L7 Phase Equilibria
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Termodinamica de HidrocarburosGeneralized Phase Equilibria

Phase EquilibriumEvolution

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

The Concept of EquilibriumTendencies toward a change are caused by a driving force of any kind
Equilibrium means the absence of any driving force, or that all forces are in exact balance.

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

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

Phase RuleIt tells us the number of independent variables required to fully characterize a systemIt does not tell us which variables to select

Generalization of the phase rule for Nc non reacting components

Phase Equilibrium and the Phase RuleThus for nonreacting systems
F = # of variables # of Independent equations relating these variables

First Law and Fundamental Thermodynamic Relationships Closed Systems The system does not exchange matter with the surroundings, but it can exchange energy. The first law is a generalization of the conservation of energy

Compression and expansion work in a gas container indicating the convention used for heat and work
Heat and Work Sign Convention

First Law and Fundamental Thermodynamic RelationshipsFor a reversible process, dQ = TdSt thus,
If the work of expansion or compression is the only kind of work allowed then:

First Law and Fundamental Thermodynamic RelationshipsReplacing work and heat expressions
Thus

First Law and Fundamental Thermodynamic RelationshipsSince
Thus one can identify,
Define: Mt = nM with M = U, H, A, G, S

Other Thermodynamic FunctionsThe relationship among these properties is:
Flow processesPhase equilibria

Differentials of Thermodynamic FunctionsExpressions similar to
The same relationships hold for the intensive properties (M = Mt /n)

Key ConceptThe (Ut ,Ht ,Ft ,Gt ,St ) are STATE properties which means independent of path.

State FunctionsPressureTemperature

Open Systems For an open system, Ut, Ht, Ft, and Gt, will also depend on the concentration of each of the components. The number of moles of each specie may change due to:
Chemical reaction within systemInterchange of matter with surroundingsInterchange and chemical reaction.

Open SystemsThe functional form of Ut, Ht, Ft, and G t for open systems are,

Open SystemsThe differential form of the thermodynamic functions are

Open SystemsThe differential form of the above equations are,

Open SystemsDefine the chemical potential of component " i " as
Most wellknown

Second Law and the Equilibrium Criteria

Second Law and the Equilibrium CriteriaThe criteria of equilibrium of a system can also be stated in terms of Ut, Ht, Ft, and Gt as follows
The internal energy, Ut, must be a minimum at constant St, Vt, and ni.The enthalpy, Ht, must be a minimum at constant St, P, and ni.The Helmholtz free energy, Ft, must be a minimum at constant T, Vt, and ni.The Gibbs free energy, Gt, must be a minimum at constant T, P, and ni.

Chemical and Phase Equilibria Criteria for an Open System Using Intensive Properties
Gas SystemLiquid systemopen

Open System: Derivation of Equilibrium ConditionsVariation of internal energy for liquid system is
Variation of internal energy for gas system is

Derivation of Equilibrium ConditionsGas + Liquid systems make a closed system and the total energy is
thus for a closed system at equilibrium

Derivation of Equilibrium ConditionsFrom mass conservation
replace in
Therefore

Auxiliary Thermodynamic Functions The mole fractions are also thermodynamic functions

Phase Equilibria ModelsCan be classified according to:
the type of fluids (hydrocarbons, alcohols, electrolytes, water and other nonhydrocarbon species)
pressure and temperature ranges of interest.

Phase Equilibria Models for Lowpressure ranges, such as those of separator and surface conditions Highpressures ranges which apply to the reservoir. Type of reservoir fluid, whether a black oil or a volatile oil, also determines the type of Phase equilibrium model

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

VLE
We will start with the simpler models first, the ones for lower pressures Single Component & Multicomponent

Residual PropertiesThat is
MR = MMigM=U, H, G, S, F (F is A in American Notation)
M:Real Property @ (T, P) of the system MR: Residual PropertyMig: Property @ (T, P) of the system evaluated as if the fluid were an ideal gas Note: there is no TR or PR

Residual PropertiesRecall for a constant composition closed system

Residual PropertiesNote that the properties used in these equations are intensive properties, that is the volume is the molar volume G and S are expressed in BTU/lbmol and BTU/lbmolR, respectively, (or in cal/gmol, cal/gmol K in the SI system of units).

Gibbs Residual EnergyAt constant temperature,
Divide by RT

Gibbs Residual EnergyFrom previous lectures we had:
Thus

Phase Equilibrium of a Single ComponentRecall

Predicted Isotherms from a cubic EOS

Phase Equilibrium Single ComponentFor constant temperature,
At equilibrium P1=P5=Ps

Phase Equilibrium of a Single ComponentBy inspection,
And also,

Maxwell Equal Area Rule

VLE in Dimensionless or Reduced FormWrite the EOS in dimensionless form using Tr=T/Tc, Pr=P/Pc, Vr=V/Vc, and the values for a and b found from the critical constraints

VLE in Dimensionless or Reduced FormFor Van der Waals EOS
with

VLE in Dimensionless or Reduced Formand

Application of Equal Area Rule
At T constant
or

VLE in Dimensionless or Reduced FormReplacing and integrating
Since P is constant,
Thus,

VLE in Dimensionless or Reduced Form

VLE in Dimensionless or Reduced FormFrom integral tables,
etc.,

VLE in Dimensionless or Reduced FormHave three equations to work with
EOS
Maxwell Equal Area Rule
and
unknowns Prs, Vrl, Vrg.

VLE at low pressuresWe will see first models that apply ONLY for low pressures

Systems of Variable Composition: Ideal BehaviorApplications to low pressuresSimplifications
the gas phase behaves as an Ideal Gas the liquid phase exhibits Ideal Solution Behavior.

Systems of Variable Composition: Ideal Behavior
The equilibrium criteria between 2 phases a and b is,

Systems of Variable Composition: Ideal BehaviorThus, at constant T and P,

Systems of Variable Composition: Ideal Behavior
Simplest VLE model (IG+IS) imply that
IG: molecular interactions are zero, molecules have no volume.
IS: forces of attraction/repulsion between molecules are the same regardless of molecular species. Volumes are additive (Amagats Law).

Forces between molecular species
A AB BA B

Ideal Gas MixtureThe pressure in a vessel containing an ideal gas mixture (n) or a single gas component (nk) is

Systems of Variable Composition: Ideal Behavior
Pk is the partial pressure of component k, and by definition

Systems of Variable Composition: Ideal BehaviorGeneralize this principle to any thermodynamic property for an ideal gas mixture

A total thermodynamic property (nU, nG, nS, nH, nF) of an ideal gas mixture is the S of the total properties of the individual species each evaluated at the T of the mixture and at its own partial pressure.

Derive Equilibrium Relations Begin with an ideal gas
The enthalpy of an ideal gas is independent of pressure, thus

Derive Equilibrium Relations
For the entropy, we must express

Derive Equilibrium RelationsRecall Maxwell Rules
For ideal gas,

Derive Equilibrium Relationsat constant temperature,

Derive Equilibrium RelationsWe also know from ideal averaging applied to entropy,

Derive Equilibrium RelationsSubstituting,
the entropy change of mixing the ideal gases is not zero

Derive Equilibrium RelationsNow, we can build the expression for the Gibbs energy for an ideal gas.
recall

Derive Equilibrium RelationsExpressed in terms of n (yk=nk/n),

Derive Equilibrium RelationsRecall,

Derive Equilibrium RelationsTherefore,

Ideal SolutionFollowing the same reasoning as for gases, we have that,
Here, Si and Gi are the properties of the pure species in the liquid state at the T and P of the mixture.

Raoults LawIt is a combination of IG + IS models. VLE for a mixture of Nc components

Raoults LawThus, at T and P,
The right hand side of this Eq. indicates pure species properties evaluated at the equilibrium T and P of the mixture

Raoults Law
As we seen before for a pure component,
So, this leads to Raoults Law!

Equilibrium RatioVaporLiquid Equilibrium ratio is defined as
There are several correlations and models for KiFrom Raults law (ideal model )
RECALL LIMITATIONS OF IDEAL MODEL

Equilibrium RatioREAD papers placed in module 3 folder for other compositionindependent kvalue models (we will have exercises using them)Compositional dependence in considered when using EOS but Kvalues become implicit

Bubble Point Evaluation Under Raoults law, the bubble point has a linear dependence with the vapor pressures of the pure components.
Once the bubble point pressure is found, the equilibrium vapor compositions are found from Raoults law.

Deviations from Raoult's lawThe dew point curve (lower black curve) in is always curved regardless whether the mixture is ideal or not. The red curves in indicate deviations from Raoult's law. When the bubble point curve is above the straight line, we will have positive deviations from Raoult's Law. When the bubble point curve is below the straight line, we will have negative deviations from Raoult's Law. This happens for nonideal mixtures and may lead to azeotropy.
P
2
P
1
T
x
1
,y
1

Dew Point CalculationAt the dew point the overall fluid composition coincides with the gas composition. That is.

Statement of EquilibriumPT123IG/IS Raoults law

Bubble Point Evaluation The bubble point pressure at a given T is

Dew Point CalculationFind DP pressure and equilibrium liquid compositions

Types of Phase Equilibria Calculations

Bubble Point Temperature given P We must follow an iterative procedure.
Bubble point temperature enters into the equation nonlinearly

Bubble Point Temperature Find TB pressure and equilibrium gas compositions

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

Bubble Point TemperatureFor wellbehaved systems (no azeotropes), the searched temperature will be bounded by the highest and lowest saturation temperature of the components in the mixture at the selected system pressure.

Bubble Point Temperature
T
2
T
1
P
x
1
,y
1
P

Bubble Point Temperature Procedure1. Evaluate and at the given pressure P, which is a saturation pressure.
2. Choose your first guess bubble point temperature as

Bubble Point Temperature Procedure3. Define a relative volatility using a reference substance such that all relative volatilities are either > 0 or < 0 (i.e. monotonically increasing or decreasing).
with the saturation pressures evaluated at the guess temperature evaluated in (2)

Bubble Point Temperature Procedure4. Expand the volatility as
with T from step 2.

Bubble Point Temperature Procedure5. Write the bubble point equation in terms of volatilities and a reference vapor pressure (lowest or highest)For a binary, you would have only one volatility
Guessed vapor pressure

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

Bubble Point Temperature Procedure
From the saturation pressure evaluated in use the Antoine equation to find a new temperature

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

Dew Point Temperature ProcedureYou can follow a very similar reasoning as the one developed for the bubble point and devise the algorithm required to solve this problem using relative volatilities

Flash CalculationsIn this type of calculations, the workhorse of reservoir simulation packages, the objective is to:
find fraction of vapor vaporized and equilibrium gas and liquid compositions given the overall mixture composition, P and T.

Flash CalculationsStart with the equilibrium equation
Material balance

Flash CalculationsNow replace either liquid or gas compositions using equilibrium equation
Here replaced xi

Flash CalculationsRearrange and sum over all compositions

Separation process

Flash CalculationsObjective function (flash function) is

Flash CalculationsThere are several equivalent expressions for the flash function
(a)
(b)
(c)
(c) is the best well behaved for numerical solution (Rachford Rice function)

Flash CalculationsOnce fv is found the equilibrium gas and liquid compositions are evaluated from
and

VLE Examples

But .Raoults model will NOT work well in these casesthen what ?

Equations of State(EOS)