Download - L7 Phase Equilibria

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Termodinamica de Termodinamica de HidrocarburosHidrocarburos

Generalized Phase Equilibria Generalized Phase Equilibria

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Phase EquilibriumPhase EquilibriumEvolution…Evolution…

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The Concept of EquilibriumThe Concept of Equilibrium

Equilibrium indicates static conditions, Equilibrium indicates static conditions, the absence of changethe absence of change In thermodynamics is taken to mean In thermodynamics is taken to mean not only the absence of change, but the not only the absence of change, but the absence absence of any tendencyof any tendency to change. to change. A system existing in equilibrium is one A system existing in equilibrium is one in which under such conditions that in which under such conditions that there is no tendency for a change to there is no tendency for a change to state to occur.state to occur.

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The Concept of EquilibriumThe Concept of Equilibrium

Tendencies toward a change are Tendencies toward a change are caused by a caused by a driving forcedriving force of any of any kindkind

Equilibrium means the absence of Equilibrium means the absence of any driving force, or that all forces any driving force, or that all forces are in exact balanceare in exact balance..

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Driving ForcesDriving Forces

Typical driving forces include:Typical driving forces include: mechanical forcesmechanical forces such as pressure on a such as pressure on a piston tend to cause energy transfer as piston tend to cause energy transfer as workwork temperature differences tend to cause temperature differences tend to cause the flow of the flow of heatheat chemical potentialschemical potentials tend to cause mass tend to cause mass transfer from one phase to another or transfer from one phase to another or cause substances to react chemically.cause substances to react chemically.

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Phase Equilibrium and the Phase Equilibrium and the Phase RulePhase Rule

In reservoir engineering applications we In reservoir engineering applications we assume that reservoir fluids are at assume that reservoir fluids are at equilibrium, equilibrium, we do not say how long the we do not say how long the equilibrium will last. equilibrium will last. As a reservoir block changes pressure due As a reservoir block changes pressure due to production (injection) we assume that to production (injection) we assume that equilibrium is reached instantly. equilibrium is reached instantly. Fluid properties in reservoir cells are Fluid properties in reservoir cells are evaluated using a sequence of connected evaluated using a sequence of connected equilibrium stages.equilibrium stages.

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Phase RulePhase Rule

It tells us the number of independent It tells us the number of independent variables required to fully variables required to fully characterize a systemcharacterize a systemIt does not tell us which variables to It does not tell us which variables to selectselect

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Generalization of the phase rule Generalization of the phase rule for for NNcc non reacting componentsnon reacting components

Number of components Number of phases Degrees of Freedom1 1 21 2 11 3 02 1 32 2 22 3 1

… … …Nc Np (Nc-Np)+2

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Phase Equilibrium and the Phase Equilibrium and the Phase RulePhase Rule

Thus for non-reacting systemsThus for non-reacting systems

F = # of variables – # of Independent F = # of variables – # of Independent equations relating these variablesequations relating these variables

2 pc NNF

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First Law and Fundamental First Law and Fundamental Thermodynamic RelationshipsThermodynamic Relationships

Closed SystemsClosed Systems The system does not exchange The system does not exchange matter with the surroundings, but matter with the surroundings, but it can exchange energy. it can exchange energy.

The first law is a generalization of The first law is a generalization of the conservation of energythe conservation of energy

dWdQdU t

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Compression and Compression and expansion workexpansion work

in a gas container in a gas container indicatingindicating

the convention the convention used for heat and used for heat and

workwork

- dW

+ dQ

Compression

+ dW

- dQ

Expansion

- dW

+ dQ

Compression

+ dW

- dQ

Expansion

Heat and Work Sign ConventionHeat and Work Sign Convention

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For a reversible process, For a reversible process, dQ = TdSdQ = TdStt

thus,thus,

If the work of expansion or If the work of expansion or compression is the only kind of compression is the only kind of work allowed then:work allowed then:

dWTdSdU tt

tPdVdW

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Replacing work and heat Replacing work and heat expressionsexpressions

ThusThus

ttt PdVTdSdU

tttt VSUU ,

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SinceSince

Thus one can identify,Thus one can identify,

TnSnU

nnV

,

PnVnU

nnS

,

tttt VSUU ,

Define: Define: MMt t = nM = nM with with M = U, H, A, M = U, H, A, G, SG, S

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Other Thermodynamic Other Thermodynamic FunctionsFunctions

The relationship among these The relationship among these properties is:properties is:

ttt PVUH

tttttt TSUTSPVHF

ttt TSHG

Flow processesFlow processes

Phase equilibriaPhase equilibria

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Differentials of Differentials of Thermodynamic FunctionsThermodynamic Functions

Expressions similar toExpressions similar to PSHHdPVTdSdH tttttt , i.e.

tttttt VTFFPdVdTSdF , i.e. TPGGdP VdTSdG ttttt , i.e.

ttt PdVTdSdU

The same relationships hold for The same relationships hold for the the intensive propertiesintensive properties (M = M(M = Mtt /n) /n)

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Key ConceptKey Concept

The The (U(Ut t ,H,Ht t ,F,Ft t ,G,Gt t ,S,St t )) are STATE are STATE properties which means properties which means independent of path. independent of path.

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State FunctionsState Functions

),( 111 TPM

Pres

sure

Pres

sure

TemperatureTemperature

),( 222 TPM

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Open SystemsOpen Systems

For an open system, For an open system, UUtt, H, Htt, F, Ftt, , and and GGtt, , will also depend on the concentration will also depend on the concentration of each of the components. of each of the components. The number of moles of each specie The number of moles of each specie may change due to:may change due to:

Chemical reaction within systemChemical reaction within systemInterchange of matter with surroundingsInterchange of matter with surroundingsInterchange and chemical reaction.Interchange and chemical reaction.

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The functional form of The functional form of UUtt, H, Htt, F, Ftt,, and and GG tt for open systems are, for open systems are,

cNtttt nnnVSUU ...,,,, 21

cNttt nnnPSHH ...,,,, 21

cNttt nnnVTFF ...,,,, 21

cNtt nnnPTGG ...,,,, 21

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The differential form of the The differential form of the thermodynamic functions arethermodynamic functions are

iinVS

N

i i

tttt dn

nUPdVTdSdU

jtt

c

,,1

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The differential form of the above The differential form of the above equations are,equations are,

iinVT

N

i i

tttt dn

nFPdVdTSdF

jt

c

,,1

iinPT

N

i i

tttt dn

nGdPVdTSdG

j

c

,,1

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Open SystemsOpen SystemsDefine theDefine the chemical potential of chemical potential of component component " i "" i " as as

in,P,Ti

ti

in,V,Ti

t

in,P,Si

t

in,V,Si

ti

j

jtjtjtt

nG ˆ

also and

nF

nH

nUˆ

Most well-knownMost well-known

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Second Law and the Second Law and the Equilibrium CriteriaEquilibrium Criteria

EquilibriumdS = 0

Time

Ent

ropy

, S

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Second Law and the Second Law and the Equilibrium CriteriaEquilibrium Criteria

The criteria of equilibrium of a system can also The criteria of equilibrium of a system can also be stated in terms of be stated in terms of UUtt, H, Htt, F, Ftt,, and and GGtt as followsas follows

The internal energy, The internal energy, UUtt,, must be a minimum at must be a minimum at constant constant SStt, , VVtt, and , and nnii..

The enthalpy, The enthalpy, HHtt,, must be a minimum at constant must be a minimum at constant SStt, P, P,, and and nnii..

The Helmholtz free energy, The Helmholtz free energy, FFtt,, must be a minimum must be a minimum at constant at constant T, VT, Vtt, and , and nnii..

The Gibbs free energy, The Gibbs free energy, GGtt, must be a minimum at , must be a minimum at constant constant T, PT, P, , and and nnii..

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Chemical and Phase Equilibria Chemical and Phase Equilibria Criteria for an Open System Criteria for an Open System Using Intensive PropertiesUsing Intensive Properties

VaporPv

Tv

niv

LiquidPl

Tl

nil

VaporPv

Tv

niv

LiquidPl

Tl

nil

Gas SystemGas System

Liquid systemLiquid systemopenopen

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Open System: Derivation of Open System: Derivation of Equilibrium ConditionsEquilibrium Conditions

Variation of internal energy for Variation of internal energy for liquid system isliquid system is

Variation of internal energy for gas Variation of internal energy for gas system issystem is v

i

N

i

vi

vvv dnnVPdnSTdnUdc

1

ˆ

li

N

i

li

lll dnnVPdnSTdnUdc

1

ˆ

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Derivation of Equilibrium Derivation of Equilibrium ConditionsConditions

Gas + Liquid systems make a closed Gas + Liquid systems make a closed system and the total energy issystem and the total energy is

thus for a closed system at equilibriumthus for a closed system at equilibrium

li

N

i

li

vi

N

i

vi dndnnVPdnSTdnUd

cc

11

ˆˆ

0ˆˆ11

li

N

i

li

vi

N

i

vi dndn

cc

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Derivation of Equilibrium Derivation of Equilibrium ConditionsConditions

From mass conservationFrom mass conservation

replace inreplace in

ThereforeTherefore

li

vi dndn

0ˆˆ11

li

N

i

li

vi

N

i

vi dndn

cc

0ˆˆ1

vi

li

N

i

vi dn

c

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Auxiliary Thermodynamic Auxiliary Thermodynamic FunctionsFunctions

The mole fractions are also The mole fractions are also thermodynamic functionsthermodynamic functions

and

11

l

li

N

i

li

li

iv

vi

N

i

vi

vi

i nn

n

nxnn

n

nycc

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Phase Equilibria ModelsPhase Equilibria Models

Can be classified according to:Can be classified according to:

the type of fluids (hydrocarbons, the type of fluids (hydrocarbons, alcohols, electrolytes, water and other alcohols, electrolytes, water and other non-hydrocarbon species)non-hydrocarbon species)

pressure and temperature ranges of pressure and temperature ranges of interest. interest.

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Phase Equilibria Models for… Phase Equilibria Models for…

Low-pressure rangesLow-pressure ranges, such as those , such as those of separator and surface conditions of separator and surface conditions High-pressures rangesHigh-pressures ranges which apply which apply to the reservoir. to the reservoir. Type of reservoir fluid, whether a Type of reservoir fluid, whether a black oil or a volatile oil, also black oil or a volatile oil, also determines the type of Phase determines the type of Phase equilibrium modelequilibrium model

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Residual PropertiesResidual Properties

Define the residual properties Define the residual properties for for mathematical conveniencemathematical convenience as the as the difference between the actual (real) difference between the actual (real) property minus the same property, property minus the same property, evaluated at the same pressure, evaluated at the same pressure, temperature, and composition, but temperature, and composition, but evaluated using the ideal gas evaluated using the ideal gas equation. equation.

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VLE VLE

We will start with the simpler We will start with the simpler models first, the ones for lower models first, the ones for lower

pressures pressures

Single Component Single Component & &

MulticomponentMulticomponent

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That isThat isMMR R = M-M= M-Migig M=U, H, G, S, FM=U, H, G, S, F ( (F F is is AA in American in American Notation) Notation)

M:M: Real Property @ (Real Property @ (T, PT, P) of the system ) of the system MMRR: : Residual PropertyResidual PropertyMMigig: : Property @ (Property @ (T, PT, P) of the system evaluated as ) of the system evaluated as if the fluid were an ideal gasif the fluid were an ideal gas

NoteNote:: there is no there is no TTRR or or PPRR

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Recall for a constant composition Recall for a constant composition closed closed systemsystem

SdTVdPdG dTSdPVdG igigig

dTSdPVdG RRR

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Note that the properties used in these Note that the properties used in these equations are equations are intensive propertiesintensive properties, , that is the volume is the molar volumethat is the volume is the molar volume GG and and SS are expressed in are expressed in BTU/lb-molBTU/lb-mol and BTU/lb-mol-R, respectively, (or in and BTU/lb-mol-R, respectively, (or in cal/g-mol, cal/g-mol K in the SI system cal/g-mol, cal/g-mol K in the SI system of units).of units).

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Gibbs Residual EnergyGibbs Residual Energy

At constant temperature,At constant temperature,

Divide by Divide by RT RT

dPVdG RR

P RG RRR

dPRTV

RTdGdP

RTV

RTdG

R

00

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Gibbs Residual EnergyGibbs Residual Energy

From previous lectures we had:From previous lectures we had:

1 , RT

PVzRTPV ig

Pz

RTV R )1(

PR

PdPz-

RTG

0

1

ThusThus

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Phase Equilibrium of a Single Phase Equilibrium of a Single ComponentComponent

RecallRecall

lvlll

vvv GGdTdP-S V dGdTdP-S V dG

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Predicted Isotherms from a Predicted Isotherms from a cubic EOScubic EOS

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Phase Equilibrium Phase Equilibrium Single ComponentSingle Component

For constant temperature,For constant temperature,

At equilibrium At equilibrium PP11=P=P55=P=P

0)( PdVPVdVdPdG

5

1

5

11155 0 dVPVPVPdG

5

115 0 )( dVPVVP

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Phase Equilibrium of a Single Phase Equilibrium of a Single ComponentComponent

By inspection, By inspection,

And also, And also,

)176531()( 15 AreaVVP

)17654321(5

1

AreaPdV

)3543()1231()176531()17654321(

AreaAreaAreaArea

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Maxwell Equal Area Rule Maxwell Equal Area Rule

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VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form

Write the EOS in dimensionless Write the EOS in dimensionless form using form using TTrr=T/T=T/Tcc, , PPrr=P/P=P/Pcc, , VVrr=V/V=V/Vcc, , and the values for and the values for aa and and bb found found from the critical constraintsfrom the critical constraints

0 ,0 2

2

cTcT VP

VP

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For Van der Waals EOSFor Van der Waals EOS

withwith

2Va

bVRTP

ccVRTa89

c

cc

PRTVb83

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andand

c

ccc RT

VPz 83

c

cc P

RTV83

2283

rc

cc

ccr

crcr VV

VRTaVVV

TRTPP

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Application of Equal Area RuleApplication of Equal Area Rule

At At TT constant constant

oror

0dG

0 )( dVPPVdVdPdG

rg

rl

V

Vrrrlrgr dVPVVP

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Replacing and integratingReplacing and integrating

Since Since P P is constant, is constant,

Thus,Thus,

rlrgrl

rgrrlrgr VVV

VTVVP 331313

ln3

8

0dP

0 dPV

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0 r

rg

rl r

rg

rl

V

Vr

T

rr

V

Vrr dV

VPVdPV

rg

rl

V

Vr

rrr

dVV)-V(

Tr-V 613

24 32

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From integral tables,From integral tables,

etc.,etc.,

bxaabxa

bbxaxdx )ln(1

)( 22

01149

131

131

1313

ln

rgrlrrlrgrg

rg

VVTVVVV

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Have three equations to work withHave three equations to work with

EOSEOS

Maxwell Equal Area RuleMaxwell Equal Area Rule

andand 0 dPV

unknowns unknowns PPrr, V, Vrlrl, V, Vrgrg..

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VLE at low pressuresVLE at low pressures

We will see first We will see first models that apply models that apply ONLY for low ONLY for low pressurespressures

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Systems of Variable Systems of Variable Composition: Ideal BehaviorComposition: Ideal Behavior

Applications to low pressuresApplications to low pressuresSimplifications Simplifications

the gas phase behaves as anthe gas phase behaves as an Ideal Ideal Gas Gas the liquid phase exhibitsthe liquid phase exhibits Ideal Ideal Solution Behavior.Solution Behavior.

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The equilibrium criteria between 2 The equilibrium criteria between 2 phases phases and and is, is,

cii Ni

TT

PP

,....2,1,ˆˆ

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Thus, at constant Thus, at constant TT and and PP,,

c

c

n

i

li

li

l

n

i

vi

vi

v

dnnGd

dnnGd

1

1

ˆ)(

ˆ)(

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Simplest VLE model (IG+IS) imply thatSimplest VLE model (IG+IS) imply that

IGIG:: molecular interactions are zero, molecular interactions are zero, molecules have no volume.molecules have no volume.

ISIS:: forces of attraction/repulsion between forces of attraction/repulsion between molecules are the same regardless of molecules are the same regardless of molecular species. Volumes are additive molecular species. Volumes are additive ((Amagat’s LawAmagat’s Law).).

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Forces between molecular Forces between molecular speciesspecies

A A B B A B

ABBBAA FFF

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Ideal Gas MixtureIdeal Gas Mixture

The pressure in a vessel The pressure in a vessel containing an ideal gas mixture (containing an ideal gas mixture (nn) ) or a single gas component (or a single gas component (nnkk) is ) is

t

kk

t

VRTnP

VnRTP

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PPkk is the partial is the partial pressure of pressure of component component k, k, and by and by definitiondefinition

kkk y

nn

PP

cN

ik PP

1

P p k

T1 T 1

n1 ,n 2, nk…, nk

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Generalize this principle to any Generalize this principle to any thermodynamic property thermodynamic property for an for an ideal gas mixtureideal gas mixture

cn

kk

igkk

ig PTMnPTnM1

),(),(

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““A total thermodynamic A total thermodynamic property (property (nU, nG, nS, nH, nU, nG, nS, nH,

nFnF) of an ideal gas ) of an ideal gas mixture is the mixture is the of the of the total properties of the total properties of the

individual species each individual species each evaluated at the evaluated at the TT of the of the mixture and at its own mixture and at its own

partial pressure.”partial pressure.”

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Derive Equilibrium RelationsDerive Equilibrium Relations

Begin with an ideal gasBegin with an ideal gas

The enthalpy of an ideal gas is The enthalpy of an ideal gas is independent of pressure, thusindependent of pressure, thus

igigig TSHG

),(),( kig

kig

k PTHPTH

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For the entropy, we must express For the entropy, we must express

),( PTSS igig

dTTSdP

PSdS

PT

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Recall Maxwell RulesRecall Maxwell Rules

For ideal gas,For ideal gas,

dTTc

dPTVdS p

P

dTT

cdP

PRdS

igpig

k

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at constant temperature,at constant temperature,

P

P

P

P

igk

igk

kk

PdRdS

PRddPPRdS

ln

ln

kk

kig

kig

k ylnRPPlnR)P,T(S)P,T(S

05/03/23 67


We also know from ideal We also know from ideal averaging applied to entropy,averaging applied to entropy,

cN

kk

igkk

ig PTSnPTnS1

),(),(

cN

kk

igkk

ig )P,T(Sy)P,T(S1

05/03/23 68


Substituting,Substituting,

the entropy change of mixing the ideal the entropy change of mixing the ideal

gases is not zerogases is not zero

cc N

kkk

N

kk

igkk

ig yyRPTSyPTS11

ln),(),(

cc N

k kk

N

k

igkk

ig

yyRSyS

1101ln

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Now, we can build the expression Now, we can build the expression for the Gibbs energy for the Gibbs energy for an ideal for an ideal gas.gas.

recallrecall

ccc N

kkk

N

k

igkk

N

k

igkk

ig yyRTPTSyTPTHyPTG111

ln),(),(),(

ijnPTii n

nG

,,

ˆ

05/03/23 70


Expressed in terms of Expressed in terms of n (yn (ykk=n=nkk/n),/n),

cc N

k

kkN

k

igk

kig

nn

nnRTG

nnPTG

11ln),(

c cc N

i

n

ikkk

N

i

igkk

ig nnnnRTGnPTnG1 11

ln)(ln),(

05/03/23 71


Recall,Recall,

cN

iinn

1

kjnn

kjnnnn

j

k

j

k

k

,1

,0

1

05/03/23 72


Therefore, Therefore,

nnn

nnnRTG

nnG k

i

ii

igi

npTi

igig

i

ij

lnlnˆ,,

iig

iig

i yRTG lnˆ

05/03/23 73

Ideal SolutionIdeal Solution

Following the same reasoning as Following the same reasoning as for gases, we have that,for gases, we have that,

iiiid xRSxS

iiiiid xxRTGxG ln

iiid

i RTxG ̂Here, Here, SSii and and GGii are the properties of the pure are the properties of the pure species in the liquid state at the species in the liquid state at the T T and and PP of the of the

mixture.mixture.

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Raoult’s LawRaoult’s Law

It is a combination of IG + IS It is a combination of IG + IS models. VLE for a mixture of models. VLE for a mixture of NNcc componentscomponents

c

idli

igvi

li

vi

N,...,i

)ˆ()ˆ(ˆˆ

1

05/03/23 75


Thus, at Thus, at TT and and PP,,

il

iiig

i xRTGyRTG lnln

),(),(ln PTGPTGxyRT ig

il

ii

i

The right hand side of this Eq. indicates pure The right hand side of this Eq. indicates pure species properties evaluated at the equilibrium species properties evaluated at the equilibrium TT

and and PP of the mixture of the mixture

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As we seen before for a pure As we seen before for a pure component,component,

So, this leads to So, this leads to Raoult’s LawRaoult’s Law!!

PPRTPTGPTG

xyRT i

iig

iil

ii

i

ln),(),(ln

0),(),( i

igii

li PTGPTG

iii PxPy

05/03/23 77

Equilibrium RatioEquilibrium Ratio

Vapor-Liquid Equilibrium ratio is defined asVapor-Liquid Equilibrium ratio is defined as

There are several correlations and models for KThere are several correlations and models for Kii

From Rault’s law (ideal model ) From Rault’s law (ideal model )

i

ii x

yK

ii

i

i KP

Pxy

RECALL LIMITATIONS OF RECALL LIMITATIONS OF IDEAL MODELIDEAL MODEL

05/03/23 78

Equilibrium RatioEquilibrium Ratio

READREAD papers placed in module 3 papers placed in module 3 folder for other composition-folder for other composition-independent k-value models (we will independent k-value models (we will have exercises using them)have exercises using them)Compositional dependence in Compositional dependence in considered when using EOS … but considered when using EOS … but K-values become implicitK-values become implicit

05/03/23 79

Bubble Point EvaluationBubble Point Evaluation

Under Raoult’s law, the bubble point Under Raoult’s law, the bubble point has a has a linear dependencelinear dependence with the vapor with the vapor pressures of the pure components.pressures of the pure components.

Once the bubble point pressure is Once the bubble point pressure is found, the equilibrium vapor found, the equilibrium vapor compositions are found from Raoult’s compositions are found from Raoult’s law.law.

05/03/23 80

Deviations from Raoult's lawDeviations from Raoult's lawThe dew point curve (lower The dew point curve (lower black curve) in is always black curve) in is always curved regardless whether curved regardless whether the mixture is ideal or not.the mixture is ideal or not. The red curves in indicate The red curves in indicate deviations from Raoult's law. deviations from Raoult's law. When the bubble point curve When the bubble point curve is above the straight line, we is above the straight line, we will have positive deviations will have positive deviations from Raoult's Law. When the from Raoult's Law. When the bubble point curve is below bubble point curve is below the straight line, we will have the straight line, we will have negative deviations from negative deviations from Raoult's Law. This happens Raoult's Law. This happens for non-ideal mixtures and for non-ideal mixtures and may lead to azeotropy.may lead to azeotropy.

P2

P1T

x1,y1

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Dew Point CalculationDew Point Calculation

At the dew point the overall fluid At the dew point the overall fluid composition coincides with the composition coincides with the gas composition. That is.gas composition. That is.

ii yz

05/03/23 82

Statement of EquilibriumStatement of Equilibrium

iii PxPy

P

T

1

23

1P

IG/IS Raoult’sIG/IS Raoult’s lawlaw

05/03/23 83

Bubble Point EvaluationBubble Point Evaluation

The bubble point pressure at a The bubble point pressure at a given given TT is is

iibpi PzPy

iibp PzP

05/03/23 84

Dew Point CalculationDew Point Calculation

Find DP pressure and equilibrium Find DP pressure and equilibrium liquid compositionsliquid compositions

iii

iii

PxPz

PxPy Px

Pz i

i

i

1

1

cN

i i

idp P

zP

05/03/23 85

Types of Phase Equilibria Types of Phase Equilibria CalculationsCalculations

Given Variables(independent)

Unknown Variables(dependent)

ProblemType

ExampleApplication

P, zi = xi T, yi Bubble Point

T, zi = xi P,yi Bubble Point Gas injection,production

P, zi = yi T,xi Dew Point

T, zi = yi P,yi Dew PointGas

Condensates,Production

P, T, zi xi, yi, fv Flash ProductionSeparation

05/03/23 86

Bubble Point Temperature Bubble Point Temperature given Pgiven P

We must follow an iterative We must follow an iterative procedure.procedure.

ii xz

Bubble point temperature Bubble point temperature enters into the equation non-linearlyenters into the equation non-linearly

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Bubble Point TemperatureBubble Point Temperature

Find TB pressure and equilibrium Find TB pressure and equilibrium gas compositionsgas compositions

TbPzPy

orTbPxPy

iii

iii

05/03/23 88


The problem is that we do not The problem is that we do not know yet at what temperature to know yet at what temperature to evaluate the pure component evaluate the pure component vapor pressures. See the following vapor pressures. See the following diagram diagram

)( bpiibp TPzP

05/03/23 89


For well-behaved systems (no For well-behaved systems (no azeotropes), the searched azeotropes), the searched temperature will be bounded by temperature will be bounded by the highest and lowest saturation the highest and lowest saturation temperature of the components in temperature of the components in the mixture at the selected system the mixture at the selected system pressure.pressure.

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T2

T1

P

x1,y1

P

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Bubble Point Temperature Bubble Point Temperature ProcedureProcedure

1. Evaluate and at the given pressure 1. Evaluate and at the given pressure PP, which is a saturation pressure., which is a saturation pressure.

2. Choose your 2. Choose your first guessfirst guess bubble point bubble point temperature astemperature as

1T

2T

ii

iii cT

baP

ln i

i

ii c

PlnabT

i

n

ii

obp TzT

c

1

)(

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3. Define a relative volatility using a reference 3. Define a relative volatility using a reference substance such that all relative volatilities are substance such that all relative volatilities are either > 0 or < 0 (i.e. monotonically increasing either > 0 or < 0 (i.e. monotonically increasing or decreasing).or decreasing).

with the saturation pressures evaluated at the guess with the saturation pressures evaluated at the guess temperature evaluated in (2)temperature evaluated in (2)

j

iij P

P

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4. Expand the volatility as 4. Expand the volatility as

with T from step 2.with T from step 2.

2

2

1

1212112 lnlnln

cTb

cTbaaPP

2

2

1

12112 exp

cTb

cTbaa

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5. Write the bubble point equation in terms of 5. Write the bubble point equation in terms of volatilities and a reference vapor pressure volatilities and a reference vapor pressure (lowest or highest)(lowest or highest)For a binary, you would have only one For a binary, you would have only one volatilityvolatility

2121222

1122211 zzPz

PPzPPzPzP

21212 zz

PP

Guessed vapor pressureGuessed vapor pressure

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ThusThus

this is your this is your first guessfirst guess saturation pressure for saturation pressure for the reference component (here “2”) at the the reference component (here “2”) at the first first guessguess temperature evaluated in step 1. temperature evaluated in step 1. From this saturation pressure use the Antoine From this saturation pressure use the Antoine equation to find an updated bubble point equation to find an updated bubble point temperature (step 1).temperature (step 1).

21212 zz

PP

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From the saturation pressure From the saturation pressure evaluated in use the Antoine evaluated in use the Antoine equation to find a new equation to find a new temperaturetemperature

222

2 cPlna

bTb

21212 zz

PP

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This new This new TT new new new new iterate until two successive iterate until two successive temperatures do not change by a temperatures do not change by a specified tolerance.specified tolerance.The Excel file provided in our WEB site The Excel file provided in our WEB site illustrates this procedure for a ternary illustrates this procedure for a ternary mixture. You can modify it and extend it mixture. You can modify it and extend it to multicomponents.to multicomponents.

12 2P

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Dew Point Temperature Dew Point Temperature ProcedureProcedure

You can follow a very similar You can follow a very similar reasoning as the one developed reasoning as the one developed for the bubble point and devise for the bubble point and devise the algorithm required to solve the algorithm required to solve this problem using relative this problem using relative volatilitiesvolatilities

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Flash CalculationsFlash Calculations

In this type of calculations, the work-In this type of calculations, the work-horse of reservoir simulation packages, horse of reservoir simulation packages, the objective is to:the objective is to:

findfind fraction of vapor vaporized and fraction of vapor vaporized and equilibrium gas and liquid compositionsequilibrium gas and liquid compositions givengiven the overall mixture composition, the overall mixture composition, P P andand T T..

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Start with the equilibrium equationStart with the equilibrium equation

Material balanceMaterial balance

iii PxPy

vivivilii fyfxfyfxz 1

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Now replace either liquid or gas Now replace either liquid or gas compositions using equilibrium compositions using equilibrium equation equation

vivi

ii fyfPPyz 1

i

i

i xP

Py

Here replaced Here replaced xxii

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Rearrange and sum over all Rearrange and sum over all compositionscompositions

vvi

ii

ffPP

zy

1

vvi

ii

ffPP

zy1

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Separation processSeparation process

zi(T1,P1)

xi(T1,P2)

yi(T1,P2)

P1 > P2T1,P2

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Objective function (flash function) Objective function (flash function) is is

01

1

vvi

iv

ffPP

zfF

)(

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Flash CalculationsFlash CalculationsThere are several equivalent expressions for the There are several equivalent expressions for the flash function flash function

(a) (a)

(b) (b)

(c) (c)

(c) is (c) is the best well behaved for the best well behaved for numerical numerical solution (Rachford- Rice solution (Rachford- Rice function)function)

01 iy

01 ix

0 ii xy

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Once Once ffvv is found the equilibrium is found the equilibrium gas and liquid compositions are gas and liquid compositions are evaluated fromevaluated from

vvi

ii

ffPP

zy

1

i

i

i xP

Pyandand

)/( IGISii kpp

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1111

cN

i iv

ikfz

1111

cN

i iv

iikfkz

cN

i iv

ii

kfkz

1 11)1(

VLE ExamplesFlash Functions and Rachford Rice

Function

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

0.00 0.20 0.40 0.60 0.80 1.00

Molar Fraction of vapor (fv)

F(fv

)

Sum XiSum YiRachford Rice

05/03/23 108

x1, y1

Ta

Tb

Tc

P1v

P2v

Pres

sure

x1, y1

Pa

Pb

Pc

T1v

T2v

Tem

pera

ture

But ….Raoult’s model will NOT work well But ….Raoult’s model will NOT work well in these casesin these cases

then what ?then what ?

05/03/23 109

Equations of Equations of StateState(EOS)(EOS)