Turbine Expanderexpander
CryogenicsCryogenics –– whywhy??
MaciejMaciej ChorowskiChorowski WroclawWroclaw UniversityUniversity ofof TechnologyTechnology FacultyFaculty ofof MechanicalMechanical andand PowerPower EngineeringEngineering
European Cryogenic Course Wroclaw 20 - 25 April, 2009 T, K
10 10 The word cryogenics was introduced by Core of the hottest stars 9 Kamerlingh Onnes and is formed from the 10 8 Greek: 10 Fusion reaction of hydrogen
7 10 Core of the Sun
6 – cold 10
5 10 – generated from TEMPERATUREVERY HIGH Plasma
4 10 Surface of the Sun
3 According to the convention adopted at the 10 Steam turbine Biological processes XIIII Congress of the International Institute of 2 10 High temperature superconductivity Boiling temperature of nitrogen Refrigeration, cryogenics treats concepts and Low temperature superconductivity 10 technologies connected to reaching and Boiling temperature of helium Superfluid helium 4 applying temperature below 120 K. 1 -1 In cryogenic temperatures: 10 -2 10
-3 - new physical phenomena are visible (liquefaction 10 Superfluid helium 3
-4 of gases, superfluidity, superconductivity); 10
-5 - all the reactions are slowed down; 10 The lowest measured temperature in the whole volume of a probe
-6 - dis-order in the matter is vanishing, noises are 10 VERY LOW TEMPERATUREVERY LOW
-7 avoided (cryo-electronics). 10 The lowest temperaure of copper nuclei
-8 European Cryogenic Course 10
CERN Geneva 2010 -9 10 Bose-Einstein condensate HistoricalHistorical developmentdevelopment ofof cryogenicscryogenics andand relatedrelated technologiestechnologies 1883 Karol Olszewski and Zygmunt Wróblewski liquefy air and its components (77 K) – Cracow, Poland 1895 Carl Linde starts LINDE AG 1898 James Dewar invents vacuum insulation and liquefies hydrogen (20,3 K), UK 1902 Georges Claudet starts L’Air Liquide 1908 Kammerlingh Onnes liquefies helium (4,2 K) and discovers superconductivity in mercury in 1911, Leiden, Holand 1950 Collins starts a serial production of helium liquefiers, USA 1986 Bednorz i Mueller discover high temperature superconductivity (the highest Tc is now of 135 K), Zurich, Switzerland 2008 Start of the ITER construction – the biggest concentrated cryogenic system 2009 Start of the LHC superconducting accelerator at CERN, Geneva (over 2000 of superconducting magnets of total length 28 km), 105 tons of He, Geneva, CERN
European Cryogenic Course CERN Geneva 2010 KarolKarol OlszewskiOlszewski andand ZygmuntZygmunt WrWróóblewskiblewski airair,, nitrogennitrogen,, oxygenoxygen liquefactionliquefaction inin 18831883
1846-1915 1845-1888
Jagiellonian University, Cracow
European Cryogenic Course CERN Geneva 2010 TechnicalTechnical gasesgases industryindustry hashas decideddecided abotabot thethe metallurgymetallurgy developmentdevelopment –– overover 100100 yearsyears ofof traditiontradition
Georges Claudet – co-founder of the L’Air Liquide in 1902
Carl von Linde – Linde AG founder in 1895
European Cryogenic Course CERN Geneva 2010 JamesJames DewarDewar andand hishis „„vacuumvacuum flasksflasks”” 1898,1898, vacuumvacuum insulationinsulation,, hydrogenhydrogen liquefactionliquefaction
1842-1923
European Cryogenic Course CERN Geneva 2010 ModernModern cryogeniccryogenic insulationinsulation systemssystems –– developmentdevelopment ofof thethe DewarDewar conceptconcept Zbiornik zewn trzny
Ciek y Przestrze gaz pró niowa
Zbiornik wewn trzny Vacuum insulation
A A
Multilayer Vacuum Insulation (MLI)
European Cryogenic Course Vacuum Insulation with Glass Spheres CERN Geneva 2010 HeikeHeike KamerlinghKamerlingh--OnnesOnnes andand hishis „„cryocryo--industryindustry”” heliumhelium liquefactionliquefaction 19081908
1853-1926
European Cryogenic Course CERN Geneva 2010 HeikeHeike KamerlinghKamerlingh OnnesOnnes DiscoveryDiscovery ofof superconductivitysuperconductivity inin mercurymercury 19111911 r.r.
European Cryogenic Course CERN Geneva 2010 CollinsCollins liquefiersliquefiers –– heliumhelium availableavailable inin laboratorieslaboratories fromfrom 19501950
Capacity of 4 – 8 LHe liter per hour
European Cryogenic Course CERN Geneva 2010 ComparisonComparison ofof NbTiNbTi superconductingsuperconducting wirewire withwith coppercopper rodsrods
Phase diagram NbTi
European Cryogenic Course CERN Geneva 2010 CryogenicsCryogenics inin LargeLarge HadronHadron ColliderCollider acceleratoraccelerator,, CERNCERN GenevaGeneva
Over 2000 of superconducting magnets of overall length exceeding 27 km Helium inventory above 100 t Temperature 1,8 K Energy 14 TeV Particles: protons
European Cryogenic Course CERN Geneva 2010 BednorzBednorz andand MuellerMueller highhigh temperaturetemperature superconductivitysuperconductivity discovereddiscovered inin 19861986
The Nobel Prize in Physics 1987 „It is too early to predict how extensive the technical applications will be, but it is quite evident that the development is being followed with keen interest by representatives of electrical power technology, by microelectronics researchers and by physicists who envisage new applications in measurement technology” Form the Nobel Prize justification IBM Zurich Research Laboratory Rüschlikon, Switzerland European Cryogenic Course CERN Geneva 2010 CriticalCritical temperaturestemperatures ofof HTSHTS superconductorssuperconductors
Group of superconductors Superconductor Critical temperature, K
YBCO YBa2Cu3Ox (Y-123) 92 BSCCO Bi2Sr2CaCu2Ox (Bi-2212) 85 Bi2Sr2Ca2Cu3Ox (Bi-2223) 110 TBCCO TlBa2Ca2CuOx (Tl-1221) 122 Tl2Ba2Ca2Cu3Ox (Tl-2223) 125 HBCCO HgBa2CuOx (Hg-1201) 94 HgBa2CaCu2Ox (Hg-1212) 117 HgBa2Ca2Cu3Ox (Hg-1223) 135 HTS superconductors can be cooled with LN2, cheap and easily available
European Cryogenic Course CERN Geneva 2010 StructureStructure ofof B2223B2223 superconductorsuperconductor
European Cryogenic Course CERN Geneva 2010 HTSHTS prototypeprototype cablescables
European Cryogenic Course CERN Geneva 2010 MeissnerMeissner effectseffects,, discovereddiscovered inin 19331933 The Meissner effect (also known as the Meissner-Ochsenfeld effect) is the expulsion of a magnetic field from a superconductor. Walther Meissner and Robert Ochsenfeld found that below the superconducting transition temperature the specimens became perfectly diamagnetic, cancelling all flux inside. The experiment demonstrated for the first time that superconductors were more than just perfect conductors and provided a uniquely defining property of the superconducting state.
Diagram of the Meissner effect. Magnetic field lines, represented as arrows, are excluded from a superconductor when it is below its critical temperature. A magnet levitating above a superconductor European Cryogenic Course (cooled by liquid nitrogen). CERN Geneva 2010 MAGLEVMAGLEV traintrain ShanghaiShanghai
European Cryogenic Course CERN Geneva 2010 OtherOther applicationsapplications ofof cryogenicscryogenics
TechnicalTechnical gasesgases industryindustry (air(air rectification,rectification, liquefaction,liquefaction, transporttransport andand storagestorage ofof gases)gases) FoodFood freezingfreezing andand storagestorage Medicine,Medicine, storagestorage ofof biologicalbiological samplessamples RocketRocket industryindustry (LH2)(LH2) PowerPower generationgeneration –– (LNG),(LNG), CO2CO2 capturecapture andand storagestorage BigBig scientificscientific facilitiesfacilities
European Cryogenic Course CERN Geneva 2010 AirAir rectificationrectification Air is a mixture of: nitrogen 78.1%, oxygen 20.9%, argon 0.93%, carbon dioxide 0.03%, neon 1.8E-3%, helium 5.2E-4%, hydrocarbons 3.5E-4%, krypton 1.1E-4%, hydrogen 5.0E- 5%, xenon 8.0E-6, ozone 1.0E-6%, radon 6.0E-18%. The main product of air separtaion are: nitrogen, oxygen, argon, neon, krypton and xenon
European Cryogenic Course CERN Geneva 2010 CryogenicCryogenic foodfood freezingfreezing
European Cryogenic Course CERN Geneva 2010 CryomedicineCryomedicine
CRYOMEDICINE
CRYOTHERAPY CRYOSURGERY
CRYOSTIMULATION TISSUE NECROSIS (to activate defensive reactions) (to destroy pathological cells)
DIRECT LOCAL WHOLE BODY SPRAY CONTACT EVAPORATING
European Cryogenic Course CERN Geneva 2010 SimulationSimulation ofof cryosurgicalcryosurgical treatmenttreatment
Cryosurgery is aimed at the tissue necrosis, cryosurgical apparatuses are supplied with LN2 or N2O
Time: 0 s 10 s 20 s 40 s
Time: 80 s 100 s 160 s 220 s
European Cryogenic Course CERN Geneva 2010 CryomedicineCryomedicine -- cryochambercryochamber
WST PNE FILTRY POWIETRZA
WYMIENNIK CIEP A FILTR POWIETRZA OSUSZACZ KRIOOCZYSZCZALNIKI POWIETRZA
N WYMIENNIK CIEP A 2 PRZEDSIONEK KOMORA W A CIWA 210 K (- 60 C) 110 K / 150 K (- 120 C / - 160 C)
ZBIORNIK CIEK EGO AZOTU
SPR ARKA Azot POWIETRZA Powietrze
European Cryogenic Course Komora CR-2002 CREATOR Wroc aw CERN Geneva 2010 CryoCryo--chamberchamber entranceentrance
European Cryogenic Course CERN Geneva 2010 CryomedicineCryomedicine inin PolandPoland nownow meansmeans overover thousandthousand cryogeniccryogenic installationsinstallations ofof differentdifferent scalescale andand yearlyyearly consumptionconsumption ofof liquiliquidd gasesgases aboveabove 5000 Mg.
AA secondarysecondary effecteffect isis aa vastvast propagationpropagation ofof thethe useuse ofof liquidliquid gagasesses andand cryogeniccryogenic cultureculture inin thethe society.society.
100 80 60 amount cryochambers of 40 20 1 1978 1989 2005 time
European Cryogenic Course CERN Geneva 2010 SeprconductingSeprconducting magnetmagnet coolingcooling andand diagnosticsdiagnostics -- NMRNMR
European Cryogenic Course CERN Geneva 2010 BiologicalBiological samplessamples storagestorage::
European Cryogenic Course CERN Geneva 2010 RocketRocket industryindustry –– liquidliquid hydrogenhydrogen
•There are three hydrogen isotopes: protonium H, deuterium D, and tritium T. •Boiling temp.: H2: 20.28 K (HD: 20.13 K) •Melting temp.: 13.96 K •vap = 1.34 kg/m3 •liq = 70.8 kg/m3 •solid = 86.7 kg/m3 •Heat of evap.: 85.7 kJ/kg •Heat of melting: 16.6 kJ/kg
•Hydrogen is used as rocket fuel
European Cryogenic Course CERN Geneva 2010 PowerPower generationgeneration -- terminalterminal LNGLNG
European Cryogenic Course CERN Geneva 2010 NaturalNatural gasgas transporttransport withoutwithout cryogenicscryogenics ??
European Cryogenic Course CERN Geneva 2010 PowerPower generationgeneration -- CoalCoal--FiredFired 110 bar CO2 OxyfuelOxyfuel BoilerBoiler ConversionConversion ~9000 t/day
Turbines 500 MWe CO2 Purification + Steam Water Compression Oxygen Air ~9000 t/day 65%13% CO2 Stack Air Furnace ID Fan Coal ~4000 t/day Flue Gas Recycle
Air FD Fan European Cryogenic Course CERN Geneva 2010 ITERITER cryogenicscryogenics –– thethe biggestbiggest concentratedconcentrated cryogeniccryogenic systemsystem inin thethe worldworld
Central
Solenoid, Nb3Sn Blanket Module Outer Intercoil Structure Vacuum Vessel
Toroidal Field Cryostat
Coil, Nb3Sn, 18
Poloidal Field Port Plug Coil, NbTi (IC Heating) Divertor
Machine Gravity Torus Supports Cryopump
European Cryogenic Course CERN Geneva 2010 ITERITER sitesite,, MarchMarch 20082008
European Cryogenic Course CERN Geneva 2010 CryogenicsCryogenics –– howhow??
MaciejMaciej ChorowskiChorowski WroclawWroclaw UniversityUniversity ofof TechnologyTechnology FacultyFaculty ofof MechanicalMechanical andand PowerPower EngineeringEngineering
European Cryogenic Course CERN, Geneval, 2010 SadiSadi CarnotCarnot andand hishis waterwheelwaterwheel analogyanalogy
WaterWater ((HeatHeat)) willwill notnot flowflow spontaneouslyspontaneously fromfrom aa lowlow levellevel ((coldcold objectobject)) toto aa highhigh levellevel ((hothot objectobject))..
High level (temperature)
Low level
European Cryogenic Course (temperature) CERN 2010 SecondSecond LawLaw ofof ThermodynamicsThermodynamics
ItIt isis notnot possiblepossible forfor heatheat toto flowflow fromfrom aa coldercolder bodybody toto aa warmerwarmer bodybody withoutwithout anyany workwork havinghaving beenbeen donedone toto accomplishaccomplish thisthis flow.flow.
European Cryogenic Course CERN 2010 T, K
10 ThirdThird LawLaw ofof ThermodynamicsThermodynamics:: 10 Core of the hottest stars 9 10
8 TheThe entropyentropy ofof allall systemssystems andand ofof allall statesstates 10 Fusion reaction of hydrogen
7 ofof aa systemsystem isis zerozero atat absoluteabsolute zerozero 10 Core of the Sun
6 10
5 10 ItIt isis impossibleimpossible toto reachreach thethe absoluteabsolute zerozero TEMPERATUREVERY HIGH Plasma 4 ofof temperaturetemperature byby anyany finitefinite numbernumber ofof 10 Surface of the Sun 3 10 Steam turbine processesprocesses Biological processes 2 10 High temperature superconductivity Boiling temperature of nitrogen Low temperature superconductivity 10 Boiling temperature of helium T S S(T, X ) 1 Superfluid helium 4
-1 X2 10
-2 10
X1 -3 10 Superfluid helium 3 TA=TB B A
-4 10
-5 10 The lowest measured temperature TC in the whole volume of a probe C -6 10 VERY LOW TEMPERATUREVERY LOW
-7 10 The lowest temperaure of copper nuclei
-8 European Cryogenic Course 10 SB=SC SA s CERN 2010 -9 10 Bose-Einstein condensate GeneralizedGeneralized coolingcooling processprocess
T X – generalized force (eg. pressure) Heat taken over in A-B transformation:
X2
Q TA s A sB X1
TA=TB B A
TC Temperature drop (TA-TC) C appears when generalized force x goes adiabaticaly to its initial value:
SB=SC SA s X2 X1
x - generalized force of conjugate variables: force - displacement Example: for conjugate variables p-V: pressure p is generalized force volume V is generalized displacement European Cryogenic Course CERN 2010 Substance Force X Displacement Y Process of entropy Process of (Conjugate Variables) decreasing temperature (ordering) lowering Gas pressure, p [Pa] Volume V , [m3] Isothermal Isentropic compression expantion Gas pressure, p [Pa] Volume V , [m3] Isothermal Isenthalpic compression expantion Paramagne Magnetic flux Magneric dipol Isothermal Adiabatic tic denisity, H [A/m] moment oM, magnetization demagnetization substance [Wb m] Dielectric Magnetic flux Electric dipol Isothermal Adiabatic de- substance denisity E, [V/m] moment P, [Cm] electrization electrisation Electron Electrical potential Electic charge Z, Electron compacting Electron dilution gas , [V] [C] Salt Chemical potential Mole number n Drying Dissolving [J/mol] Rod Mechanical force F Length l Compression Stress relaxation
Polymer Mechanical force F Length l Tension (fiber Tention relaxation elongation) European Cryogenic Course CERN 2010 Substance Force X Displacement Y Process of entropy Process of (Conjugate Variables) decreasing temperature (ordering) lowering Gas pressure, p [Pa] Volume V , [m3] Isothermal Isentropic compression expantion Gas pressure, p [Pa] Volume V , [m3] Isothermal Isenthalpic compression expantion Paramagne Magnetic flux Magneric dipol Isothermal Adiabatic tic denisity, H [A/m] moment oM, magnetization demagnetization substance [Wb m] Dielectric Magnetic flux Electric dipol Isothermal Adiabatic de- substance denisity E, [V/m] moment P, [Cm] electrization electrisation Electron Electrical potential Electic charge Z, Electron compacting Electron dilution gas , [V] [C] Salt Chemical potential Mole number n Drying Dissolving [J/mol] Rod Mechanical force F Length l Compression Stress relaxation
Polymer Mechanical force F Length l Tension (fiber Tention relaxation elongation) European Cryogenic Course CERN 2010 TemperatureTemperature dropdrop byby isentropicisentropic changechange ofof XX parameterparameter
T X2 S S(T, X ) X1
TA=TB B A
TC C C S S C S S dS D T dT D T dX 0 E T U X E X UT
SB=SC SA s
C S S D T C dT S E X U C S S T C S S s D T D T E dX U C S S D T ? ? s E X U D T E T U X T E T U X
European Cryogenic Course CERN 2010 TemperatureTemperature dropdrop byby isentropicisentropic changechange ofof XX parameterparameter
T C S S X2 D T C dT S E X UT X1 s D T TA=TB B A E dX U C S S s D T E T U X
TC C
SB=SC SA s C S S C S S c TD T X C dT S E X U c X dT Tds D T T s D T E T U X T E dX Us cX
C Y S C S S C Y S TD T D T D T C dT S E T U From Maxwell relations: D T X E EuropeanX UT CryogenicE CourseT U X s CERN 2010 E dX Us cX IsentropicIsentropic expansionexpansion withwith externalexternal workwork
T p 1 p2 h1 1 compression
h2 expansion K 2'
2
European Cryogenic Course CERN 2010 S IsentropicIsentropic expansionexpansion withwith externalexternal workwork C S S D T C S S C S S C dT S E p U dS D T dT D T dp 0 D T T D T s D T E T U p E p U dp C S S T E US D T E T U p
C S S cp C S S C v S D T D T D T E T U T D p T T p E UT E U p C v S TD T C dT S E T U T D T p s D dp T cp cp E US
European Cryogenic Course CERN 2010 IsentropicIsentropic expansionexpansion withwith externalexternal workwork C v S TD T C dT S E T U T Pv RT D T p s D dp T cp cp E US
T AZOT HEL 1 We get: s 10 bar 1 bar 10 bar 1 bar TT p 2 1 301 K 1 300 K 298 K 2 1 222 K 3 C S T p 193 K 3 After integration 2 D 2 T T p 155 K 1 E 1 U K 4 120 K 4 K European Cryogenic Course CERN 2010 SS PistonPiston expanderexpander
34
2 5
1 6 p1 p2
GAZ GAZ Europeanp1, T 1, h Cryogenic1 Course p2, T 2, h 2 CERN 2010 TurbineTurbine expanderexpander
GAZ p1, T 1, h 1
GAZ p2 T2 h2
European Cryogenic Course CERN 2010 TurbineTurbine expanderexpander
32 1 Cryogenic helium turbo-expander: 1 – gas turbine, 2 – gas bearing, 3 – gas work extractor European Cryogenic Course CERN 2010 IsentropicIsentropic expansionexpansion withwith externalexternal workwork –– isentropicisentropic efficiencyefficiency ofof thethe expanderexpander
T p1 p2 h1
1
h2 h h ' K 1 2 2' s h h 2 1 2
European Cryogenic Course CERN 2010 S AdiabaticAdiabatic demagnetizationdemagnetization
European Cryogenic Course CERN 2010 Substance Force X Displacement Y Process of entropy Process of (Conjugate Variables) decreasing temperature (ordering) lowering Gas pressure, p [Pa] Volume V , [m3] Isothermal Isentropic compression expantion Gas pressure, p [Pa] Volume V , [m3] Isothermal Isenthalpic compression expantion Paramagne Magnetic flux Magneric dipol Isothermal Adiabatic tic denisity, H [A/m] moment oM, magnetization demagnetization substance [Wb m] Dielectric Magnetic flux Electric dipol Isothermal Adiabatic de- substance denisity E, [V/m] moment P, [Cm] electrization electrisation Electron Electrical potential Electic charge Z, Electron compacting Electron dilution gas , [V] [C] Salt Chemical potential Mole number n Drying Dissolving [J/mol] Rod Mechanical force F Length l Compression Stress relaxation
Polymer Mechanical force F Length l Tension (fiber Tention relaxation elongation) European Cryogenic Course CERN 2010 AAdiabaticdiabatic demagnetizationdemagnetization ooff paramagneticparamagnetic substancessubstances
T To S S (T , H ) H0
H4 C S S C S S H3 H dS D T dT D T dH 0 2 H 1 H0 E T U E H U 2 X T Tp 1 C S S D T Tk C dT S E H U 3 D T T s s E dH U s C S S S2 = S 3 S1 D T S E T U X
European Cryogenic Course CERN 2010 AdiabaticAdiabatic demagnetizationdemagnetization ofof paramagneticparamagnetic substancessubstances
C S S C S S S S (T , H ) dS D T dT D T dH 0 E T U H E H UT
C S S D T C dT S E H U D T T s E dH U C S S s D T E T U H
European Cryogenic Course CERN 2010 AdiabaticAdiabatic demagnetizationdemagnetization ofof paramagneticparamagnetic substancessubstances
C S S c D T H Tds cH dT E T U H T
C S S C M S D T o D T and M H E H UT E T U H Equivalent to the equatio
European Cryogenic Course of state for gases CERN 2010 C S S C M S D T D T HowHow dodo wewe getget o E H UT E T U H the formulas? the formulas? M H
dU TdS HdM dU dQ pdV o HdM o
Potencja Gibbsa: G U o HM TS
2G 2G S M C S C S D T o D T TH HT E H UT E T U H
European Cryogenic Course CERN 2010 GeneralGeneral equationequation toto calculatecalculate thethe magnetocaloricmagnetocaloric coefficientcoefficient ifif thethe relationrelation governinggoverning thethe M,HM,H andand TT isis knownknown::
C S S C M S T D T oT D T C dT S E H UT E T U H s D T E dH U s cH cH
European Cryogenic Course CERN 2010 M H
C Curie law (valid for not very high magnetic fields): T
CH 2 C dT S CH T T o o 0 k o 1 Final temperature afetr s D T c T 2 dH c T H o demagnetization: E U s H
H C M S CH 2 Cooling capacity: Q T O D T dH o E T U T 0 H 2
European Cryogenic Course CERN 2010 MagneticMagnetic refrigeratorrefrigerator –– principleprinciple ofof operation,operation, AA –– initialinitial coolingcooling ofof thethe sample,sample, BB –– isothermalisothermal magnetization,magnetization, CC –– removalremoval ofof thethe heatheat exchangeexchange gas,gas, DD –– adiabaticadiabatic demagnetization;demagnetization; 11 –– paramagneticparamagnetic substance,substance, 22 –– samplesample chamber,chamber, 33 –– valve,valve, 44 –– magnetmagnet
ABCD 3
4
2 1
CIEK Y CIEK Y CIEK Y CIEK Y HEL HEL HEL HEL
H=0 HEuropean Cryogenic Course H H=0 CERN 2010 ExamplesExamples ofof paramagneticparamagnetic substancessubstances
European Cryogenic Course CERN 2010 Magnetic stack - example,
European Cryogenic Course CERN 2010 MagneticMagnetic refrigeratorrefrigerator
European Cryogenic Course CERN 2010 MultistageMultistage magneticmagnetic refrigeratorrefrigerator
European Cryogenic Course CERN 2010 GeneralGeneral wayway toto lowerlower thethe temperaturetemperature
A. TheThe entropyentropy mustmust bebe aa functionfunction ofof twotwo parameters:parameters: temperaturetemperature TT andand X,X, wherewhere XX isis aa generalizedgeneralized forceforce (gas(gas pressure,pressure, magneticmagnetic fifieldeld ).). AtAt thethe roomroom temperaturetemperature thethe entropyentropy mustmust bebe highhigh enough.enough. B. TheThe entropyentropy cancan bebe loweredlowered byby externalexternal workwork donedone onon thethe bodybody (ga(gass compression,compression, paramagneticparamagnetic magnetization)magnetization) inin isothermalisothermal process.process. C. ByBy isentropicisentropic processprocess thethe temperaturetemperature ofof thethe bodybody willwill gogo down.down.
T 1 10 Minimum entropy X2 storage in function of
0 10 temperature AA X1 TA=TB B A
-1 B 10 TC C Entropia zredukowana S/R
-2 10 -3 -2 -1 0 1 2 3 European Cryogenic Course 10 10 10 10 10 10 10
SB=SC SA CERNs 2010 Temperatura, K