05/03/23 1
Termodinamica de Termodinamica de HidrocarburosHidrocarburos
Generalized Phase Equilibria Generalized Phase Equilibria
05/03/23 3
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.
05/03/23 4
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..
05/03/23 5
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.
05/03/23 6
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.
05/03/23 7
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
05/03/23 8
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
05/03/23 9
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
05/03/23 10
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
05/03/23 11
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
05/03/23 12
First Law and Fundamental First Law and Fundamental Thermodynamic RelationshipsThermodynamic Relationships
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
05/03/23 13
First Law and Fundamental First Law and Fundamental Thermodynamic RelationshipsThermodynamic Relationships
Replacing work and heat Replacing work and heat expressionsexpressions
ThusThus
ttt PdVTdSdU
tttt VSUU ,
05/03/23 14
First Law and Fundamental First Law and Fundamental Thermodynamic RelationshipsThermodynamic Relationships
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
05/03/23 15
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
05/03/23 16
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)
05/03/23 17
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.
05/03/23 18
State FunctionsState Functions
),( 111 TPM
Pres
sure
Pres
sure
TemperatureTemperature
),( 222 TPM
05/03/23 19
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.
05/03/23 20
Open SystemsOpen Systems
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
05/03/23 21
Open SystemsOpen Systems
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
05/03/23 22
Open SystemsOpen Systems
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
05/03/23 23
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
05/03/23 24
Second Law and the Second Law and the Equilibrium CriteriaEquilibrium Criteria
EquilibriumdS = 0
Time
Ent
ropy
, S
05/03/23 25
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..
05/03/23 26
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
05/03/23 27
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
05/03/23 28
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
05/03/23 29
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
05/03/23 30
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
05/03/23 31
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.
05/03/23 32
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
05/03/23 33
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.
05/03/23 34
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
05/03/23 35
Residual PropertiesResidual Properties
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
05/03/23 36
Residual PropertiesResidual Properties
Recall for a constant composition Recall for a constant composition closed closed systemsystem
SdTVdPdG dTSdPVdG igigig
dTSdPVdG RRR
05/03/23 37
Residual PropertiesResidual Properties
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).
05/03/23 38
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
05/03/23 39
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
05/03/23 40
Phase Equilibrium of a Single Phase Equilibrium of a Single ComponentComponent
RecallRecall
lvlll
vvv GGdTdP-S V dGdTdP-S V dG
05/03/23 42
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
05/03/23 43
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
05/03/23 45
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
05/03/23 46
VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form
For Van der Waals EOSFor Van der Waals EOS
withwith
2Va
bVRTP
ccVRTa89
c
cc
PRTVb83
05/03/23 47
VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form
andand
c
ccc RT
VPz 83
c
cc P
RTV83
2283
rc
cc
ccr
crcr VV
VRTaVVV
TRTPP
05/03/23 48
Application of Equal Area RuleApplication of Equal Area Rule
At At TT constant constant
oror
0dG
0 )( dVPPVdVdPdG
rg
rl
V
Vrrrlrgr dVPVVP
05/03/23 49
VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form
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
05/03/23 50
VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form
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
05/03/23 51
VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form
From integral tables,From integral tables,
etc.,etc.,
bxaabxa
bbxaxdx )ln(1
)( 22
01149
131
131
1313
ln
rgrlrrlrgrg
rg
VVTVVVV
05/03/23 52
VLE in Dimensionless or VLE in Dimensionless or Reduced FormReduced Form
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..
05/03/23 53
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
05/03/23 54
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.
05/03/23 55
Systems of Variable Systems of Variable Composition: Ideal BehaviorComposition: Ideal Behavior
The equilibrium criteria between 2 The equilibrium criteria between 2 phases phases and and is, is,
cii Ni
TT
PP
,....2,1,ˆˆ
05/03/23 56
Systems of Variable Systems of Variable Composition: Ideal BehaviorComposition: Ideal Behavior
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)(
05/03/23 57
Systems of Variable Systems of Variable Composition: Ideal BehaviorComposition: Ideal Behavior
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).).
05/03/23 59
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
05/03/23 60
Systems of Variable Systems of Variable Composition: Ideal BehaviorComposition: Ideal Behavior
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
05/03/23 61
Systems of Variable Systems of Variable Composition: Ideal BehaviorComposition: Ideal Behavior
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
),(),(
05/03/23 62
““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.”
05/03/23 63
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
05/03/23 64
Derive Equilibrium RelationsDerive Equilibrium Relations
For the entropy, we must express For the entropy, we must express
),( PTSS igig
dTTSdP
PSdS
PT
05/03/23 65
Derive Equilibrium RelationsDerive Equilibrium Relations
Recall Maxwell RulesRecall Maxwell Rules
For ideal gas,For ideal gas,
dTTc
dPTVdS p
P
dTT
cdP
PRdS
igpig
k
05/03/23 66
Derive Equilibrium RelationsDerive Equilibrium Relations
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
Derive Equilibrium RelationsDerive Equilibrium Relations
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
Derive Equilibrium RelationsDerive Equilibrium Relations
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
05/03/23 69
Derive Equilibrium RelationsDerive Equilibrium Relations
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
Derive Equilibrium RelationsDerive Equilibrium Relations
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
Derive Equilibrium RelationsDerive Equilibrium Relations
Recall,Recall,
cN
iinn
1
kjnn
kjnnnn
j
k
j
k
k
,1
,0
1
05/03/23 72
Derive Equilibrium RelationsDerive Equilibrium Relations
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.
05/03/23 74
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
Raoult’s LawRaoult’s Law
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
05/03/23 76
Raoult’s LawRaoult’s Law
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
05/03/23 81
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
05/03/23 87
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
Bubble Point TemperatureBubble Point Temperature
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
Bubble Point TemperatureBubble Point Temperature
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.
05/03/23 91
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
)(
05/03/23 92
Bubble Point Temperature Bubble Point Temperature ProcedureProcedure
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
05/03/23 93
Bubble Point Temperature Bubble Point Temperature ProcedureProcedure
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
05/03/23 94
Bubble Point Temperature Bubble Point Temperature ProcedureProcedure
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
05/03/23 95
Bubble Point Temperature Bubble Point Temperature ProcedureProcedure
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
05/03/23 96
Bubble Point Temperature Bubble Point Temperature ProcedureProcedure
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
05/03/23 97
Bubble Point Temperature Bubble Point Temperature ProcedureProcedure
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
05/03/23 98
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
05/03/23 99
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..
05/03/23 100
Flash CalculationsFlash Calculations
Start with the equilibrium equationStart with the equilibrium equation
Material balanceMaterial balance
iii PxPy
vivivilii fyfxfyfxz 1
05/03/23 101
Flash CalculationsFlash Calculations
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
05/03/23 102
Flash CalculationsFlash Calculations
Rearrange and sum over all Rearrange and sum over all compositionscompositions
vvi
ii
ffPP
zy
1
vvi
ii
ffPP
zy1
05/03/23 104
Flash CalculationsFlash Calculations
Objective function (flash function) Objective function (flash function) is is
01
1
vvi
iv
ffPP
zfF
)(
05/03/23 105
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
05/03/23 106
Flash CalculationsFlash Calculations
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
05/03/23 107
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 ?
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