resengch13

CONTENTS

1 INTRODUCTION1.1 Importance of the Critical Point1.2 Equilibrium Ratio

2 EMPIRICAL PREDICTION METHODS FOREQUILIBRIUM BEHAVIOUR2.1 Black-Oil System2.2 Correlation of Experimental Data Through K

Values2.2.1 Method of Calculating Convergence Pressure

3 METHODS BASED ON THERMODYNAMICPRINCIPLES3.1 Methods Based on Empirical Equations of

State of Fluid Phase Theory3.2 Prediction of Vapour-Liquid Equilibrium

4. THE COMPOSITION DATA REQUIREMENTSFOR THE APPLICATION THERMODYNAMICEQUILIBRIUM PREDICTION METHODS TOMULTI-COMPONENT HYDROCARBONMIXTURES

1313Equilibrium Ratio Prediction and Calculation

2

LEARNING OBJECTIVES

Having worked through this chapter the Student will be able to:

Comment briefly on the approach to vapour equilibrium ( VLE ) calculationsin terms of ; the black oil approach, empirical K values and EOS..

Describe briefly the application of convergence pressure based K values inVLE calculations.

Be able to determine the correct convergence pressure for VLE application .

Comment on the evolution of EOS in VLE application . ( The student would notbe expected to reproduce the equations.)

Describe briefly the application of EOS in VLE calculations.

Department of Petroleum Engineering, Heriot-Watt University 3


VAPOUR-LIQUID EQUILIBRIUM PREDICTIONS OF PHYSICALPROPERTIES

1 INTRODUCTION

In the previous chapter we introduced the concept of vapour-liquid equilibrium andits role in attempting to predict the behaviour of multi-component hydrocarbonmixtures in the two phase region, in particular the ability to predict phase densities andcomposition as a function of pressure and temperature. In the chapter we alsoreviewed some of the basic equations and procedures which are at the foundation ofthese multicomponent calculations

In petroleum engineering applications the situation is complicated in that often verylittle detailed information is available on the hydrocarbon mixture. That is, eitherindividual components are not identified, or, because it is a multi-component mixturecertain component concentrations are difficult to measure accurately even thoughthey can greatly influence the properties of the mixture.

There are two distinctive features of vapour equilibrium calculations in the context ofpetroleum engineering compared to other sectors like process engineering. In processengineering the calculations are mainly related to either a single process unit or a fewin series, whereas in reservoir engineering the calculations are often part of multi gridblock reservoir simulation where the numbers of process units can be considered inhundreds or even thousands. In process engineering a full compositional descriptionis available for the prime objective is to determine compositional split to obtainspecific compositional objectives. In petroleum engineering however, the objectiveis not to determine full compositional description of produced phases, but todetermine phase volumes and phase properties. The challenge is to carry this out witha minimal composition description without compromising the quality of physicalproperty predictions.

In addition for petroleum engineering applications the calculation method usedshould be:

(1) applicable to multi-component mixtures;(2) be accurate in predicting thermodynamic equilibrium and volumetric properties

over a wide range of conditions of temperature and pressure and specifically beaccurate around and beyond the critical point;

(3) preferably require only pure component data or binary data which is eitherreadily available from the literature or derivable from available data.

1.1 Importance of the Critical PointThe prediction of the true critical properties of a multi-component system is animportant aspect of the general problem of predicting the overall phase behaviour ofthe system. The critical state is the unique condition about which the liquid and vapourphases are defined, and hence it has theoretical as well as practical significance. Inhydrocarbon processing and producing operations, a knowledge of the criticalcondition is of particular significance because many of these operations take placeunder conditions which are at or near the bubble point or upper dew point regions andare frequently accompanied by isobaric or isothermal retrograde phenomena. Fluid

4

property predictions and design calculations in this region are often the most difficultto make, and a knowledge of a precise location of the critical point for the system understudy is of the utmost assistance.

From a theoretical point of view, the derivatives of many of the thermodynamic andtransport properties take on a special significance as the critical state is approached.In an empirical way the critical state has formed an integral part of many usefulgeneralised correlations such as those based on the theorem of corresponding statesor the convergence pressure concept in vapour-liquid equilibrium calculations.

In many ways the characteristics of the critical state that make it theoretically andpractically significant are also the characteristics that make it one of the more difficultconditions to measure experimentally. The very fact that density differences betweenphases vanish, that the rate of volume change with respect to pressure approachesinfinity, or that infinitesimal temperature gradients can be responsible for a transitionfrom 100% liquid to 100% vapour all make the critical condition one of the moredifficult to measure or observe accurately. For obvious economic reasons, it is acondition that cannot be obtained by experiment in any practical way for the manysystems for which it is required. Consequently, many attempts have been made todevelop methods for predicting the critical properties based on generalised empiricalor semi-empirical procedures.

1.2 Equilibrium RatioIn the previous chapter we presented the concept of equilibrium ratio, the measure ofhow a component is distributed between different phases however we didnt explainhow these distribution values are obtained.

The literature abounds with methods and equations associated with phase equilibriumprediction. The methods can be broadly classified as:

(1) methods which involve empirical curve fitting to experimental data;(2) methods which are based on thermodynamic principles.

2 EMPIRICAL PREDICTION METHODS FOR EQUILIBRIUMBEHAVIOUR

There are a range of approaches to expressing the vapour liquid distribution behaviourof reservoir fluid hydrocarbon systems, as outlined below.

2.1 Black-Oil SystemThe black-oil system as described in the chapter on liquid properties treats the fluidsas two components, stock tank oil and solution gas. This concept and associatedcalculations were covered in that chapter.

2.2 Correlation of Experimental Data Through K ValuesA complicating factor in the application of equilibrium ratios in the context ofreservoir fluids is that the distribution of a component between phases is not onlyinfluenced by the temperature and pressure but is also influenced by the composition.Calculating therefore how a unique fluid like that from a new exploration well



performs in these separation calculations is not straightforward, since no one beforewould have been able to carry out tests on the fluid!!

An approach is required that the distribution of each component of the mixturebetween the liquid and vapour phases be experimentally determined for a range oftemperatures and pressures. The resultant measurements are expressed as an equilibriumratio Kj, for component j defined as:

where: Ky

xjj

j

= (1)

Kj

= equilibrium ratioy

j= mole fraction of component j in the vapour phase

xj

= mole fraction of component j in the liquid phase

This approach is really only useful for light hydrocarbons. The data is usuallyexpressed graphically as a plot of K

j versus pressure for a constant temperature.

The oil industry has relied on experimentally determined equilibrium ratios althoughincreasingly over recent years the move has been towards Equation of State derivedK values. Clearly the empirical K values are obtained from known mixtures, thechallenge is to determine the applicability of these empirical values to new mixtures.

At high pressures the equilibrium ratio is a function of temperature, pressure and alsothe composition of the mixture. This compositional influence is of great significance.Hence, pure component K-values measured for one mixture cannot be accuratelyapplied to another mixture. Figure 1 and 2 present the K values for a heavy oil and atthe other extreme a condensate at a temperature of 200F. The K values converge tounity at 5,000 psia and 4,000 psia respectively. This point is called the convergencepressure. Close examination of these two sets of K values demonstrate that thesedistribution coefficients do not have the same values at particular pressures andtemperatures, confirming this compositional influence on K values.

The effect of composition on the K values is achieved by applying the concept ofconvergence pressure. The concept of convergence pressure arose from the observationthat for light hydrocarbon mixtures the isothermal component K-values converge tounity at a specific pressure known as the convergence pressure Pc, see Figures 3 fora binary mixture and 4 for a light oil . This convergence pressure is a measure of thecomposition of the mixture and can be calculated from a knowledge of the effectiveboiling point of the lightest and heaviest components in the mixture.

If the temperature at which the equilibrium ratios are presented is the criticaltemperature then the convergence pressure would be the critical pressure.

At a given temperature, as the system pressure increases, the K-values of allcomponents of the system converge to unity when the system pressure reaches theconvergence pressure. In other words, it is the pressure for a system at a giventemperature when vapour-liquid separation is no longer possible. Naturally, it isequally impossible to have a vapour-liquid separation at a given temperature in whichthe system pressure is greater than the convergence pressure.

6

100

10

10 100 1,000 10,000

K=1

0.1

0.01

0.01

0.02

0.04

0.060.08

0.1

0.2

0.4

0.60.8

1

2

4

68

10

20

40

Pressure, psia

100 1,000 10,000

Pressure, psia

M

ethaneEthanePropaneButanesPentanesHexanes

Heptanes and Heavier

MethaneEthane

PropaneButanes

PentanesHexanes


Equ

ilibr

ium

Rat

io, K

Figure 2

Equilibrium Ratios at

200F for a Condensate

Fluid (Amyx Bass &

Whiting)1

Figure 1

Equilibrium Ratios at

200F for Low-Shrinkage

Oil(Amyx Bass & Whiting)1



Component 1

CONSTANT TEMPERATURE

Component 2

In PcIn P

In Kj

Kj - 1.0

10.0

1.0

10 100 1,000 10,000

0.1

0.01

Pressure, psia

Variations of EquilibriumConstant With Pressure

at 200 FMethaneEthanePropane

ButanesPentanesHexanes


Equ

ilibr

ium

Con

stan

t, K

Figure 3

Variation of K Values with

pressure for a typical

Binary Hydrocarbon

System

Figure 4

Variation of K Values with

pressure for a crude oil

containing Light

Hydrocarbons

8

The apparent convergence pressure is related to the composition of the fluid. Thisempirical basis has found favour in the industry for many years as a source of K valuedata despite the ability to calculate the information based on equations of state.However with the increasing availability of computational power and greater confidencein equations of state the use of convergence pressure based K values is diminishing.

A comprehensive set of equilibrium ratios is published in the GPSA Engineering DataBook 2. The K values for single components are presented in the NGAA book for arange of convergence pressures. K values for a convergence pressure of 5,000 psia areat the back of this chapter for 2 pure components. Figure 9 and 10.

2.2.1 Method of Calculating Convergence PressureAs was indicated in figures 2 and 3 different mixtures present different K values asrepresented by different convergence pressures. The challenge is therefore to selectthe convergence pressure value appropriate to the fluid for which we are seeking todetermine its phase separation characteristics.

The procedure is to convert our mixture into a two component mixture based onmethane then seek to identify a compound which has the same physical properties asour heavier C2+ component

The chart, Figure 5 from the NGPSA Manual, can be used for this2. The method is totake as the lightest component present in any significant quantity (1.0% or greater ina raw mixture; it is usually methane) as the light component of a two componentsystem. The heavy component is estimated from the composition of the remainingcomponents. A visual estimate is usually good enough. Join the heavy and lightcomponent together as shown on the Figure 5 and read off P

K against the operating

temperature.

It has been established that the convergence pressure of systems as encounteredgenerally in natural gas processes is a function of the temperature and the compositionof the liquid phase. This presumes that the liquid composition had already been knownfrom a flash calculation using a first approximate guess for convergence pressure.Therefore, the method of calculating convergence pressure is an iterative procedure.This calculation is suggested:

Step 1: Assume the liquid phase composition or make an approximation. (If thereis no guide, use the total feed composition).

Step 2: Identify the lightest hydrocarbon component which is present at least 0.1mole % in the liquid phase.

Step 3: Calculate the weight average critical temperature and critical pressure for theremaining heavier components to form a pseudo binary system. (A shortcutapproach good for most hydrocarbon systems is to calculate the weight average Tconly).

Step 4: Trace the critical locus of the binary consisting of the light component andpseudo-heavy component. When the averaged pseudo heavy component is betweentwo real hydrocarbons, interpolation of the two critical loci must be made.



Ken

sol

nC17n

C18

Nitrogen

Methane

Ethane

Propyl

ene

Propa

ne

I-Buta

ne

N-Bu

tane I

-Pen

tane N

-Pen

tane

N-He

xane

N-He

ptane

N-Oc

tane

N-De

cane

n-H

exad

ecan

e

Ethyle

ne

C2-n

C10

C2-n

C6 C

3-nC

6H 2

-nC 6

C1-K

enso

l

C3-n

C7

nC4-

nC10

H2O

H2S

NH

3S

O2

CO

2

nC9

nC7-

nC19

nC7-

nC23

nC5-

nC24

nC5-

nC22

nC5-

nC18

nC5-

nC16

nC11

nC12

nC13

nC14

nC15

Ben

zene

Tolu

ene

Met

hyl-

Cyc

lohe

xane

C1-n

C7

C1-

nC9

C1-n

C10

H 2-C 3

H 2-C 2

H 2-C 2

Trip

le P

hase

Loc

us

Trip

le P

hase

Loc

us

20,0

00

10,0

009,

000

8,00

07,

000

6,00

0

5,00

0

4,00

0

3,00

0

2,00

0

1,00

090

080

070

060

0

500

400

300

200

100 -3

00-2

00-1

000

100

200

300

400

500

600

700

800

900

Convergence Pressure, PSIA

Con

verg

ence

Pre

ssur

es fo

r H

ydro

carb

ons

(Crit

ical

Loc

us)

Tem

pera

ture

, F

Figure 5

Convergence pressure for

Hydrocarbons.2

10

1C,enahteM

20,000

10,0009,0008,0007,0006,000

5,000

4,000

3,000

2,000

1,000900800700600

500

400

300

200

100

-100 0 100 200 300 400 500 600 700 800 900

Con

verg

ence

Pre

ssur

e, P

SIA

Temperature, F

daehrevOrebrosbA

edurCtnenitnoC-diM'urht1C

edurCtnenitnoC-diM

edurCetallitsiD'urht

1C

enilosaG'urht1C

161.tw.loMliO

naeL

etallitsiDedurC

enilosaGthgiL

Absorbers, Crude Flashing Towers

Step 5: Read the convergence pressure (ordinate) at the temperature (abscissa)corresponding to that of the desired flash conditions, from Figure 5.

Step 6: Using the convergence pressure determined at Step 5, together with thesystem, obtain K-values for the components from the appropriate convergence-pressure K-charts.

Step 7: Make a flash calculation with the feed composition and the K-values fromStep 6.

Step 8: Repeat Steps 2 through 7 until the assumed and calculated convergencepressures check within an acceptable tolerance, or until the two successive calcula-tions give the same light and pseudo heavy components check within an acceptabletolerance.

When the convergence pressure so determined is between the values for which chartsare provided, interpolation between charts may be necessary depending on how closethe operating pressure is to the convergence pressure.

If K-values do not change rapidly with PK(P

K>>P) then the set of charts nearest to

calculated Pic may be used. Otherwise, a crossplot of K values versus PK all at constanttemperature and pressure, must be constructed for interpolation.

For those components characterised as a C7+

fraction Katz has suggests using a K valueof 15% of that of C

7+. Danesh3 makes reference to other correlations to estimate the

critical properties of C7+

fractions. McCain6 has also presented pseudo criticalproperties of C

7+ as a function of molecular weight and specific gravity, his correlation

figures are shown below in figure 7.

Figure 6

Pseudocritical properties of

Heptane Plus. (NGPSA 5th.

edition 1957)



500

450

400

350

300

250

200

150

100100 150 200 250 300

100 150 200 250 300

Molecular weight of heptanes plus

Molecular weight of heptanes plus

1700

1600

1500

1400

1300

1200

1100

1000

900

Pse

udoc

ritic

al te

mpe

ratu

re,

RP

seud

ocrit

ical

tem

pera

ture

, R

Specific gravity of heptanes plus = 1.0

.95

.90

.85

.80

.75

.70

1.0 = Specific gravity of heptanes plus

.95

.90

.85

.80

.75

.70

The following example has been presented by McCain in his text6. and is helpful asa worked example in determining the appropriate convergence pressure charts

Example: McCain4. The gas-liquid equilibrium of a high-shrinkage crude oil hasbeen calculated. The composition of the liquid phase formed at 75F and 100 psia isgiven below. A convergence pressure of 2000 psia was used to determine theequilibrium ratios for the calculations. What value of convergence pressure shouldhave been used for this mixture?

Figure 7

Pseudocritical properties of

Heptane Plus6.

12

Component Composition of liquid mole fraction

Methane 0.0356Ethane 0.0299Propane 0.0919i-butane 0.0170n-butane 0.0416i-pentane 0.0198n-pentane 0.0313Hexane 0.0511Heavier* 0.6818 1.0000

* Molecular weight and gravity of C7+ assumed to be 268 and 0.886

SolutionFirst, the composition of the liquid must be expressed as weight fraction.

Component Composition of Molecular Weight xMj Composition of liquid mole weight Mj liquid, weight fraction xjMj/xjMj

C1 0.0356 16.0 0.5696 0.0029C2 0.0299 30.1 0.9000 0.0046C3 0.0919 44.1 4.0528 0.0206i-C4 0.0170 58.1 0.9877 0.0050n-C4 0.0416 58.1 2.4170 0.0123i-C5 0.0198 72.2 1.4296 0.0073n-C5 0.0313 72.2 2.2599 0.0115*C6 0.0511 86.2 4.4048 0.0224*C7+ 0.6818 263 179.3134 0.9133 1.0000 196.3348 0.9999

* Molecular weight and gravity of C7+ assumed to be 263 and 0.886 respectively

Second, adjust weight fraction to exclude methane and calculate weighted-averagecritical properties.

Component Composition Critical R Critical psia excluding temperature wjtcj pressure, wjpcj methane, wj R Tcj psia pc psia pcj

C2 0.0046 549.8 2.53 707.8 3.26C3 0.0207 665.7 13.78 616.3 12.76i-C4 0.0050 734.7 3.67 529.1 2.65n-C4 0.0123 765.3 9.41 550.7 6.77i-C5 0.0073 828.8 6.05 490.4 3.58n-C5 0.0115 845.4 9.72 488.6 5.623C6 0.0225 899.3 20.23 445.4 10.02*C7+ 0.9160 1360 1245.76 240 219.84 0.9999 wt avg Tc= wt avg pc= 1311R, 851F 265 psia

* Critical properties of C7+ from figure 7.



The calculated values of weight averaged Tc and P

c are close to Kensol and the mid

continent crude point of figures 5 and 6. The location of the temperature 75F with themethane - component (Kensol or mid - continental crude) is at a convergence pressureof around 10,000 psia, a value much greater than the assumed value of 2,000. Thecalculator needs to be repeated using the higher convergence pressure related K valuedata. The calculation procedure will converge when the estimated convergencepressure is the same as the calculated convergence pressure.

3 EQUATION OF STATE BASED EQUILIBRIUM CALCULATIONS

3.1 Methods Based on Empirical Equations of State of Fluid Phase TheoryThe thermodynamic properties of a pure fluid may be represented by an equation ofstate of the generalised form:

f (P,V,T) = 0

where the pressure P, temperature T and molar volume V are related by a mathematicalfunction. Most equations of state have been developed by fitting an analyticalexpression to pure component PVT data. To extend the application of the developedequation to mixtures, the parameters of the equation must take into account thecomposition of the mixture and for simplicity require only the insertion of purecomponent data. Since most equations are effectively only mathematical models, theequations tend in general to be more complex, that is contain a large number ofparameters, as the required level of accuracy increases. The basic assumption in thedevelopment of an equation of state is that at a critical point:

PV

T

=2PV2

T

= 0

Equations of state can be used for the following purposes:

1 representation of PVT data to assist data smoothing and improve interpolation;

2 prediction of vapour-liquid equilibria of mixtures especially at high pressures;

3 prediction of gas phase properties of pure fluids and their mixtures using aminimum amount of experimental data.

In the gas property chapter we reviewed the topic of equations of state. Currentlyalthough a number of different equations could be used, the industry favours two, thePeng Robinson equation of state and the Soave modification of the Redlich Kwongequation of state.

3.2 Prediction of Vapour-Liquid EquilibriumAn equation of state capable of predicting behaviour in the liquid phase and vapourphase is sufficient in itself for vapour-liquid equilibrium predictions. Unfortunatelyas we have indicated although modifications to the Van der Waals type equation havepredicted vapour properties the equations in general have not been so accurate inpredicting the liquid behaviour.

14

Many vapour-liquid prediction methods have therefore gone to an equation of state,for example the Redlich-Kwong equation, for the prediction of the vapour phasefugacity and have employed a liquid theory to evaluate the liquid phase properties.

We will now go through the steps in the application of equation of states in vapourliquid equilibrium calculations. It is worth noting that the steps described are thosewhich would be taking place within each grid block of a compositional reservoirsimulator. The implication therefore is a very large computational load for largenumbers of grid blocks.

The application of equation of states is based on the fact that atequilibrium, the fugacity of the gas and liquid phases are identical i.e:

fiv = f

iL (2)

where fiv and f

iL are the fugacities of component in the vapour and liquid phases

respectively.The ratio of the fugaciy to pressure is called the fugacity coefficientwhere

i = f

i/(pz

i) (3)

where zi is the composition of the component in the system.

The fugacities can be expressed for a vapour and liquid therefore by:

fiL =

i x

ip

and:f

iv =

i y

ip

p - is the system pressure;y

i & x

i- are the mole fractions of i in the vapour and liquid phase.

Therefore:

Kx p

y p

yni

iL

ig

iL

i

ig

i

i

i

= = =( )

f

f

The fugacity coefficients for the liquid and gas can be calculated using the followingequation

RTRTV

pn

dV zii T V n

V

i

=

1ln ln

, ,

(4)

McCain5 and also Ahmed6 give good descriptions of the application of EOS inequilibrium calculations.

At the present time the preferred EOS are the 3 parameter Peng-Robinson (PR) 8,9 andthe Soave-Redlich Kwong (SRK)10. Other equations exist and are more accurate inpredicting some properties. The considerable investment in binary interaction



parameters for the preferred equations is such that there is a reluctance to use somerecently developed EOSs. Danesh's3 text gives a good review of the EOS.

Since the applications of the equations are applied to mixtures, mixing rules arerequired to determine the values for the parameters in the particular EOS being used.

We will use the Peng Robinson equation as our basis but others could be used.

The Peng Robinson equation is

pRT

V ba

V V bmT

m m

=

+( )) ( )= b V bm

(5)

The equation is set up for both liquid and gas using the following mixing rules tocalculate b, and aT . The rules are presented in the context of gas ie. y, clearly forliquids, x values are used.

and= b y bi

ii

(6)

( ) ( ).= a y y a a kT ij

ji

Ti Tj iy0 5 1 (7)

where kij are the binary interaction coefficients, and k

ii = k

jj = o and k

ij = k

ji.

The value of bi and a

Ti for the individual components are calculated as follows

.=bRTPi

ci

ci

0 07780 (8)

and aTi

= aci

i and (9)

.=aR T

Pcici

ci

2 2

0 45724 (10)

i is a temperature dependant factor where

.. .= + ( )m Ti ri0 5 0 51 1 0 (11)where

m i i= + 0 37476 1 54226 0 269922. . . (12)

The Peng Robinson equation can be written as a cubic equation in terms of z, thecompressibility factor,

16

z3 - (1-B)z2 + (A-2B -3B2)z - (AB - B2 -B3) =0 (13)

where

= =AaP

R TB

bPRT2 2

(14)

The solution to the above equation gives three roots for z. The highest value is theliquid phase z factor and when the equation is solved using vapour compositions thenthe lowest root is the z factor for the vapour.

If the Peng Robinson equation is combined with the fugacity coefficient equation thefollowing equation results for the fugacity coefficient of each component.

ln ln ln.

.

.i i i iz B z B

AA B

z B

z B= ( ) + ( ) ( )

+ +( ) +( )

12

2 1

2 11 5

1 5

1 5 (15)

Where B'i = b

i/b (16)

and

= ( )

A a a y a ki T Ti i Ti ij

12 10 5 0 5

1

. .(17)

Following all these steps independently for the liquid and gas phases the fugacities ofthe gas and liquid phases can be calculated.

fLi

= xip

Li and f

vi = y

ip

vi

When fLi

=fvi then equilibrium is achieved and calculations are complete.

Having presented these equations we will now describe the process to calculate Kvalues , vapour liquid equilibrium ratios and compositions given a system composition,temperature and pressure.

Step 1: Estimate the K values of the system. The Wilson11 equation is good for thispurpose.

Kpp

TTi

cii

ci=

+( )

exp .5 37 1 1 (18)

wi = acentric factor for component

i

Step 2: Carry out vapour equilibrium calculations using estimated K values usingthe iterative procedure outlined previously. That is estimate the V/L ratio and iterateuntil convergence is obtained, that is when compositions sum to unity. We now haveliquid and vapour compositions to use in equation of state calculations.



Step 3: Using liquid compositions, calculate the A & B values for the EOS and thensolve the z-value form of the EOS, to determine z. The lowest root (value) is the zvalue for the liquid.

Step 4: Calculate the compositional coefficients A'i and B'

i for the liquid components

and calculate the fugacity coefficients of the components of the liquid

Step 5: Repeat steps 3 & 4 using the vapour phase compositions.

Step 6: Calculate fgi and f

Li f

gi=y

ip

i and f

Li= x

ip

I . Check if f

gi= f

Li.

If this value is greater than 10-12 then the whole process has to be repeated from step1, except that the K values used are the calculated K values arising from step 5 i.e.

Rather than set up the tolerance check on fugacityequivalence the tolerance can be

i = f

Li - f

gi

based on K values.A value of of 0.001 can be used for the sum of the errors.

KiLi

gi

= (19)

iiE

iC

iE

iC

K K

K K=

( ) 2(20)

The iteration is complete when these tolerance limits are met and the compositions ofthe repective phases are those which have been been determined at the last iteration.Calculations can then proceed to provide volumetric and density data for therespective phases.

Danesh3 has given a flow diagram for the above flash calculation.

An example follows to illustrate the calculation process.

18

NO

YES

NO

YES

Start

End

Input zi, P, T,Component Properties

Estimate Ki, Using Eq.(18)

Calculate xi, yi, Using Figs. 20-22 (Ch 12)

Adjust Ki = Ki old (fiL/fiV)

Is(1- fiL/fiV)2



No. Component 1 2 3 41 Methane 0.00002 n-Pentane 0.0236 0.00003 n-Decane 0.0489 0.0000 0.00004 n-Hexadecane 0.0600 0.0070 0.0000 0.0000

4. THE COMPOSITION DATA REQUIREMENTS FOR THE APPLICATIONOF THERMODYNAMIC EQUILIBRIUM PREDICTION METHODS TOMULTI-COMPONENT HYDROCARBON MIXTURES

In designing a versatile phase equilibrium prediction computer program, the availableexperimental data for the mixture whose performance is to be simulated, must beconsidered. Most laboratory analyses of crude oil are predominantly concerned withthe composition. This is normally reported as a percentage weight. This presents fewproblems for lighter hydrocarbons such as methane which can have no isomers nor forother light hydrocarbons which do have isomers, for example butane, since thenumber of hydrocarbons having the same number of carbon atoms is relatively few.However, for heavier hydrocarbons the number of hydrocarbons which possess thesame number of carbon atoms can be very large due to the presence of not only alphaticor straight chain compounds but also of cyclic compounds such as aromatics ornaphthenes, or combinations of these structural types. Obviously the structure ofmolecules greatly influences the forces between them and hence their deviations fromideality. This is most significant when binary interaction parameters are employed inthe mixing. The analysis of the heavier components in crudes is further complicatedby the fact that most of the individual components are present in very small quantitieswhich makes them difficult to identify and quantify.

Ideally what is required is a full analysis of a crude oil in which all components arequantified and identified so that the effects on the mixture behaviour due to theirstructural and chemical properties can be taken into consideration. The phaseequilibrium prediction method which is employed, must match the quality and extentof the experimental data available, for example, the use of a highly accurate methodmight be invalidated if the composition analysis has been carried out on a basis ofgeneralised crude fractions such as boiling range. However, this is where flexibilitymust be built into the program.

At present there is a tendency to increase the extent to which a crude oil compositionis analysed and this can only serve to make phase equilibrium prediction moreaccurate. Condensate well fluids, however, are still generally reported on a basis ofa heptanes plus weight fraction, that is the collective weight fraction of heptane andheavier components, because these heavier components are generally only present inminute quantities.

It is thus desirable that any proposed phase equilibrium method should be able toutilise detailed experimental data based on identifying the individual components ofa crude oil as well as data derived from a more generalised approach as discussedabove.

20

Most of the work carried out on fluid phase equilibria has concerned itself solely withbinary or tertiary systems and occasionally mixtures containing pure componentswhose behaviour is quite well understood. Much of the published work on theadaptation of experimental data for prediction techniques has been related to petroleumrefinery operations.

Figure 9

K value chart for Ethane.



500400300250200

10080

60

4020

0-20-40

-60

Temperature F

10 30 50 100 300 500 1,000 3,000 10,0002 4 6 7 8 9 2 4 6 7 8 9 2 4 6 7 8 9

10 30 50 100 300 500 1,000 3,000 10,0002 4 6 7 8 9 2 4 6 7 8 9 2 4 6 7 8 9 1,0009876

5

4

3

2

1009876

5

4

3

2

109876

5

4

3

2

1.09876

5

4

3

2

0.19876

5

4

3

2

1,0009876

5

4

3

2

1009876

5

4

3

2

109876

5

4

3

2

1.09876

5

4

3

2

0.19876

5

4

3

2

Pressure, PSIA

Pressure, PSIA

K= y/x K= y/x

ETHANECONV. PRESS. 5000 PSIA

Plotted from 1947 tabulationof G. G. Brown, University of Michigan. Extrapolated and drawn by The Fluor Corp. Ltd.in 1957.

22

10 30 50 100 300 500 1,000 3,000 10,0002 4 6 7 8 9 2 4 6 7 8 9 2 4 6 7 8 9

10 30 50 100 300 500 1,000 3,000 10,0002 4 6 7 8 9 2 4 6 7 8 9 2 4 6 7 8 9 1,0009876

5

4

3

2

1009876

5

4

3

2

109876

5

4

3

2

1.09876

5

4

3

2

0.19876

5

4

3

2

1,0009876

5

4

3

2

1009876

5

4

3

2

109876

5

4

3

2

1.09876

5

4

3

2

0.19876

5

4

3

2

Pressure, PSIA

Pressure, PSIA

K= y/x K= y/x

OCTANECONV. PRESS. 5000 PSIA

Temperature F

500450

400380360340320300280260240220200

180

160

140

120

100

80

60

40

20

0

-20

Plotted from 1947 tabulationof G. G. Brown, University of Michigan. Extrapolated and drawn by The Fluor Corp. Ltd.in 1957.



Solution to Exercise

EXERCISE

Calculate the liquid and vapour phase composition when the mixture with thecomposition given in Table 1 is flashed to 2000 psia and 100C. Use the Peng-Robinson equation of state with binary interaction parameters given in Table 2.Table 1: Multicomponent system.

Component Composition mole fractionMethane 0.55100n-Pentane 0.11400n-Decane 0.14600n-Hexadecane 0.18900

Table 2: Binary interaction parameters of Peng- Robinson Eq.

No. Component 1 2 3 41 Methane 0.00002 n-Pentane 0.0236 0.00003 n-Decane 0.0489 0.0000 0.00004 n-Hexadecane 0.0600 0.0070 0.0000 0.0000

SOLUTION

Step 0: Calculate the coefficients of components in the mixture, using the properties(critical temperature T

c and critical pressure P

c) of pure compounds:

.=acR T

Pici

ci

2 2

0 45724(10)

.=bRTPi

ci

ci

0 07780 (8)

If 0.5 : m= 0.3796 + 1.485 - 0.1644 2 - 0.016673

w - accentric factor from thermodynamic property tables for pure components

. .= + )m Ti r1 0 1 0 ((( )2 (11)

ai = c

i

i(9)

Figure 10

K value chart for Octane.

24

Component Tci Pci i aci i ai bi K atm acentric factorC1 190.6 45.39 0.0115 2.4632 0.7112 1.7518 .0268n-C5 469.7 33.26 0.2515 20.4230 1.1686 23.8663 .0902n-C10 617.7 20.82 0.4923 56.4253 1.5327 86.4825 0189n-C16 723 13.82 0.7174 116.4577 1.9176 223.3236 .3340

Step 1: Select initial values for equilibrium ratios (K-values) and calculate the trialcomposition of liquid and vapour phases at equilibrium. The procedure is iterative asfollows.

Iteration OneInitial K-values can be calculated from Wilson Equation:

KPP

ExpTTi

cii

ci= )( +( ) ) . . . )5 37 1 0 1 0 (15)

Component Feed (Zi) K-Values (Ki)-Wilson Eq.Methane 0.55100 4.7644n-Pentane 0.11400 4.290 E-02n-Decane 0.14600 7.9917 E-04n-Hexadecane 0.18900 1.7768 E-05

Step 2: Determine the trial value of phase fraction (vapour fraction here) by solvingthe vapour equilibrium equation:

ZLV

KVi

Li l

N

+=

=

(5)

Vapour Fraction (Vf) = 0.4378

Use the determined vapour fraction (Vf) to calculate the liquid and the vapour phase

composition from material balance:

Component Feed (Zi) Liquid (xi) Vapour (yi)Methane 0.55100 0.2081 0.9914n-Pentane 0.11400 0.1962 0.0084n-Decane 0.14600 0.2595 0.0002n-Hexadecane 0.18900 0.3362 0.000006

Step 3, 4 and 5: Calculate composition dependent coefficients for compressibilityfactor (Z - factor) calculations for both liquid and vapour:

Liquid:

a x x a a kiJ l

N

i l

N

j i j ij= ( ) ( )== .

.1

0 5(7)

b x bi ii l

N

== (8)



Vapour: a yiJ l

N

i

===

ll

N

j i j ijy a a k ( ) ( )0 5

1.

. (7)

b x bi ii l

N

== (8)

AaP

R T= 2 2

BbPRT

=

(14)

Phase a b A BLiquid 74.88118 0.18470 10.86872 0.82089Vapour 1.83396 0.02737 0.26619 0.12165

Calculate Z - factors of the liquid and vapour phases (Peng - Robinson EOS):

Z3 - (1 - B)Z2 + (A - 2B - 3B2)Z - (AB - B2 - B3) = 0. (13)

Vapour Phase, take the highest root: Zv = 0.92034

Liquid Phase, take the smallest root: Zl = 0.96602

Step 6: Calculate the fugacity of each component in the liquid and vapour phase:

ln ln( ) ( )

ln( )( )

. .ii

i j j ijj l

N

Z B Z lbb

AB a

a y a l k

Z BZ B

= + ( )

+ +

=2 2

12

2 12 1

0 5 0 5

(15)

Fugacity of components in the vapour phase: iv

i ivf y P=

Fugacity of components in the Liquid phase: il

i ilf x P=

Liquid VapourComponent i

l fil i

v fiv

Methane .23811E+01 .67429E+02 .90058E+00 .12150E+03n-Pentane .66312E-01 .17708E+01 .36516E+00 .41838E+00n-Decane .19023E-02 .67189E-01 .14686E+00 .41455E-02n-Hexadecane .43075E-04 .19707E-02 .51625E-01 .41965E-04

End of Iteration One

Next Iteration:i

il

K =iiv

(16)

Iterate till the fugacity of every component in the liquid phase is equal to that of vapourphase (iterate from Step 2.):

26

Final Iteration

Component Feed (Zi) K-Values (Ki) Methane 0.55100 2.44097n-Pentane 0.11400 0.19991n-Decane 0.14600 1.6774 E-02n-Hexadecane 0.18900 1.25099 E-03

Then: Vapour Fraction (Vf) = 0.2710


Calculate composition dependent coefficient for z - factor calculations for both liquidand vapour:

Liquid:

a x x a a ki

J l

N

i l

N

j i j ij= ( ) ( )== .

.1

0 5 (7)

i ii l

N

b y b==

(8)

Vapour:

a yiJ l

N

i

===

ll

N

j i j ijy a a k ( ) ( )0 5

1.

. (4)

i ii l

N

b y b==

(8)

AaP

R T= 2 2 B

bPRT

= (14)

Phase a b A BLiquid 48.03656 0.14799 6.97233 0.6577Vapour 2.11326 0.02929 0.30673 0.1301

Calculate z - factors of Liquid and Vapour and Vapour Phase (Peng - Robinson EOS):

Z3 - (1 - B)Z2 + (A - 2B - 3B2)Z - (AB - B2 - B3) = 0. (3)

Vapour Phase, take the highest root: Zv = 0.90039

Liquid Phase, take the smallest root: Zl = 0.81539

Calculate the fugaciy of each component in the liquid and vapour phase:



ln ln( ) ( )

ln( )( )

. .ii

i j j ijj l

NiZ B Z l

bb

AB a

a y a l kbb

x

Z BZ B

= + ( )

+ +

=2 2

12

2 12 1

0 5 0 5

Fugacity of components in the vapour phase: iv

i ivf y P=

Fugacity of components in the liquid phase:

il

i ilf x P=

Liquid VapourComponent i

l fil i

v fiv

Methane .22032E+01 .11881E+03 .90269E+00 .11882E+03n-Pentane .66155E-01 .13105E+01 .33093E+00 .13106E+01n-Decane .19873E-02 .53832E-01 .11847E+00 .53831E-01n-Hexadecane .45319E-04 .15983E-02 .36226E-01 .1598E-02

Molecular volume VmRTP

2 2

PPMRT

2

2

Phase 2 Vm gr mole/cc Ma g/mole/cc Ps/ccLiquid 099039 0.05451 103.86 0.56613Vapour .81539 0.04936 18.165 0.08967

Compositions are for final iteration


28

REFERENCES

1. Amyx, J.W, Bass, D.M, Whiting, RL, Petroleum Reservoir Engineering, McGrawHill. New York 1960

2. Gas Processors Suppliers Association. Engineering Data Book 9th Edition. Tulsa

3. Danesh, A, PVT and Phase Behaviour of Petroleum Reservoir Fluids. 1998Elsevier. pp 66-77

4. Wilson,G: A Modified Redlich-Kwong EOS,Application to General PhysicalData Calculations,Paper Nu 15C, presented at the AIChE 65th National Meeting(May 1968)

5. McCain, W.D. The Properties of Petroleum Fluids Penwell. 1st. Edition 1973Starling, K.E., A.I.Ch.E.Jnl, 1972, 18, 6, 1184-1189.

6. McCain, W.D. The Properties of Petroleum Fluids. Pennwell, 2nd Ed., 1990, p414-436.

7. Ahmed, T. Hydrocarbon Phase Behaviour, Gulf Pub., 1989.

8. Peng, D.Y., Robinson, D.B., I.E.C. Fundamentals 1976, 15, 1, 59-64.

9. Jhaveri B.S. and Youngren G.K. Three-Parameter Modification of the Peng-Robinson Equalising State to Improve Volumetric Predictions SPE 13118 (1984)

10. Soave, G., Chem. Eng. Sci., 1972, 27, 1197-1203.

11. Wilson, G.M., "A Modified Redlich - Kwong EOS Application to GeneralPhysical Data Calculations." Paper No 15c presented at the AIChE 65th NationalMeeting (May 1968) AI ChE.

resengch13

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