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Thermodynamics for CryogenicsRam C. DhuleyUSPAS – Cryogenic EngineeringJune 21 – July 2, 2021

Goals of this lecture Revise important definitions in thermodynamics– system and surrounding, state properties,derived properties, processes Revise laws of thermodynamics and learn how toapply them to cryogenic systems Learn about ‘ideal’ thermodynamic processesand their application to cryogenic systems Prepare background for analyzing real-worldhelium liquefaction and refrigeration cycles.2Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Introduction Thermodynamics deals withthe relations between heatand other forms of energysuch as mechanical,electrical, or chemical energy. Thermodynamics has its basisin attempts to understandcombustion and steam powerbut is still “state of the art” interms of practical engineeringissues for cryogenics,especially in processefficiency.James Dewar (invented vacuum flask in 3/1.8814903Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Definitions Thermodynamic system is the specified region in which heat,work, and mass transfer are being studied. Surrounding is everything else other than the system. Thermodynamic boundary is a surface separating the systemfrom the surrounding.BoundarySystemSurrounding4Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Definitions (continued)Types of thermodynamic systems Isolated system has no exchange of mass or energy with itssurrounding Closed system exchanges heat but not mass Open system exchanges both heat and mass with its surroundingThermodynamic State Thermodynamic state is the condition of the system at a giventime, defined by ‘state’ properties More details on next slides Two state properties define a state of a “homogenous” system Usually, three state properties are needed to define a state of anon-homogenous system (example: two phase, mixture)5Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Common state properties 6T – temperatureP – pressure (force per unit area)v – specific volume (inverse of density)U – internal energy of the closed systemu – internal energy per unit mass (specific internal energy)H – enthalpy U PVh – enthalpy per unit mass (specific enthalpy)S – entropys – entropy per unit mass (specific entropy)Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Common derived properties Some important thermodynamic properties are definedfrom others, such as the heat capacities, cv , and cP ucv Tv hcp Tp Since typically two properties define the state of the purefluid in thermodynamics, equations generally have twoindependent variables Derivatives then are partial derivatives with respect to oneindependent variable with the other held constant7Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Thermodynamic Path and ProcessThermodynamic Path A path is series of intermediate statesthat a system takes while going fromone (initial) state to another (final)stateThermodynamic Process Process defines the type of path taken by the system while goingfrom initial to final state Isothermal – constant temperature Isobaric – constant pressure Isochoric – constant volume Isentropic – constant entropy Isenthalpic – constant enthalpy8Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Equation of stateEquation of state (EOS) is a relation between the stateparameters that a system obeys while changing from onestate to another.An example is Ideal gas EOS, given by pv RTwhere R is universal gas constant/gas molar mass-good approximation for gases at low pressure andhigh temperature (far away from critical p and T)- ‘ideality’ can be evaluated from compressibility factorZ pv/RT (an ideal gas has Z 1)Several empirical EOS have been formulated for real gases,examples are Van der Waals EOS, Peng Robinson EOS9Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Compressibility factor for helium 5 % non-idealityPv/RT300 K 1 % non-ideality20 K 2 % non-idealityPressure (atm)Helium has 10% non-ideality down to 8 K10Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

More about ideal gases The internal energy of an ideal gas is a function oftemperature alone, Du cv DT where cv is constant Since h u Pv, so h u RT for an ideal gas,enthalpy is also a function of temperature alone foran ideal gas, Dh c pDT where cp is constant.11Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Reversible processA theoretical process in which state of a system ischanged in a series of infinitesimal steps such that atevery step the system holds equilibrium with itssurrounding. At every step, the change can be “undone”to restore the system to its prior state.In the context of refrigeration/cryogenics, a reversibleprocess consumes the least amount of mechanical power(work) for compression produces maximum mechanical power (work) duringexpansion12Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Reversible processProcess done inlarge, discreet steps(irreversible)Process doneinfinitesimal steps(reversible)13Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

The laws of thermodynamics Zeroth law – Two systems in thermal equilibriumwith a third system are in thermal equilibrium witheach other First law – The total energy of a system is conservedduring a thermodynamic process Second law – The entropy of a system increases inall real processes and is conserved in reversibleprocesses Third law – The entropy of a pure substance incomplete thermodynamic equilibrium becomeszero at a temperature of absolute zero14Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

The laws of thermodynamicsWmech(m, P, v, T )outQ(Text )P,V , T ,U (T ), S (T )zout(m, P, v, T )inzinFirst law: dUv2v2 Q Wmech m h gz m h gz dt22in out Second law:dSQ ms ms S gen with S gen 0dtText inoutNotes: Time derivative terms are zero for steady state processes Mass flow terms are zero for closed systems Sign convention: Inward positive, outward negative15Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Isothermal compressionReversible, constant temperature compression - requires theleast amount of mechanical work to pressurize a fluid Sets an upper limit on the efficiency of compression, aka ‘isothermalefficiency’Isothermal compressorQR ,revm, Pin , TWin ,Tm, Pout , TLet’s apply laws of thermodynamics to determinework of isothermal compression at steady state Neglect changes in kinetic and potential energyAssume ideal gas, Pv RTFirst law:dU 0 QR ,rev Win,T m(hin hout )dt 0 for ideal gas QR ,revdS 0 m( sin sout ) S genSecond law:dtT 0 for a reversible processCombine, re-arrange, Win,T QR,rev m( sin sout )Tand use ideal gas EOS: mRT ln( pout / pin )16Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Isothermal compressionAn example from Fermilab The second stage screw compressor at Fermilab’s MTFcompresses 200 grams/sec helium from about 2.6 barto 15 bar For helium R 2.078 J/gK, so the ideal work at 300 Kisææ 15 öæJ ögrams ö 220kWç 2.078 (300K ) ln ç ç 200è 2.6 øègK øsec øè With typical power consumption of 800 HP 600 kW,the isothermal efficiency is about 37%17Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

A real helium compressor Oil-flooded screw compressors are nowstandard A typical pressure ratio is about 4:1, so twostages are used in a typical helium plant toget a 15:1 to 20 :1 pressure ratio18Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Isentropic expansionReversible expansion of a fluid without exchange of heat The expansion produces mechanical work at the expense ofenthalpy loss, which results in coolingProduces maximum change in enthalpy and sets an upper limit onthe efficiency of cooling via expansion, aka ‘isentropic efficiency’Isentropic expanderm, Pin , TinWout , sLet’s apply laws of thermodynamics to determinework of isentropic expansion at steady state Neglect changes in kinetic and potential energyAssume ideal gas, Pv RTFirst law:m, Pout , ToutdU 0 Q Wout , s m(hin hout )dt 0 for isentropic processSecond law: each term is identically zeroRearrange first law and use ideal gas EOS:Wout , s m(hin hout ) mc p (T1 T2 )19Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Isentropic expansionIsentropic efficiency Isentropic expansion efficiency is defined asDhrealDhisentropcwhere Δh hin – hout Δhreal will always be less than Δhisentropc so efficiencywill be less than 100% For real expanders, 65% to 85%20Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Thermodynamically ideal refrigeratorType 1: Refrigeration load is isothermal (constant T) The medium to be cooled is at constant temperatureQrej ectTambProcesses: 1 - 2 – isothermal compression2 - 3 – isentropic expansion3 - 4 – isothermal heat absorption (ref. load)4 - 1 – isentropic compressionApply first law to the entire cycle:Cyclic process: u 0 Qnet Wnet ,inNet work input:TrefQrefWnet ,in Tamb ( s2 s1 ) Tref ( s4 s3 ) (Tamb Tref )( s4 s3 ) QnetRefrigeration power: Qref Tref (s4 s3 )21Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Thermodynamically ideal refrigeratorType 1: Refrigeration load is isothermal (constant T) The medium to be cooled is at constant temperatureCoefficient of performance (COP):Qrej ectTamb Defined as the ratio of refrigeration load to netinput mechanical work (i.e., benefit/cost)COPisoT ,ideal TrefQrefQrefWnet ,in TrefTamb TrefThe COP of an ideal gas, ideal isothermal loadrefrigerator: depends only on ambient and load temperatures does not depend on the fluid or its pressure represents the highest refrigeration power onecan get for a given work inputEg: COPisoT ,ideal 0.013 forTref 4 K and Tamb 300 KAt least 77 W ( 1/0.013) of mechanical power input isneeded at 300 K to obtain 1 W refrigeration at 4 K22Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Thermodynamically ideal refrigeratorType 2: Refrigeration load is non-isothermal Temperature of the medium increases as refrigeration load is absorbedTypically, the pressure remains constant (isobaric ref. load)Qrej ectTambProcesses: 1 - 2 – isobaric heat absorption (ref. load)2 - 3 – isentropic compression3 - 4 – isothermal compression4 - 1 – isentropic expansionApply first law to the entire cycle:Cyclic process: u 0 Qnet Wnet ,inNet work input:QrefWnet ,in Qnet Qrej ect Qref Tamb ( s4 s3 ) Qref Tamb ( s2 s1 ) Qref23Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Thermodynamically ideal refrigeratorType 2: Refrigeration load is non-isothermal Temperature of the medium increases as refrigeration load is absorbedTypically, the pressure remains constant (isobaric ref. load)Qrej ectApply first law to the entire cycle:Refrigeration power:Tamb2211Qref Tds (dh vdP) h2 h1Coefficient of performance (COP):COPisoP ,ideal QrefWnet ,inh2 h1 Tamb ( s2 s1 ) (h2 h1 )If the fluid is an ideal gas,COPisoP ,ideal 24Qref(T2 / T1 ) 1(Tamb / T1 ) ln(T2 / T1 ) ((T2 / T1 ) 1)Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Isenthalpic expansionA simple, commonly used method to produce cooling Expansion through a valve does no work, and neitheradds nor removes energy– Process is isenthalpic Enthalpy of the perfect gas is a function oftemperature alone– Isenthalpic process of perfect gasdoes not change the temperature¶T ö 0¶P øhFor ideal gases Real fluids may change temperature via an isenthalpicexpansion– Joule-Thomson effect25Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Isenthalpic expansionJoule Thompson expansion In real fluids, including helium,isenthalpic expansion may providetemperature drop– Starting P and T need to be on the ‘left’side of inversion curve where (dP/dT) h 0dP ö 0dT øhdP ö 0dT øh Joule-Thomson expansion through avalve is said to be through a “J-T valve”– Not as efficient of isentropic expansion– But very convenient and easy (no movingparts)– Used frequently as the final expansionstage in real helium refrigerators, liquefiers26Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

Source of statepropertiesfor thermodynamicanalysesIsothermal compressionIsentropicexpansionTemperature entropychart for heliumIsenthalpicexpansionH constantP constant2-phase dome27Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)ρ constant

Source of statepropertiesfor thermodynamicanalyses28NIST ThermophysicalProperties of uid/)Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

How to use the NIST webbook?29Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

How to use the NIST webbook?30Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

How to use the NIST webbook?Data from this table can be copied to a spreadsheet or exportedas a tab-delimited file.31Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

References1. R. Barron, “Cryogenic Systems”, Oxford University Press; 2 ndedition (June 13, 1985).2. NIST database: Thermophysical Properties of Fluid 3. A nice explanation of reversible processes https://chem.libretexts.org/Bookshelves/General Chemistry/Map%3A Principles of Modern Chemistry (Oxtoby et al.)/Unit 4%3A Equilibrium in Chemical Reactions/13%3A Spontaneous Processes and Thermodynamic Equilibrium/13.4%3A Entropy Changes in Reversible Processes32Ram C. Dhuley USPAS Cryogenic Engineering (Jun 21 - Jul 2, 2021)

represents the highest refrigeration power one can get for a given work input,, ref ref isoT ideal net in amb ref QT COP W T T At least 77 W ( 1/0.013) of mechanical power input is needed at 300 K to obtain 1 W refrigeration at 4 K for and TK ref COP 4 TK