Cryogenics Introduction
Cryogenics Introduction
Cryogenics Introduction
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An Introduction to Cryogenics
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AN INTRODUCTION TO CRYOGENICS
Ph. Lebrun
President, Commission A1 “Cryophysics and Cryoengineering” of the IIR
Accelerator Technology Department, CERN, Geneva, Switzerland
CONTENTS
Ph. Lebrun
President, Commission A1 “Cryophysics and Cryoengineering” of the IIR
Accelerator Technology Department, CERN, Geneva, Switzerland
Abstract
This paper aims at introducing cryogenics to non-specialists. It is not a
cryogenics course, for which there exists several excellent textbooks
mentioned in the bibliography. Rather, it tries to convey in a synthetic form
the essential features of cryogenic engineering and to raise awareness on key
design and construction issues of cryogenic devices and systems. The
presentation of basic processes, implementation techniques and typical
values for physical and engineering parameters is illustrated by applications
to helium cryogenics.
Table 1
Characteristic temperatures of cryogenic fluids [K]
∗ λ point
1
Figure 1 130 000 m3 LNG carrier with integrated Invar® tanks (source Gaztransport)
Figure 2 Cryogenic air separation plant with heat exchanger and distillation column towers (source Air
Products)
2
Figure 4 Automotive liquid hydrogen fuel tank
The quest for low temperatures however finds its origin in early thermodynamics, with Amontons’s
gas pressure thermometer (1703) opening the way for the concept of absolute zero inferred a century
later by Charles and Gay-Lussac, and eventually formulated by Kelvin. It is however with the advent
of Boltzmann’s statistical thermodynamics in the late nineteenth century that temperature – until then
a phenomenological quantity - could be explained in terms of microscopic structure and dynamics.
Consider a thermodynamic system in a macrostate which can be obtained by a multiplicity W of
microstates. The entropy S of the system was postulated by Boltzmann as
S = kB ln W (1)
with kB ≃ 1.38 10-23 J/K. This formula, which founded statistical thermodynamics, is displayed on
Boltzmann’s grave in Vienna (Figure 5).
Figure 5 L. Boltzmann’s grave in the Zentralfriedhof, Vienna, bearing the entropy formula
3
Adding reversibly heat dQ to the system produces a change of its entropy dS, with a proportionality
factor T which is precisely temperature
dQ
T = (2)
dS
Thus a low-temperature system can be defined as one to which a minute addition of heat produces a
large change in entropy, i.e. a large change in its range of possible microscopic configurations.
Boltzmann also found that the average thermal energy of a particle in a system in equilibrium at
temperature T is
E ~ kB T (3)
Consequently, a temperature of 1 K is equivalent to a thermal energy of 10-4 eV or 10-23 J per particle.
A temperature is therefore low for a given physical process when kB T is small compared to the
characteristic energy of the process considered. Cryogenic temperatures thus reveal phenomena with
low characteristic energy (Table 2), and enable their application when significantly lower than the
characteristic energy of the phenomenon of interest. From Tables 1 and 2, it is clear that “low-
temperature” superconductivity requires helium cryogenics: several examples of helium-cooled
superconducting devices are shown in Figure 6. Considering vapour pressures of gases at low
temperature (Figure 7), it is also clear that helium must be the working cryogen for achieving “clean”
vacuum with cryopumps.
Table 2
Characteristic temperatures of low-energy phenomena
2. CRYOGENIC FLUIDS
2.1 Thermophysical properties
The simplest way of cooling equipment with a cryogenic fluid is to make use of its latent heat of
vaporization, e.g. by immersion in a bath of boiling liquid. As a consequence, the useful temperature
range of cryogenic fluids is that in which there exists latent heat of vaporization, i.e. between the triple
point and the critical point, with a particular interest in the normal boiling point, i.e. the saturation
temperature at atmospheric pressure. This data is given in Table 1. In this introduction to cryogenics,
we will concentrate on two cryogens: helium which is the only liquid at very low temperature, and
nitrogen for its wide availability and ease of use for pre-cooling equipment and for thermal shielding.
4
a) b)
c) d)
Figure 6 Helium-cooled superconducting devices a) Large Hadron Collider at CERN, b) 5 MW HTS ship
propulsion motor (AMS), c) ITER experimental fusion reactor, d) whole-body MRI system (Bruker)
1.E+04
1.E+03
He
1.E+02 H2
1.E+01 Ne
1.E+00 N2
1.E-01 Ar
O2
1.E-02
CH4
Psat [kPa]
1.E-03
CO2
1.E-04 H2O
1.E-05
1.E-06
1.E-07
1.E-08
1.E-09
1.E-10
1.E-11
1.E-12
1 10 100 1000
T [K]
5
To develop a feeling about properties of these cryogenic fluids, it is instructive to compare them
with those of water (Table 3). In both cases, but particularly with helium, applications operate much
closer to the critical point, i.e. in a domain where the difference between the liquid and vapour phases
is much less marked: the ratio of liquid to vapour densities and the latent heat associated with the
change of phase are much smaller. Due to the low values of its critical pressure and temperature,
helium can be used as a cryogenic coolant beyond the critical point, in the supercritical state. It is also
interesting to note that, while liquid nitrogen resembles water as concerns density and viscosity, liquid
helium is much lighter and less viscous. This latter property makes it a medium of choice for
permeating small channels inside superconducting magnet windings and thus stabilizing the
superconductor.
Table 3
Properties of helium and nitrogen compared to water
Table 4
Vaporization of liquid helium and liquid nitrogen at normal boiling point under 1 W applied heat load
Boil-off measurements constitute a practical method for measuring the heat load of a cryostat
holding a saturated cryogen bath. In steady conditions, i.e. provided the liquid level in the bath is
maintained constant, the boil-off m& vap precisely equals the vapor flow m & out escaping the cryostat,
which can be warmed up to room temperature and measured in a conventional gas flow-meter. At
decreasing liquid level though, part of the vapor will take the volume in the cryostat previously
occupied by the liquid which has vaporized, and the escaping flow will be lower than the boil-off.
More precisely, if the boil-off vapor is taken at saturation in equilibrium with the liquid
⎛ ρ ⎞
m& out = m& vap ⎜⎜1 − v ⎟⎟ < m& vap (4)
⎝ ρl ⎠
6
The escaping gas flow measured must therefore be corrected upwards to obtain the true boil-off.
From values of saturated liquid to vapor density ratios in Table 3, this correction factor is only 1.006
for nitrogen and can therefore be neglected. For helium though, it amounts to 1.16 and must clearly be
taken into account.
The term C (T - Tv) adding to Lv in the denominator brings a strong attenuation to the specific liquid
requirement, provided there is good heat exchange between the solid and the escaping vapor.
Calculated values of specific liquid cryogen requirements for iron are given in Table 5, clearly
demonstrating the interest of recovering the sensible heat of helium vapor, as well as that of pre-
cooling equipment with liquid nitrogen.
Table 5
Volume [l] of liquid cryogens required to cool down 1 kg of iron
7
lower temperature, the use of He II imposes that at least part of the cryogenic circuits operate at sub-
atmospheric pressure, thus requiring efficient compression of low-pressure vapor and creating risks of
dielectric breakdown and contamination by air in-leaks.
Thermo-physical properties of cryogenic fluids are available from tables, graphs and software
running on personal computers, a selection of which is listed in the bibliography.
10000
SOLID
1000 SUPER-
λ LINE CRITICAL
Pressure [kPa]
He II He I
CRITICAL
100 POINT
PRESSURIZED He II SATURATED He I
(Subcooled liquid)
10
VAPOUR
SATURATED He II
1
0 1 2 3 4 5 6
Temperature [K]
This equation also defines the thermal conductivity k(T) of the material, which varies with
temperature. Conduction along a solid rod of length L, cross section A spanning a temperature range
[T1, T2], e.g. the support strut of a cryogenic vessel, is then given by the integral form
T2
A
Q= (8)
L ∫ T1 k (T ) dT
Thermal conductivity integrals ∫TT2 k (T ) dT of standard materials are tabulated in the literature. A few
1
examples are given in Table 6, showing the large differences between good and bad thermal
conducting materials, the strong decrease of conductivity at low temperatures, particularly for pure
8
metals, and the interest of thermal interception to reduce conductive heat in-leak in supports. As an
example, the thermal conductivity integral of austenitic stainless steel from 80 K to vanishingly low
temperature is nine times smaller than from 290 K, hence the benefit of providing a liquid nitrogen
cooled heat sink on the supports of a liquid helium vessel. The lower thermal conductivity values of
non-metallic composites, combined with their good mechanical properties, makes them materials of
choice for low heat-inleak structural supports (Figure 9).
Table 6
Thermal conductivity integrals of selected materials [W/m]
Figure 9 Non-metallic composite support post with heat intercepts for LHC superconducting magnets
3.2 Radiation
Blackbody radiation strongly and only depends on the temperature of the emitting body, with the
maximum of the power spectrum given by Wien’s law
with Stefan-Boltzmann’s constant σ ≃ 5.67 10-8 W m-2 K-4. The dependence of the radiative heat flux
on the fourth power of temperature makes a strong plea for radiation shielding of low-temperature
vessels with one or several shields cooled by liquid nitrogen or cold helium vapor. Conversely, it
makes it very difficult to cool equipment down to low temperature by radiation only: in spite of the
2.7 K background temperature of outer space (and notwithstanding the Sun’s radiation and the Earth’s
albedo which can be avoided by proper attitude control), satellites or interplanetary probes can make
use of passive radiators to release heat only down to about 100 K, and embarked active refrigerators
are required to reach lower temperatures.
9
Technical radiating surfaces are usually described as “gray” bodies, characterized by an
emissivity ε < 1
Q = ε σ A T4 (11)
The emissivity ε strictly depends on the material, surface finish, radiation wavelength and angle of
incidence. For materials of technical interest, measured average values are found in the literature [5], a
subset of which is given in Table 7. As a general rule, emissivity decreases at low temperature, for
good electrical conductors and for polished surfaces. As Table 7 shows, a simple way to obtain this
combination of properties is to wrap cold equipment with aluminum foil. Conversely, radiative
thermal coupling requires emissivity as close as possible to that of a blackbody, which can be achieved
in practice by special paint or adequate surface treatment, e.g. anodizing of aluminum.
Table 7
Emissivity of some technical materials at low temperature
The net heat flux between two “gray” surfaces at temperature T1 and T2 is similarly given by
Q = E σ A (T14 - T24) (12)
with the emissivity factor E being a function of the emissivities ε1 and ε2 of the surfaces, of the
geometrical configuration and of the type of reflection (specular or diffuse) between the surfaces. Its
precise determination can be quite tedious, apart from the few simple geometrical cases of flat plates,
nested cylinders and nested spheres.
If an uncooled shield with the same emissivity factor E is inserted between the two surfaces, it
will “float” at temperature Ts given by the energy balance equation
Qs = E σ A (T14 - TS4) = E σ A (TS4 - T24) (13)
Solving for Ts yields the value of Qs = Q / 2: the heat flux is halved in presence of the floating shield.
More generally, if n floating shields of equal emissivity factor are inserted between the two surfaces,
the radiative heat flux is divided by n + 1.
3.3 Convection
The diversity and complexity of convection processes cannot be treated here. Fortunately, in the
majority of cases, the correlations established for fluids at higher temperature are fully applicable to
the cryogenic domain [6], and reference is made to the abundant technical literature on the subject. In
10
the case of forced convection, one should keep in mind that the high density and low viscosity of
cryogenic fluids often result in flows with high Reynolds number Re and hence strong convection. The
Nüsselt number Nü which characterizes the efficiency of convective heat transfer relative to
conduction in the fluid, is an increasing function of the Prandtl Pr and Reynolds numbers, respectively
representing the ratio of mass to heat transport, and the ratio of inertial to viscous forces
Nü = f (Pr, Re) (14)
The case of natural convection at low temperature however deserves particular mention, as this
mechanism, usually weak at room temperature except on very large scales, becomes dominant in
cryogenic equipment. In this case, the Nüsselt number is an increasing function of the Prandtl and
Grashof Gr numbers, with the latter representing the ratio of buoyancy to viscous forces
Nü = f (Pr, Gr) (15)
For gases, while Pr is about constant and independent of temperature, Gr is proportional to the heated
volume, temperature difference and coefficient of volume thermal expansion which scales as 1/T in the
ideal case. As a consequence, there may exist in helium cryostats strong natural convection processes
with Grashof numbers up to the 1012 range, i.e. higher than those encountered in the general
circulation of the Earth’s atmosphere. This has been used by hydrodynamics specialists to study
turbulent convection in extreme conditions. The cryogenic engineer sees it as a powerful mechanism
for cooling equipment and homogenizing its temperature.
Note that the thermal conductivity k(T) of the gas is independent of pressure.
When ℓ >> d at low residual pressure, the molecular regime prevails and the heat transfer
between two surfaces at temperatures T1 and T2 is given by Kennard’s law
Q = A α(T) Ω P (T2 - T1) (17)
where Ω is a parameter depending upon the gas species, and α is the “accommodation coefficient”
representing the thermalisation of molecules on the surfaces; its value depends on T1, T2, the gas
species and the geometry of the facing surfaces. Note that the conductive heat flux in molecular
regime is proportional to pressure P and independent of the spacing between the surfaces (and
therefore not amenable to the concept of thermal conductivity). Typical values of heat flux by gas
conduction at cryogenic temperature are given in Table 8.
11
Table 8
Typical values of heat flux to vanishingly low temperature between flat plates [W/m2]
Figure 10 Prefabricated MLI blankets being installed around an accelerator superconducting magnet
Of particular interest is the case of operation in degraded vacuum, where the heat in-leak by
molecular conduction is directly proportional to the residual pressure. The presence of a multilayer
system which segments the insulation space into many cells thermally in series significantly contains
the increase in heat in-leak to the low-temperature surface (Table 8). In this respect, the multilayer
system is no longer used for its radiative properties, but for the reduction of molecular gas conduction.
12
In the extreme case of complete loss of vacuum in a liquid helium vessel, MLI also efficiently limits
the heat flux which would otherwise be very high due to condensation of air on the cold wall, thus
alleviating the requirements for emergency discharge systems.
Cross-section A
.
m vapour flow
Cp(T)
T T
x
Qcon
Tbath
Qbath LHe
Figure 11 Vapour cooling of necks and supports with perfect heat exchange
Assuming steady state and perfect heat exchange between the escaping vapor and the solid, the energy
balance equation reads
dT
k (T ) A = Qv + m& C (T − Tv ) (18)
dx
where Qv is the heat reaching the liquid bath and m& is the vapor mass flow-rate. In the particular case
of self-sustained vapor cooling, i.e. when the vapor mass flow-rate m& precisely equals the boil-off
from the liquid bath,
Qv = Lv m& (19)
Combining equations (18) and (19) and integrating yields the value of Qv
A T0
k (T )
Qv = dT (20)
L ∫ Tv 1 + (T − T ) C
v
L v
The denominator of the integrand clearly acts as an attenuation term for the conduction integral.
Numerical results for helium and a few materials of technical interest appear in Table 9. If properly
used, the cooling power of the vapor brings an attenuation of one to two orders of magnitude in the
conductive heat in-leak.
13
Table 9
Attenuation of heat conduction between 290 K and 4 K by self-sustained helium vapor cooling [W/cm]
Vapor cooling can also be used for continuous interception of other heat loads than solid
conduction. In cryogenic storage and transport vessels with vapor-cooled shields, it lowers shield
temperature and thus reduces radiative heat in-leak to the liquid bath. In vapor-cooled current leads, a
large fraction of the resistive power dissipation by Joule heating is taken by the vapor flow, in order to
minimize the residual heat reaching the liquid bath [7].
A worked-out example of how these diverse thermal insulation techniques are implemented in a
real design is given in reference [8].
Q0 Qi
≥ (22)
T0 Ti
In equation (22), the equality applies to the case of reversible process. From the above
Qi
Wi ≥ T0 − Qi (23)
Ti
This expression can be written in three different ways. Introducing the reversible entropy
variation ΔSi = Qi/Ti
Wi ≥ T0 ΔSi - Qi (24a)
Another form isolates the group (T0/Ti – 1) as the proportionality factor between Qi and Wi, i.e.
the minimum specific refrigeration work
14
⎛T ⎞
Wi ≥ Qi ⎜⎜ 0 − 1⎟⎟ (24b)
⎝ Ti ⎠
As Carnot has shown in 1824, this minimum work can only be achieved through a cycle constituted of
two isothermal and two adiabatic transforms (Carnot cycle). All other thermodynamic cycles entail
higher refrigeration work for the same refrigeration duty.
A third form of equation (23) is
Wi ≥ ΔEi (24c)
This introduces the variation of exergy ΔEi = Qi (T0/Ti – 1), a thermodynamic function representing the
maximum mechanical work content (Gouy’s “énergie utilisable”) of a heat quantity Qi at temperature
Ti, given an environment at temperature T0 .
T0= 300 K
Q0
R W : mechanical
work
Qi
Ti
Equation (24b) enables to calculate the minimum mechanical power needed to extract 1 W at
4.5 K (saturated liquid helium temperature at 1.3 bar pressure, i.e. slightly above atmospheric) and
reject it at 300 K (room temperature), yielding a value of 65.7 W. This is the power that would be
absorbed by a refrigerator operating on a Carnot cycle between 4.5 K and 300 K. In practice, the best
practical cryogenic helium refrigerators have an efficiency of about 30 % with respect to a Carnot
refrigerator, hence a specific refrigeration work of about 220.
Cryogenic refrigerators are often required to provide cooling duties at several temperatures or in
several temperature ranges, e.g. for thermal shields or continuous heat interception (see paragraph 3.6
above). Equation (24b) can then be applied to the cooling duty at every temperature and every
elementary mechanical power Wi summed or integrated in the case of continuous cooling. This also
allows comparison of different cooling duties in terms of required mechanical work.
15
The heat quantities Qcondens and Qprecool exchanged at constant pressure are given by the enthalpy
variations ΔHcondens and ΔHprecool. With T0 = 300 K and the entropy and enthalpy differences taken from
thermodynamic tables, one finds Wliq = 6628 W per g/s of helium liquefied. Given the minimum
specific mechanical work of 65.7 at 4.5 K, this yields an approximate equivalence of about 100 W at
4.5 K for 1 g/s liquefaction. More precisely, a liquefier producing 1 g/s liquid helium at 4.5 K will
absorb the same power (and thus have similar size) as a refrigerator extracting about 100 W at 4.5 K,
provided they both have the same efficiency with respect to the Carnot cycle. For machines with
mixed refrigeration and liquefaction duties, this equivalence can be approximately verified by trading
some liquefaction against refrigeration around the design point and vice versa.
Compressor Compressor
HP LP HP LP
T0= 300 K T0= 300 K
R
4.5 K 4.5 K
Q1 Q1
S S
4.2 23.1
4.2 J.g-1.K-1 J.g-1.K-1 J.g-1.K-1
(a) (b)
Figure 13 Helium refrigerator (a) vs. liquefier (b)
16
Figure 14 A hypothetical Carnot cycle for helium liquefaction
17
T
P1
A P2 (< P1)
B3
B1 H
isenthalpic
(Joule-Thomson valve)
B'2
isobar
(heat exchanger) B2
adiabatic (expansion engine)
isentropic
S
18
Figure 17 Cryogenic turbo-expander (source Linde)
These elementary cooling processes are combined in practical cycles, a common example for
helium refrigeration is provided by the Claude cycle and its refinements. A schematic two-pressure,
two-stage Claude cycle is shown in Figure 18: gaseous helium, compressed to HP in a lubricated
screw compressor, is re-cooled to room temperature in water-coolers, dried and purified from oil
aerosols down to the ppm level, before being sent to the HP side of the heat exchange line where it is
refrigerated by heat exchange with the counter-flow of cold gas returning on the LP side. Part of the
flow is tapped from the HP line and expanded in the turbines before escaping to the LP line. At the
bottom of the heat exchange line, the remaining HP flow is expanded in a Joule-Thomson valve and
partially liquefied.
19
LogT
1 stage
compression
heat rejection
2 stages
compression
ambient temperature heat rejection ambient
temperature
heat exchange
heat exchange
polytropic
heat expansion
exchanger polytropic
expansion heat exchange
levels
heat exchange
heat exchange
isenthalpic
expansion
isenthalpic
expansion
heat injection
two-phase area heat injection s
Figure 18 Schematic example of two-pressure, two-stage Claude cycle: T-S diagram (left) and flow scheme (right)
Large-capacity helium refrigerators and liquefiers operate under this principle, however with
many refinements aiming at meeting specific cooling duties and improving efficiency and flexibility
of operation, such as three- and sometimes four-pressure cycles, liquid nitrogen pre-cooling of the
helium stream, numerous heat exchangers, many turbines in series or parallel arrangements, Joule-
Thomson expansion replaced by adiabatic expansion in a “wet” turbine, cold compressors to lower the
refrigeration temperature below 4.5 K. A view of such a large plant appears in Figure 19.
The capital cost of these complex machines is high, but scales less than linearly with
refrigeration power, which favors large units. Operating costs are dominated by that of electrical
energy, typically amounting to about ten percent of the capital cost per year in case of quasi-
continuous operation. For overall economy, it is therefore very important to seek high efficiency,
which is also easier to achieve on large units. For a review of these aspects, see reference [10].
5. CONCLUSION
This brief paper has presented the basic ideas and principles of the most important aspects of
cryogenics, i.e. cryogenic fluids, heat transfer, thermal design and refrigeration. It has also provided
the reader with typical numerical values of the relevant parameters, enabling him to perform order-of-
magnitude estimates and apply his engineering judgement. There is of course much more to say on
each of these topics, some of which have significantly developed over the years and still constitute
areas of technical progress. Many other subjects not addressed here also pertain to cryogenic
engineering, such as materials at low temperature, storage, handling and transfer of fluids, two-phase
flow and discharge, vacuum and leak-tightness technology, instrumentation (in particular
thermometry), process control, impurity control and safety. In all cases, the interested reader is
referred to the selected bibliography for detailed information and to the proceedings of the cryogenic
engineering conferences for recent developments.
20
a) Compressor station
b) Coldbox
21
ACKNOWLEDGEMENTS
The substance and the form of this course owe much to discussions with many colleagues who have
provided useful inputs, particularly L. Tavian at CERN. While equations, tables and figures were
carefully integrated in the text by E. Delucinge, any error which may remain in the document is
evidently mine.
SELECTED BIBLIOGRAPHY
Conference proceedings
Proceedings of the Cryogenic Engineering Conferences (CEC) and International Cryogenic Material
Conferences (ICMC), in Advances in Cryogenic Engineering (multiple publishers).
Proceedings of the International Cryogenic Engineering Conferences (ICEC) (multiple publishers).
22
S. Turner (ed.), Proceedings of the CAS “Superconductivity in particle accelerators”, Hamburg,
CERN 96-03, Geneva (1996).
S. Russenschuck & G. Vandoni (eds.), Proceedings of the CAS “Superconductivity and cryogenics for
accelerators and detectors”, Erice, CERN-2004-008, Geneva (2004).
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(2004) 375.
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[5] W. Obert et al., Emissivity measurements of metallic surfaces used in cryogenic applications,
Adv. Cryo. Eng. 27 (1982) 293.
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23