PETE 310PETE 310Lectures # 25 -26Lectures # 25 -26

Chapter 12 Chapter 12

Gas-Liquid Gas-Liquid EquilibriumEquilibrium

Gas-Liquid Equilibrium Gas-Liquid Equilibrium Ideal BehaviorIdeal Behavior

Applications to low pressuresApplications to low pressuresSimplifications Simplifications

the gas phase behaves as anthe gas phase behaves as an Ideal Gas Ideal Gas

the liquid phase exhibitsthe liquid phase exhibits Ideal Ideal Solution BehaviorSolution Behavior

Ideal BehaviorIdeal Behavior

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

cii Ni

TT

PP

,....2,1,ˆˆ

Equilibrium ConditionsEquilibrium Conditions

The last criteria implies The last criteria implies “tendency of a component to “tendency of a component to be in phase be in phase or or is balanced” is balanced” – “net mass transfer across – “net mass transfer across phases is zero”phases is zero”

cii N,....2,1i,ˆˆ

Ideal Behavior Model Ideal Behavior Model

Gas phase behaves as an ideal gas Gas phase behaves as an ideal gas (IG), and liquid phase behaves as an (IG), and liquid phase behaves as an ideal solution (IS). ideal solution (IS).

These assumptions imply thatThese assumptions imply thatIG:IG: molecular interactions are zero, molecular interactions are zero,

molecules have no volume.molecules have no volume.IS:IS: forces of attraction/repulsion forces of attraction/repulsion

between molecules are the same between molecules are the same regardless of molecular species. regardless of molecular species. Volumes are additive Volumes are additive (Amagat’s Law).(Amagat’s Law).

Forces between molecular Forces between molecular speciesspecies

A A B B A B

ABBBAA FFF

Statement of EquilibriumStatement of Equilibrium

iii PxPy

P

T

1

23

1P

IG/IS Raoult’s lawIG/IS Raoult’s law

Types of VLE CalculationsTypes of VLE Calculations

Temperature x1, y1

Pre

ssu

re

P2v

TaCP1

Bubble Curv

e

Dew

Cur

ve

Flash

Liquid

Vapor

P1v P1

v

CP2

P2v

Ta0 1

Recall Molar CompositionsRecall Molar Compositions

By convention liquid compositions By convention liquid compositions (mole fractions) are indicated with an (mole fractions) are indicated with an xx and gas compositions with a and gas compositions with a yy..

liquidnn

nx

21

11

gasnn

ny

21

11

withwith

In generalIn general

Mathematical Mathematical RelationshipsRelationships

vl fyfxz 111 vv fyfxz 111 1 )(

lv

vv nnnn

nnf

2121

21

)(

11

1 1

xy

xzfv

ii

iiv xy

xzf

Depletion PathDepletion PathIsothermal Reservoir Depletion Process for aIsothermal Reservoir Depletion Process for a

Reservoir Oil with 2 ComponentsReservoir Oil with 2 Components

Temperature

Pre

ssu

re

PD

PB

T = Ta

Ta

CPM

z1 = fixed

z1 y1x1 10

AB

C

z1=overall mole fraction of [1], y1=vapor mole fraction of [1], x1=liquid mole fraction of [1]

Quantitative Phase Quantitative Phase Equilibrium ExerciseEquilibrium Exercise

P-xy Diagram

0

400

800

1200

1600

2000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Composition (%C1)

Pre

ssu

re (

psi

a)

T=160F

Bubble Point Evaluation Bubble Point Evaluation (Ideal Behavior Model)(Ideal Behavior Model)

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

iibpi PzPy

iibp PzP

Bubble Point from Raoult's Bubble Point from Raoult's lawlaw

P 2

P 1 T

x 1 ,y 1

zzii=x=xii

Bubble Point EvaluationBubble Point Evaluation

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

Dew Point CalculationDew Point Calculation

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

ii yz

Dew Point Calculation Dew Point Calculation (Ideal Behavior Model)(Ideal Behavior Model)

Find DP pressure and Find DP pressure and equilibrium liquid equilibrium liquid compositionscompositions

iii

iii

PxPz

PxPy P

x

P

z i

i

i

1

1

cN

i i

idp

P

zP

Dew Point from Raoult's lawDew Point from Raoult's law

P 2

P 1 T

x 1 ,y 1

zzii=y=yii

Flash CalculationsFlash Calculations

In this type of calculations the In this type of calculations the objective is to:objective is to:

findfind fraction of vapor vaporized fraction of vapor vaporized (fv)(fv) and equilibrium gas and and equilibrium gas and liquid compositionsliquid compositions

givengiven the overall mixture the overall mixture composition, composition, P P andand T T..

Flash Calculations Flash Calculations (Ideal Behavior (Ideal Behavior Calculations)Calculations)

Start with the equilibrium Start with the equilibrium equationequation

Material balanceMaterial balance

iii PxPy

vivivilii fyfxfyfxz 1

Flash CalculationsFlash Calculations

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

vivi

ii fyfP

Pyz 1

i

i

i xP

Py

Here replaced Here replaced xxii

Flash CalculationsFlash Calculations

Rearrange and sum over all yiRearrange and sum over all yi

vvi

ii

ffPP

zy

1

vvi

ii

ffPP

zy

1

Separation processSeparation process

zi(T1,P1)

xi(T1,P2)

yi(T1,P2)

PP11 > > PP22

T1,P2

Flash CalculationsFlash Calculations

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

01

1)(

vvi

iv

ffPP

zfF

This is 1/ki – ideal equilibrium ratioThis is 1/ki – ideal equilibrium ratio

Flash CalculationsFlash CalculationsThere are several equivalent There are several equivalent

expressions for the flash function expressions for the flash function

(a) (a)

(b) (b)

(c) (c)

01 iy

01 ix

0 ii xy

Flash CalculationsFlash Calculations

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

vvi

ii

ffPP

zy

1

i

i

i xP

Pyandand

Vapor Pressure Models Vapor Pressure Models (Antoine Equation)(Antoine Equation)

1. Constants depend upon the 1. Constants depend upon the component – Different Unitscomponent – Different Units

ii

iii cT

baP

ln

1111

cN

i iv

i

kf

z

1111

cN

i iv

ii

kf

kz

cN

i iv

ii

kf

kz

1 11

)1(

Example in our web site excel file VLE_310Example in our web site excel file VLE_310

Flash 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(f

v)

Sum Xi

Sum Yi

Rachford Rice