------------------ ----------------------------------------------------
  Pressure/          Description
  temperature drop   

  ∆P~1~/∆T~6~        Along the length of the inner triplet corrugated
                     magnet heat exchanger tube

  ∆P~2~/∆T~7~        Through the shell side of J-T heat exchanger

  ∆P~3~/∆T~8~        Through the octant transfer line elevation change

  ∆P~4~/∆T~9~        Through the octant transfer line to the cold
                     compressor station
  ------------------ ----------------------------------------------------

Inputs to the model include the inner triplet system heat load,
geometric parameters of the cold mass and external heat exchanger, and
pipe sizes.

The model also includes a scaling factor for the inner triplet dynamic
heat load. A scaling factor of 0.5 corresponds to the LHC I nominal
luminosity; a scaling factor of 1 corresponds to the LHC I ultimate
luminosity.

**∆T~1~ -- ∆T~4~: Temperature drops through pressurized He II**

The first four temperature drops considered occur within the pressurized
He II and are governed by the superfluid steady-state heat transport
equation of Equation 1 \[2\]:

$\text{q`` } = \left( \text{-f}^{\text{-1}}(T)\frac{\text{dT}}{\text{dx}} \right)^{\text{1/3}}$
(1)

where q\`\` is the heat flux \[W/cm^2^\], -f^-1^(T) \[W^3^/cm^5^-K\] is
the He II heat conductivity function, and dT/dx \[K/cm\] is the
temperature gradient.

**∆T~5~: Temperature drop across the external heat exchanger tube wall**

The temperature drop across the heat exchanger tube wall is calculated
with three heat transfer resistances in series. These resistances are
based on the Kapitza conductance on the outside of the tube wall, the
thermal conductivity of the wall material (copper), and the Kapitza
conductance on the inside of the tube wall. The Kapitza coefficient for
this material is Ch~K~ = 892.8 W/m^2^-K^4^ \[3\].

**∆T~6~: Temperature drop along the inner triplet magnet heat exchanger
tube**

The pressure drop in the inner triplet magnet heat exchanger tube is
calculated piecewise along its length using the standard, single-phase
pressure drop equation. A factor of 4 is added for the corrugations, and
another factor of 1.5 is included based on experience with the Heat
Exchanger Test Unit (HXTU).

The calculated pressure drop is converted to a temperature drop with
Equation 2:

$\Delta\text{T } = \frac{\Delta P}{\left( \text{16}\text{.}\text{77 T - 24}\text{.}\text{35} \right)}$
(2)

where T is the temperature \[K\], ∆P is the calculated pressure drop
\[Pa\], and ∆T is the corresponding temperature drop \[mK\].

# ∆T~7~: Temperature drop through the shell side of the J-T heat exchanger

The vapor produced in the magnet heat exchanger goes through the shell
side of the J-T heat exchanger and passes over finned tubes to precool
the 4.6 K/3 bar supercritical helium supply.

To scale the vapor pressure drop as it passes over the finned tubes,
Equations 3 \[4\] and 4 are used:

$\text{f } \propto \left( \overset{\text{.}}{m} \right)^{\text{-0}\text{.}3}$
(3)

$\Delta\text{P } \propto \text{ f }\left( \frac{{\overset{\text{.}}{m}}^{2}}{\rho} \right) \propto \left( \frac{{\overset{\text{.}}{m}}^{1\text{.}7}}{\rho} \right)$
(4)

where f is the friction factor for flow over finned tube banks,
$\overset{\text{.}}{m}$is the mass flow rate, ρ is the vapor density,
and ∆P is the calculated pressure drop. The system parameters for a
scaling factor of 1 -- Q = 205.5 W, T = 1.863 K -- are used as a
baseline for scaling to a new operating point.

The temperature drop corresponding to this pressure drop is calculated
with Equation 2.

**∆T~8~: Temperature drop due to octant elevation change**

The pressure drop due to elevation change in the octant is constant at
62 Pa, the value included in the original inner triplet model. The
corresponding equivalent temperature drop decreases slightly as the
inner triplet heat load increases.

### ∆T~9~: Temperature drop through the octant transfer line

Pressure drop through the octant transfer line is calculated by Equation
5:

$\Delta\text{P } = \text{ 360 }\left\lbrack \text{1 } + \text{ 0}\text{.}\text{082 }\left( \text{Scaling Factor} \right) \right\rbrack^{2}$
(5)

where ∆P is the pressure drop \[Pa\] and the Scaling Factor has a value
of 0.5 for LHC I nominal luminosity, a value of 1 for LHC I ultimate
luminosity, etc. This equation indicates that at LHC I ultimate
luminosity, the return flow due to the inner triplet dynamic heat load
is 8.2% of the flow returning from the rest of the octant. This
percentage increases proportionally as the dynamic heat load increases,
and the pressure drop increases with the square of the total mass flow
rate.

The temperature drop corresponding to this pressure drop is calculated
with Equation 2.

**T~10~: Temperature at the cold compressors**

The cold compressors are able to reach a minimum pressure of 15 mbar, or
an equivalent saturation temperature of 1.775 K.

At ultimate luminosity (scaling factor of 1), the existing model
indicates that the inner triplet dynamic heat load is about 8% of the
octant heat load. Scaling up the inner triplet dynamic heat load by a
factor of six, for example, would then mean the inner triplet dynamic
heat load is about 50% of the octant heat load. It is assumed that the
15 mbar cold compressor inlet pressure could be maintained if the 1.9 K
system heat load increases by 50%.

# Scaling Results

To run the model, a scaling factor is provided. Temperature drops
through the cryogenics system are then calculated. Figure 4 is a plot of
calculated system temperatures vs. inner triplet total (static +
dynamic) heat load to 1.9 K.

![](media/image4.wmf){width="5.574305555555555in"
height="4.102083333333334in"}

**[Figure 4]{.underline}** Calculated system temperatures vs. inner
triplet total heat load to 1.9 K.

At a total heat load of about 400 W, the calculated temperature at the
magnet thermal center begins to increase rapidly and presents an
operating limit to the LHC I inner triplet cryogenics system. It is
important to note that Figure 4 presents temperatures from the cold mass
cooling holes to the cold compressors. Additional temperature drops from
the magnet coils to the cold mass cooling holes must also be considered
and could present additional limitations.

Limits of the inner triplet design versus limits of the LHC cryogenics
system can also be differentiated. Temperatures T~7~-T~10~ are dictated
by characteristics of the LHC cryogenics system: cold compressor
capacity, QRL return line size and elevation change, and QRL service
module J-T heat exchanger. These factors are considered fixed for the
purposes of this document. These characteristics dictate the available
inner triplet cooling temperature margin in order for the magnets to
remain in superfluid at a given inner triplet total heat load. This
temperature margin, defined as T~λ~ -- T~7~, is plotted as a function of
inner triplet heat load in Figure 5.

![](media/image5.wmf){width="3.6881944444444446in" height="2.59375in"}

**[Figure 5]{.underline}** Inner triplet temperature margin vs. inner
triplet heat load.

Comparing temperature drops can point to limitations in the LHC I inner
triplet cryogenics system. Figure 6 plots temperature drops in the
pressurized He II system as a function of the inner triplet total heat
load. All of these temperature drops begin to grow exponentially at heat
loads of 400-600 W. Any magnet design for an upgraded inner triplet must
therefore address issues such as cooling hole sizing, interconnect pipe
sizing, crossover pipe sizing and frequency, and heat exchanger annulus
sizing.

![](media/image6.wmf){width="4.357638888888889in"
height="3.303472222222222in"}

## Figure 6 Pressurized He II temperature drop vs. inner triplet total heat load to 1.9 K.

Similarly, Figure 7 plots temperature drops in the subatmospheric He
system as a function of the inner triplet total heat load. The largest
equivalent temperature drop is due to flow losses in the QRL transfer
line return to the cold compressors. As was stated previously, any
upgrades of the cryogenics system beyond the inner triplet is considered
outside the scope of this document. The second largest temperature drop
is across the heat exchanger corrugated tube wall. An upgraded inner
triplet could address this using extended surfaces (fins) or using a
larger fraction of the tube wall for heat transfer from the pressurized
liquid He II to the saturated liquid He II. The fourth largest
temperature drop is along the length of the inner triplet heat
exchanger. This could be improved through heat exchanger pipe sizing and
design (e.g., corrugated tube vs. straight tube).

![](media/image7.wmf){width="4.348611111111111in"
height="3.1555555555555554in"}

## Figure 7 Subatmospheric He temperature drop vs. inner triplet total heat load to 1.9 K.

### Conclusions

Heat load scaling of the LHC I inner triplet cryogenics model has been
completed. The resulting system temperature profile shows that for any
given inner triplet total heat load, the LHC cryogenics system presents
a temperature margin within which the magnets must operate.

The calculated temperature drops point toward areas of further study.
The entire pressurized He II system will require careful analysis.
Temperature drop across the magnet heat exchanger pipe wall will also
need to be investigated, as will pressure drop along the heat exchanger
pipe.

#### References

\[1\] T. Peterson, FNAL (2002).

\[2\] S. Van Sciver, Helium Cryogenics, Plenum Press (1986), p. 145.

\[3\] C. Darve, Y. Huang, T. Nicol, and T. Peterson, "He II Heat
transfer through a Corrugated Tube -- Test Report," Fermilab Technical
Division Note TD-99-064 (1999).

\[4\] W. Rohsenow, J. Hartnett, and Y. Cho, Handbook of Heat Transfer,
McGraw-Hill (1998), p. 17.87.