Heat Transfer
G.Vandoni
CERN, AT Division
G.Vandoni, Heat Transfer
Academic Training 2005
1
A detour in basic thermodynamics
A refrigerator extracts heat at a temperature T below ambient
and rejects it at a Tambient.
Second law of thermodynamics :
(Ta  T )
W Q
T
W refrigeration work
Q heat to extract at T
and reject at Ta
Minimize thermal loads:
boundary temperatures fixed, heat transfer rate
minimization seeked
Maximize heat extraction:
heat transfer rate fixed, minimize temperature difference
G.Vandoni, Heat Transfer
Academic Training 2005
2
The 3 modes of heat transfer
Conduction: heat transported in solids or fluids at rest

Q  k (T ) A grad T
FOURIER’s law:
Convection: heat transport produced by flow of fluid

Q  hA(Tw  T f )
Convection exchange:

Radiation: heat carried by electromagnetic radiation
Stefan-Boltzmann’s law:
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Q  A(Th4  Tc4 )
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3
Electrical analogy



Valid in the three cases for a small DT (linearization of
Stefan-Boltzmann’s law)
series/parallel impedances
Basis for modelling and numerization above 1D
T1
Q
T2
S
Qk
(T2  T1 )
l
1 l
Rth 
k S
V1
I
I
1
Relec
V2
S
(V2  V1 )
l
l

S
thermal
impedance
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Academic Training 2005
4
Cryogenic heat transfer modes
PeakNucleate
BoilingFlux
increase of Re for
decreasing T
increase of Gr for
decreasing T, h~T-1/2
k~T0.7
T3
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5
Time-independent conduction
 x2 dx 

G   
x1 A( x ) 


1D, constant A
dT

Q  k (T ) A
dx

Th
Tc
Th
1
A Th

Q   k (T )dT
L Tc
Tt
L
A
k (T )dT  material' s property
Tc
300
Heat flux reduction by intermediate
temperature thermalization:
250
200
150
Temperature profile T(x) of st.steel bar
with thermalization 2/3 of length at 80K
100
50
0
0
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0.2
0.4
0.6
0.8
1
6
Intermediate heat interception
Stainless steel
T
Th
pure Copper
300 K
T
L
A
77 K
4K
Tt
x
x
Tc
Purely conductive T(x) profile over the whole length
Thermalization (=fixing the temperature) at Tt
Larger Q evacuated at Tt, but smaller at Tc =>
optimization possible with exergy function
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7
Thermal conductivity integrals
Tc=4 K

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Reduction of heat flow to the cold boundary temperature
by thermal interception at intermediate temperature
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8
Time-dependent conduction
Energy conservation
difference
between heat
entering and
leaving dv
  T  
T
k
  Q h  C
x  x 
t
internal
heat source
density
dv
rate of temperature
increase
(thermal inertia)
Diffusivity D=k/C characterizes the propagation of a thermal
transient…
ro2
~
D
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…through a characteristic time depending on
the object’s dimension ro
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9
Diffusivity and time regimes
T
ro2

D
x
x
late regime
x
early regime
ro
Late regime:
exponential decay
ro2
t 
D
ro2
t 
D
T (t )  T  (To  T ) exp(  t )
 = hS/( C V) time constant of the
system
0
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2
4
6
8
t
10
10
Internal versus external resistance
Under some circumstances, the decay is
exponential starting from t=0
Q surf
A
 hDTs
surface
thermal
resistance
Qvol kDTvol

A
ro
T
x
internal
thermal
resistance
x
hro
 1
Biot number: Bi 
k
Lumped capacitance model
applies starting t=0
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Exponential
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ro
T (t ) T
11
Conductivity of solids
-> form for pure and alloyed metals
-> st.steel
-> increase with T
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12
Conductivity of solids

Heat carriers: phonons (k~T3) and electrons (k~T)

Good electrical conductors = good thermal conductors

Hinder heat transmission at low T ?
(but not the best ones !)
DEFECTS
difference between pure and alloyed
effect of modification of the defect content: magnetic impurities,
annealing, cold work

Hinder heat transmission at high T ? Phonon-phonon
Phonon-electron
no difference between pure and alloyed metals
T behaviour well known
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13
Metal’s conductivity
 (T )k (T )  LLorentzT
k (T )  k ( RRR, T )
Wiedemann-Franz:
free-electron metal
RRR parametrization
(next slide)
Superconductor’s conductivity
Electronic above Tc, phononic below Tc:
Pb:
knormal/ksupra=45/T2 In : knormal/ksupra=1/T2
=> Thermally switch between conducting and isolating by
applying a magnetic field>critical field…
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14
RRR parametrization of k(T)
K (W / m.K )  (W0  Wi  Wio ) 1
Thermal Conductivity of copper
k [W/m K]
10000
RRR 
RRR=80
RRR=120
RRR=180
r 
0.0003
0.634
 
RRR
W0 
RRR=230
1000
 (273 K )
 (4 K )


T
P1T P 2
Wi 
1  P1P 3T ( P 2 P 4 ) e
W *W i
Wi 0  P 7 0
W0  W i
100
1
10
T [K]
100
1000
Valid over a broad range of RRR, ~10% exactness

r
P1
P2
P3
P4
P5
P6
P7
 P5 


 T 
P6
0.634 / RRR
 / 0.0003
1.7541E-08
2.763
1.1020E+03
-0.165
70
1.756
0.838 / r0.1661
A similar parametrization also available for (RRR,T)
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15
Diffusivity of common materials
Cv(T) decreases
faster than k(T):
small equilibration
times at low T
D=k/Cv
Diffusivity larger
for conductors
than insulators
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16
Specific heat of structural materials
Cv heat capacity per kg mole
approximately described by the
Debye function
T
C v  9 R
q D



3
q /T

0
x 4e x
dx
x
2
(e  1)
qD Debye temperature, a
material’s property
Nb: Tc/qD=0.04
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17
Conductivity of gases: 2 regimes
mean free path
L
vs
wall distance L
  115 
molecular:

p
T
M
[Pa.s],[Pa],[cm]
viscous:
q proportional to p
q independent from p
q independent from L
q=kSDT/L
k predicted by kinetic
theory of gases
P [Pa]
10-2
100
Ar
0.63 cm
6.3 10-5
N2
1.8
1.8 10-4
He
0.60
6.0 10-5
1
2
1  8 RT 
k  
 C v
3  M 
k~T0.7
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18
Viscous regime
Thermal conductivity k [Wcm-1 K-1] @ 1 atm
T [K]
4He
H2
N2
300
80
20
1.56 10-3
0.64 10-3
0.26 10-3
1.92 10-3
0.6 10-3
0.16 10-3
2.60 10-4
0.76 10-4
5
0.10 10-3
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Academic Training 2005
19
Molecular regime: Kennard’s law
   1  R 

 
Q  A1 
   1  8 
Cp/Cv
1/ 2
p
MT
T2  T1 
R ideal gas constant
 accomodation
coefficient
Q [mW/cm2]
01, degree of thermal equilibrium between molecules and
wall, ~0.7-1 for heavy gases.
1.E+00
H2
He
1.E-01
 1 2

N2
A
 2   1 (1   2 ) 1
1.E-02
A2
for simple geometries,
(parallel plates, coaxial
cylinders, spheres)
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1.E-03
1.E-06
1.E-05
300K->77K
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1.E-04
p [m m Hg]
20
Contact resistance
DT
RH 
Q / AC

- phonon scattering (Kapitza)
- spot-like contact points
Features:







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Temperature discontinuity at
the interface:
Proportional to FORCE, not to pressure (constant
spot area, number of contact points increases
with force)
For metals, saturates above 30N @ 300K
Hysteresis upon loading cycles (plastic
deformations)
Can be reduced by fillers, grease, In, coatings
For el. conductors, Rh~Rel
Rh-1=Kh increases with T then saturates
Approximately proportional to microhardness/k
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21
Contact resistances
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22
Thermal switches
SCOPE:
Good thermal contact for cooldown
BUT
Thermal insulation once cold
REALIZATION:




heat
sink
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Exchanger gas: long time for evacuation
Gas heat exchanger: short time for evacuation
Superconducting switch (Pb or In)
Polycristalline graphite: k~T3 up to 100K
device
Switch from normal (thermally
conducting) to superconducting
(thermally insulating) with
applied magnetic field
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23
RADIATION
Any surface T>0K absorbs () and emits () energy as
electromagnetic radiation:
depending on direction and wavelength
incident P
reflected P
transmitted P
The whole incident
radiation is absorbed: =1
absorbed P
Energy conservation
1
Opaque medium
1
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BLACK-BODY:
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24
Black-body radiation
Planck’s law for energy flux
emitted by a cavity [W/cm3]
E b ,
C1 5

exp( C 2 / T )  1
Wien’s law
 max 
2898
[ m / K ]
T
300 K
10 m
80 K
36 m
Integral over :
Stefan-Boltzmann’s law
for black body
5500 K 0.4-0.7 m
(sun)
(visible)
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q  T 4
=5.67 10-8 W m-2
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25
Heat exchange between two black surfaces
4
4

Q  A1F12 (T2  T1 )
A1, T1, A2, T2
Geometrical FORM FACTOR F12
F12=
(radiation leaving A1 intercepted by A2) /
(radiation leaving A1 in all directions)
=
integral of solid angle under which A1 sees A2
F12 tabulated for several useful geometries
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26
From a blackbody to a real body
Definition of (total hemispherical) emissivity   1:
black-body
q  T 4
q  T 4
real-body
Monochromatic directional emissivity   ( , T , ) 
APPROXIMATIONS
I  ( , T , )
I b ,
grey-body
diffuse-body
( independent of )
( independent of q)
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27
Kirchoff’s law
From energy conservation in a cavity:
For black-body and diffuse grey body: (T) =  (T)
Practical use:
 can be estimated from  provided the incident radiation
and the surface have the same temperature
In reality, (,q,T) ≠  (,q,T)
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28
Electrical analogy for real (diffuse/grey) surfaces
q12
T14
T24
q1
q2
1  1
A1 1
internal resistance
of the surface to
black-body emission
motor
q12 
flux
1
A1F12
1 2
A2 2
resistance between
two blackbodies
 T14  T24 
1  1
1
1 2


A1 1 A1F12 A2 2
Blackbody form
factors can be used
for real diffusegrey surfaces
total thermal impedance
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29
Heat transfer between 2 real surfaces
q  A112 (T24  T14 )
12
effective emissivity
(emissivities + view factor)
 1 2
 2  (1   2 ) 1
Parallel plates
Spheres and long cylinders
self-contained, not
concentrical/coaxial
(A1<A2)
 1 2
2 
A1
(1   2 ) 1
A2
A2>>A1 equivalent to A2 black: black-body radiation
fills the cavity between the two surfaces and is
collected by A1 proportionnally to 1
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30
Emissivity and materials
Real emissivities depend on direction and wavelength




Drude law for
ideal metal
 T 
  , T   0.365

Polished metals: small 
Insulators: large 
  (T): for real metals, ~T at low T
Coatings: since  related to surface, not bulk, resistance, =>
lower limit on thickness of reflectors (1 above ~40nm)
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31
Emissivity and materials –2-
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32
Radiative heat transfer in cryogenics
4
4
q   (Twarm
 Tcold )
Blackbody radiation from 290 K to 80 K:
399 W/m2
Blackbody radiation from 290 K to 4.2 K :
401 W/m2
negligible effect of Tcold
Blackbody radiation from 290 K to 4.2 K:
401 W/m2
Blackbody radiation from 80 K to 4.2 K :
2.3 W/m2
reduction of heat flux by one cooled screen
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33
Floating radiation screens
Floating =
not actively cooled, they operate at a temperature determined
by heat balance
Tw
Tc
   4
4
(Tw  Tc )
 q  
 2 

T
1    4
4
q  
(Tw  Tc )
2  2 
1    4
4
q 

(Tw  Tc )
n 1  2   
1 4
T  (Tw  Tc4 )
2
4
4
4
T

T
4
c
Ti  Tc4  w
i 1
n
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34
Multi-layer Insulation
Stacking of “reflectors” separated by insulating “spacers”
reflector
spacer
blanket
Reflector: low emittance
radiation shield
polyester film, 300-400 A pure Al
coating, usually double face
Spacer: insulating, lightweight
material
paper, silk, polyester net
1. Heat transfer parallel to the layers ~1000 times greater than
normal to the layers
thermal coupling between blanket edges and construction elements may
dominate heat rate.
2. Heat transfer very sensitive to layer density
single local compression affects the T profile over the entire blanket,
substantially degradating heat loss (factors 2-3 more !)
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35
Effective conductivity
k=aT+ bT3
Heat transfer rate
q=k/e DT, e = thickness
W/m2
MLI: effective conductivity
Optimal density: 10-20 cm-1
layers/cm
Low boundary
temperature:
77 K-> 4K
High boundary
temperature:
300 K-> 77K
G.Vandoni, Heat Transfer
heat transfer rate determined by aT, not by
radiation
1 single aluminized foil is sufficient in high vacuum
in bad vacuum, MLI provides sufficient
insulation
heat transfer rate determined by radiation
important reduction with layer’s number
bad vacuum: radiation dominates anyway
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MLI: number of layers
30 layers,
300K-> 77K,
0.5 W/m2
10 layers,
77K-> 4K,
20 mW/m2
N = 15 cm-1
Tc= 4.2K,  = 0.03
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37
MLI and residual pressure
MLI constitutes a supplementary
protection against vacuum rupture, only
at low boundary temperature: at high
boundary temperature, radiation
dominates anyway
interstitial
gas:
nitrogen
300 K -> 77 K
77 K -> 4.2 K
300 K -> 77 K
77 K -> 4.2 K
300 K -> 77 K
77 K -> 4.2 K
300 K -> 77 K
77 K -> 4.2 K
Kennard’s law
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38
Passive cooling by radiators

Radiation cooling to a cold screen -> cool down without contact

Requires large surface-to-volume ratio + large emissivity

Black silicon paints compatible with high vacuum from the
space industry (cooling of CERN antiproton collector’s mobile
electrodes)
Cooling in space applications
towards the cosmic background
radiation at 2.7K
Figure: the NGST (next generation
space telescope) solar screen
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39
Free and forced CONVECTION
Q / A  q  h(Ts  T f )
Q transferred heat, A surface area
h:
heat transfer coefficient, function of
fluid properties, flow velocity and
channel geometry
Scope:
determine h
Tf
Ts
Analysis: dimensionless groups, EMPIRICAL correlations
Free (natural) convection : the fluid movement is due to expansion
upon heating, reduction of density and buoyancy (kettle, fireplace)
Forced convection: the fluid is set into movement by external action
(pressure difference, mechanical action, elevation difference)
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40
Convection exchange coefficient
Boiling, water
q  h(Ts  T f )
Boiling organic liquids
Convective heat transfer in
cryogenic fluids not different from
any other, except He II
Condensation, water vapors
Condensation, organic vapors
Liquid metals, forced convection
Boiling HeI, N2 peak nucleate flux
(PNBF):
104 W/m2K
Water, forced convection
Organic liquids, forced convection
Gases 200atm, forced convection
Gases 1atm, forced convection
Gases, natural convection
1
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10
102
103
h (W/m2K)
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104
105
106
41
Dimensionless groups
Group
Name
Re
Reynolds
Pr
flow character
Prandtl
fluid
characteristics
Nu
Nusselt
Definition
Physical interpretation
Vd / 
inertia force/viscous force
C p / k
momentum transport/
thermal diffusivity
hd / k
convection exchange/
conduction exchange
defines convection exchange
Gr
Grashof
Ra
Rayleigh
gDTd 3  2 /  2
buoyancy force/viscous force
like Re for free convection
Gr Pr
d=characteristic dimension, ex. tube diameter or hydraulic diameter, =dynamic
viscosity, =volume expansivity, k=thermal conductivity, DT=temperature difference,
=density, Cp=specific heat at constant pressure, h=heat transfer coeff , g= gravity
acceleration
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Reynolds number and flow character
Re 
Vd

 density, V fluid average velocity, d
hydraulic diameter,  dynamic viscosity
Inertia forces compared to viscous forces
Viscous forces are
stabilizing:
Inertial forces are
de-stabilizing:
laminar flow
turbulent flow
Laminar:
low heat transfer coefficient
Turbulent:
high heat transfer coefficient
In free convection, Gr plays the role of Re: buoyancy versus viscosity
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43
Free (natural) convection
General relation
Nu = function(Gr,Pr)
Nu = a (Gr .Pr) n= a . Ra
n
Configuration
regime
limits
a
n
vertical, free surface
laminar
turbulent
5.103<Ra<109
109<Ra<1013
0.59
0.13
¼
1/3
Ra<103
1.18
1/8
laminar
103<Ra<2.107
0.54
¼
turbulent
2.107<Ra<1013
0.14
1/3
Empirical form
horizontal, free surface
d to be used to calculate: diameter (horizontal cylinder), height (vertical
plates/cylinders), smallest exchange dimension (horizontal plates), distance between
walls (enclosures)
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44
Free convection in gases and air
common gases:
h~p½,
important increase at
low temperature
h~T-½.
cold helium gas (80K, 1 bar): Nu~3.65 (laminar)
Air close to ambient conditions
vertical plates
 DT 
h  1.4 

d


horizontal
plates
 DT 
h  1 .3 

 d 
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1/ 4
Watt m-2 K-1
1/ 4
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Watt m-2 K-1
45
1phase forced convection
Empirical relation
Nu = f (Re,Pr) = aF Rem Prn
Configuration
regime
limit
a
m
n
F
horizontal
plate
laminar
103<Re<105
0.66
½
1/3
1
turbulent
3 105<Re
0.036
0.8
1/3
1
103<Re<2.1 103
Re Pr D/L >10
1.86
1/3
1/3
(D/L)1/3
Re Pr D/L >10
RePrD/L>2.4 105
0.023
0.8
0.33 1+(D/L)0.7
horizontal tube laminar
annular space
turbulent
Colburn formula
Sieder & Tate formula
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46
Steps to solve a convection problem
1.
Calulate Re to determine flow character: laminar/turbulent
hydraulic calculation of pressure drop dp v 2 4 f

f=Fanning, function of Re
dx
2 d
2.
Evaluate Pr (fluid characteristics)
3.
4.
Choose the appropriate formula for Nu -> h
In doubt about the importance of free convection:
calculate Gr
G.Vandoni, Heat Transfer
Academic Training 2005
47
Boiling heat transfer in He I
Increase of heat transfer up to a
Peak Nucleate Boiling Flux:
He I: 1 W cm-2 @ 1K superheat
N2 :
10 W cm-2 @ 10K
H2O: 100 W cm-2 @ 30K
Hysteresis: cooling path not the
same as warming path
Positive consequence for safety:
limit to the highest flux
released by a warm object
(quenching magnet, human skin)
G.Vandoni, Heat Transfer
Academic Training 2005
48
Two-phase convection
heat transfer = bubble formation and motion near the walls +
direct sweeping of the heated surface by the fluid
Instabilities of density-wave type:
pressure waves increase locally the heat transfer rate, the fluid
expands
=> decrease in conductivity and heat transfer rate
How to avoid them:
-Maintain
low vapor quality
-Not too large differences in elevation
-No downstream flow restrictions
-Introduce upstream flow restrictions
G.Vandoni, Heat Transfer
Academic Training 2005
-> destabilizing
-> stabilizing
49
Refrigeration properties of cryogens
Working domain
close to critical
point:
properties of
liquid and vapor
phase are similar
low vaporization
heat
Low viscosity
hence excellent
leaktightness
required for He
He
N2
H2O
Normal boiling point
4.2
77
373
Critical temperature
5.2
126
647
Critical pressure
2.3
34
221
Liquid density/ Vapor
density*
Heat of vaporization *
7.4
175
1600
[Jg-1]
20.4
199
2260
Liquid viscosity *
[poise]
3.2
152
283
Enthalpy increase
between T1 and T2
T1 = 4.2 K
T2 = 77 K
384
-
-
T1 = 4.2 K
1157
-
T2 = 300 K
*at normal
boiling point
G.Vandoni, Heat Transfer
228
highly effective for selfsustained vapor cooling!
Academic Training 2005
50
Shielding potential of cold vapours
Pure conduction heat
losses evacuated at the
coldest temperature
Self sustained vapour cooling: vapour
flow generated only by heat leak is
used to cool the device
Th
L
A
Tc
A T2

Q   k (T )dT
L T1
G.Vandoni, Heat Transfer
Heat evacuation across a small DT
thermodynamically much more
efficient
Academic Training 2005
51
Shielding potential of cold vapours
heat balance, perfect exchange
k (T ) A
dT
Q  k (T ) A
 m C p (T  Tl )
dx
dT
 m C p (T  Tl )
dx
self-sustained evaporation of fluid
Q  m L
v
Tl
A T2
k (T )dT

Q 
L Tl 1  (T  Tl )C p / Lv
Q
A T2

Q   k (T )dT
L l
Th.conductivity integral
[W cm-1]
[W cm-1]
ETP copper
OFHC copper
128
110
1620
1520
Aluminium 1100
39.9
728
AISI 300 st.steel
0.92
30.6
G.Vandoni, Heat Transfer
Academic Training 2005
52
Phase diagram of helium
FORCED FLOW
small inventory, no instability
bi-variant, high p, JT heating
POOL BOILING
constant T, irrespective of q
PNBF, large quantities of cryogen
FORCED FLOW
JT cooling, good heat transfer,
small liquid inventory
flow instabilities, small (p,T) range
high HT, low T, large Cv
refrigeration cost, sub-atm pipes
high HT, low T
dielectric breakdown, sub atm,
large gas volume
G.Vandoni, Heat Transfer
Academic Training 2005
53
Typical heat inleaks in a cryostat
…between flat plates, at vanishingly low temperature
[W/m2]
Black-body radiation from 290 K
400
Black-body radiation from 80 K
2.3
Residual gas conduction (100mPa helium) from 290 K
19
Residual gas conduction (1mPa helium) from 290 K
0.19
Multi-layer insulation (30 layers) from 290 K, residual pressure below
1mPa
Multi-layer insulation (10 layers) from 80 K, residual pressure below 1mPa
0.5-1.5
Multi-layer insulation (10 layers) from 80 K, residual pressure 100mPa
0.2
0.05
Radiation screening
Insulation vacuum
MLI at high (>30 layers) and low (10 layers) boundary temperature
G.Vandoni, Heat Transfer
Academic Training 2005
54
Heat inleaks in an accelerator
Cryostat heat inleak
Resistive dissipation
Beam-induced losses
radiation to cold surface
superconductor splices
synchrotron radiation
cold mass supports
wall resistance
beam-image currents
warm-to-cold
feedthroughs
instrumentation
beam-gas inelastic
scattering
AC losses
beam losses
G.Vandoni, Heat Transfer
Academic Training 2005
55