MODELACIMODELACIÓÓN MATEMN MATEMÁÁTICA Y COMPUTACIONAL TICA Y COMPUTACIONAL DE PROPAGACIDE PROPAGACIÓÓN DE ONDAS DE CHOQUE EN N DE ONDAS DE CHOQUE EN

MATERIALES CON TRANSICIONES DE FASEMATERIALES CON TRANSICIONES DE FASE

Vladimir Tchijov Vladimir Tchijov Gloria Cruz LeGloria Cruz Leóón n

SuemiSuemi RodrRodrííguez Romoguez RomoFelipe de JesFelipe de Jesúús Vargas Torress Vargas Torres

Seminario de Modelación Computacional 2006

Facultad de Estudios Superiores CuautitlFacultad de Estudios Superiores Cuautitláánn

CONTENIDOCONTENIDO

IntroducciIntroduccióón: n: ¿¿QuQuéé son ondas de choque?son ondas de choque?CompresiCompresióón de agua y hielo por ondas de choquen de agua y hielo por ondas de choqueTermodinTermodináámica de Hmica de H22OOModelo cinModelo cinéético de transiciones de fase mtico de transiciones de fase múúltiples en ltiples en hielohieloVerificaciVerificacióón del modelon del modeloOndas de choque unidimensionales en hieloOndas de choque unidimensionales en hieloConclusionesConclusiones

¿¿QuQuéé son ondas de choque? son ondas de choque? (agua como ejemplo)(agua como ejemplo)

Relaciones de Relaciones de RankineRankine--HugoniotHugoniot

CompresiCompresióón de materiales por ondas de choque:n de materiales por ondas de choque:un problema multidisciplinarioun problema multidisciplinario

•• Experimentos naturalesExperimentos naturales• Fisicoquímica y termodinámica• Modelación matemática• Simulación computacional

Nosotros estudiamos el agua y sus fases sNosotros estudiamos el agua y sus fases sóólidas. lidas. ¿¿Por quPor quéé??

DesafDesafííos cientos cientííficosficos

Aplicaciones cientAplicaciones cientííficas y tecnolficas y tecnolóógicas: congelacigicas: congelacióón de alimentos; n de alimentos; preservacipreservacióón de materiales bioln de materiales biolóógicos; ingeniergicos; ingenieríía espaciala espacial

InternationalInternational AssociationAssociation onon thethe PropertiesProperties ofof WaterWater andand SteamSteam(IAPWS)(IAPWS)

Nuestro grupo de trabajo:Nuestro grupo de trabajo:

Dr. Vladimir TchijovDr. Vladimir TchijovDra. Dra. SuemiSuemi RodrRodrííguez Romoguez RomoDra. Gloria Cruz LeDra. Gloria Cruz LeóónnDr. Ricardo Dr. Ricardo BaltazarBaltazar AyalaAyalaM. en C. Felipe de JesM. en C. Felipe de Jesúús Vargas Torress Vargas TorresM. en C. JosM. en C. Joséé LuisLuis Garza RiveraGarza RiveraDr. Oleg Nagornov (Rusia)Dr. Oleg Nagornov (Rusia)

Colegas de Alemania, Francia, Estados UnidosColegas de Alemania, Francia, Estados Unidos

ShockShock compressioncompression ofof iceice

ShockShock compressioncompression ofof ice: ice: ExperimentsExperiments

•• Larson, 1984: Ice at 263 K; stresses up to 3.6 GPa

• Gaffney et al., 1994: Ice at 263 K ; stresses up to 3 GPa• Stewart, 2001: Ice at 100 K; stresses up to 5.2 GPa• Dolan & Gupta, 2003: Water at 300 K; stresses up to 3.5 GPa

Time Time scalescale: 10: 10--77 –– 1010--66 ss

ShockShock compressioncompression ofof ice ice (Larson, 1984)(Larson, 1984)

Experimental Experimental arrangementarrangement ofof shockshock--wavewavecompressioncompression ofof ice: Stress ice: Stress gaugesgauges

((GaffneyGaffney & & SmithSmith, 1994), 1994)

ShockShock--wavewave compressioncompression ofof waterwater((DolanDolan & & GuptaGupta, 2003), 2003)

Experimental Experimental observationsobservations revealreveal multiplemultiple nonnon--equilibriumequilibrium phasephase transitionstransitions in ice in ice subjectedsubjected totoshockshock compressioncompression

The main goal of our research is the The main goal of our research is the modelingmodeling ofof thethephenomenaphenomena ofof phasephase changeschanges in in shockshock--compressedcompressed ice ice basingbasing onon::

ThermodynamicsThermodynamics ofof liquidliquid waterwater andand itsits solidsolid--statestatepolymorphspolymorphs (ices (ices IhIh, II, III, V, VI, VII , II, III, V, VI, VII andand VIII)VIII)KineticKinetic modelmodel ofof multiplemultiple phasephase changeschanges in ice in ice compressedcompressed by by shockshock waveswavesComputerComputer simulationsimulation ofof shockshock--wavewave propagationpropagationthroughthrough anan ice ice samplesample

ThermodynamicsThermodynamicsofof ordinaryordinary waterwater substancesubstance

HH22O O phasephase diagramdiagram

230 320 410 500Temperature (K)

0

1

2

3

4

5

Pre

ssur

e (G

Pa)

VIIVIII

VI

VII III

Stable water

Ih

ExampleExample: Ice III (: Ice III (initialinitial cluster)cluster)

ExampleExample: Ice III (final cluster): Ice III (final cluster)

ExampleExample: Ice V (: Ice V (initialinitial cluster)cluster)

ExampleExample: Ice V (final cluster): Ice V (final cluster)

EquationsEquations ofof statestate forfor icesices

Ice Ice IhIh1. 1. R. Feistel, W. Wagner, A new equation of state for HR. Feistel, W. Wagner, A new equation of state for H22O ice O ice IhIh, , J. Phys. Chem. Ref. DataJ. Phys. Chem. Ref. Data 3535 (2006) 1021(2006) 1021--1047.1047.2. 2. R. Feistel, W. Wagner, V. Tchijov, C. R. Feistel, W. Wagner, V. Tchijov, C. GuderGuder, Numerical , Numerical implementation and oceanographic application of the Gibbs implementation and oceanographic application of the Gibbs potential of ice, potential of ice, Ocean Ocean SciSci.. 11 (2005) 29(2005) 29--38. 38.

Free Fortran/Basic/C++ software is available!Free Fortran/Basic/C++ software is available!

Ice IIIce IIG. Cruz G. Cruz LeLeóónn, S. , S. RodrRodrííguezguez RomoRomo, V. Tchijov, , V. Tchijov, Thermodynamics of highThermodynamics of high--pressure ice polymorphs: ice II, pressure ice polymorphs: ice II, J. Phys. J. Phys. Chem.Chem. SolidsSolids 6363 (2002) 843(2002) 843--851.851.

EquationsEquations ofof statestate forfor icesices

Ices III, VIces III, VV. Tchijov, R. V. Tchijov, R. BaltazarBaltazar Ayala, G. Cruz Ayala, G. Cruz LeLeóónn, O. Nagornov, , O. Nagornov, Thermodynamics of highThermodynamics of high--pressure ice polymorphs: ices III and pressure ice polymorphs: ices III and V,V, J. Phys. J. Phys. Chem. SolidsChem. Solids 6565 (2004) 1277(2004) 1277--12831283..

Ice VIIce VIV. Tchijov, J. Keller, S. V. Tchijov, J. Keller, S. RodrRodrííguezguez RomoRomo, , O.NagornovO.Nagornov, Kinetics , Kinetics of phase transitions induced by shockof phase transitions induced by shock--wave loading in ice,wave loading in ice, J. J. Phys. ChemPhys. Chem. . 101 (101 (1997) 62151997) 6215--62186218 ..

EquationsEquations ofof statestate forfor icesices

Ice VIIIce VIIYingweiYingwei FeiFei, Ho, Ho--kwangkwang Mao, Mao, R.J.HemleyR.J.Hemley, Thermal , Thermal expansivityexpansivity, , bulk modulus, and melting curve of H2O bulk modulus, and melting curve of H2O –– ice VII to 20 GPa, ice VII to 20 GPa, J. J. Chem. Chem. PhysPhys. . 9999 (1993) 5369(1993) 5369--53735373..

HeatHeat capacitycapacity ofof highhigh--pressurepressure icesicesV. Tchijov,V. Tchijov, Heat capacity of highHeat capacity of high--pressure ice polymorphs, pressure ice polymorphs, J. J. Phys. Chem. SolidsPhys. Chem. Solids 6565 (2004) 851(2004) 851--854854..

EquationsEquations ofof statestate forfor waterwater

LiquidLiquid waterwater

1. 1. W. Wagner, A. W. Wagner, A. PruPrußß, The IAPWS Formulation 95 for the thermodynamic , The IAPWS Formulation 95 for the thermodynamic properties of ordinary water substance for general and scientifiproperties of ordinary water substance for general and scientific use, c use, J. Phys. J. Phys. Chem. Ref. DataChem. Ref. Data 3131 (2002) 387(2002) 387--535535. . ((≡≡ IAPWSIAPWS--95, the international standard 95, the international standard for scientific and industrial use)for scientific and industrial use)

2. 2. O. V. Nagornov, V. E. O. V. Nagornov, V. E. ChizhovChizhov (Tchijov), Thermodynamic properties of ice, (Tchijov), Thermodynamic properties of ice, water, and a mixture of the two at high pressures, water, and a mixture of the two at high pressures, J. J. ApplAppl. Mech. and . Mech. and TechnTechn. . PhysPhys. . 3131 (1990) 378(1990) 378--385.385.

3.3. V. Tchijov, Analysis of the equationsV. Tchijov, Analysis of the equations--ofof--state of water in the metastable state of water in the metastable region at high pressures, region at high pressures, J. Chem. PhysJ. Chem. Phys. . 116116 (2002) 8631(2002) 8631--8632.8632.

KineticKinetic modelmodel ofof multiplemultiple phasephasetransitionstransitions in icein ice

KineticKinetic modelmodel ofof phasephase transitionstransitions in icein ice

Fractions of phases Fractions of phases xxii ≥≥ 00 in a nonin a non--equilibrium mixture (equilibrium mixture (ii = = IhIh, , II, III, V, VI, VII, liquid water):II, III, V, VI, VII, liquid water):

Kinetic equations:Kinetic equations:

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑

1ii

x =∑

( )iji ij

j i

dxdt

γ γ≠

= −∑

KineticKinetic modelmodel ofof phasephase transitionstransitions in icein ice

Model:Model:

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑

( )1 exp ( ) , 0 ( , )

0, 0 ( , )i ij ij ij i j

ij

i j

x A P f T B x and T P

x or T Pγ

⎧ ⎡ ⎤− − − ≥ ∈Ω⎪ ⎣ ⎦=⎨= ∉Ω⎪⎩

where:where:ΩΩjj is the region of stability of a phase is the region of stability of a phase jj,,

AAijij, , BBijij are constants.are constants.

VerificationVerification ofof thethe modelmodel((““00--dimensionaldimensional”” problem)problem)

““00--dimensionaldimensional”” problem of shock compression of iceproblem of shock compression of ice

Specific volume and internal energy of a mixture:Specific volume and internal energy of a mixture:

Pressure law:Pressure law:

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑

( , )i ii

V x V P T=∑

( , )i ii

E x E P T=∑

( , )i ii

E x E P T=∑

, 0( )

(2 ), 2* * *

* * * *

P t t t tP t

P t t t t t⋅ ≤ ≤⎧

=⎨ ⋅ − ≤ ≤⎩

““00--dimensionaldimensional”” problem of shock compression of iceproblem of shock compression of ice

Adiabatic compression:Adiabatic compression:

Kinetic equations:Kinetic equations:

Initial conditions:Initial conditions:

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ ( , )i ii

E x E P T=∑ ( ) 0dE dVP tdt dt

+ =

( ) 0dE dVP tdt dt

+ = ( )1 7( ), , ,..., , wdT F P t T x x xdt

⇒ =

( )iji ij

j i

dxdt

γ γ≠

= −∑

tt = 0 = 0 ⇒⇒ TT = = TT00, , xx11 = 1, = 1, xx22 = 0, = 0, ……, , xx77 = 0, = 0, xxww = 0= 0

LoadingLoading//unloadingunloading pathspaths onon thethe PP--VV diagramdiagram((PP**=3.6 GPa, =3.6 GPa, tt**=8=8××1010--77ss))

LoadingLoading//unloadingunloading pathspaths onon thethe PP--TT diagramdiagram((PP**=3.6 GPa, =3.6 GPa, tt**=8=8××1010--77ss))

FractionsFractions ofof waterwater andand ice ice phasesphases((PP**=3.6 GPa, =3.6 GPa, tt**=8=8××1010--77ss))

LoadingLoading//unloadingunloading pathspaths onon thethe PP--VV diagramdiagram((PP**=3.6 GPa)=3.6 GPa)

LoadingLoading//unloadingunloading pathspaths onon thethe PP--TT diagramdiagram((PP**=3.6 GPa)=3.6 GPa)

OneOne--dimensional dimensional shockshock waveswaves in icein ice

OneOne--dimensional dimensional shockshock waveswaves in icein ice

Specific volume and internal energy of a mixture:Specific volume and internal energy of a mixture:

Balance equations in Balance equations in LagrangianLagrangian variables variables tt, , zz

((tt > 0, 0 < > 0, 0 < zz < < LL00)) ::

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑

( , )i ii

V x V P T=∑

( , )i ii

E x E P T=∑

( , )i ii

E x E P T=∑

1 10, 0, 0V u u P E VPV t z V t z t t∂ ∂ ∂ ∂ ∂ ∂

− = + = + =∂ ∂ ∂ ∂ ∂ ∂

OneOne--dimensional dimensional shockshock waveswaves in icein ice

Kinetic equations:Kinetic equations:

Boundary and initial conditions:Boundary and initial conditions:

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ ( , )i ii

E x E P T=∑

( )iji ij

j i

dxdt

γ γ≠

= −∑

0 0 0(0, ) , ( , ) ( 0)u t u P L t P t= = ≥

0 0

0 0 0

( ,0) , ( ,0) ,( ,0) , ( ,0) (0 )

V z V E z EP z P T z T z L

= =

= = ≤ ≤

0( ,0)i ix z x= (i = Ih, II, III, V, VI, VII, liquid)

El software:El software:

CFDLIB (Computational Fluid Dynamics CFDLIB (Computational Fluid Dynamics Library)Library) –– poderoso paquete de dinpoderoso paquete de dináámica de mica de fluidos computacional escrito en fluidos computacional escrito en FortranFortran 7777

VisualizadorVisualizador de de ccáálculoslculos original original escritoescrito en en C++ C++

1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ 1.ii

x =∑ ( , )i ii

E x E P T=∑

CalculatedCalculated densitydensity ofof shockshock--compressedcompressed iceice((pistonpiston velocity velocity uu00=650 m/s)=650 m/s)

CalculatedCalculated stress in stress in shockshock--compressedcompressed ice ice ((pistonpiston velocity velocity uu00=650 m/s)=650 m/s)

CalculatedCalculated particleparticle velocityvelocity atat twotwo gaugesgauges (1 (1 –– 3.139 mm, 2 3.139 mm, 2 –– 6.268 mm). Piston velocity 6.268 mm). Piston velocity uu00=650 m/s.=650 m/s.

ConclusionsConclusions

ExperimentsExperiments onon shockshock compressioncompression ofof ice ice indicateindicatemultiplemultiple nonnon--equilibriumequilibrium phasephase transitionstransitions..

Reliable EOS for liquid water and its solidReliable EOS for liquid water and its solid--state state polymorphs has been developed in the last years, two polymorphs has been developed in the last years, two of them being now the international standards.of them being now the international standards.

TheThe developeddeveloped kinetickinetic modelmodel permitspermits reproducingreproducing withwithreazonablereazonable accuracyaccuracy thethe complexcomplex phenomenaphenomena ofof phasephasechangeschanges in ice in ice compressedcompressed by by shockshock waveswaves..