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    Pure Substances

    An equation

    of

    state in the sp

    iri

    t of

    va

    n der

    W

    aal

    s

    will produce

    isotherms as shown in Figure 15-1for apure sub

    s

    tance. Notice that the

    isotherms at and above the critical temperature look very much like the

    corresponding experimental isotherms

    of

    Figure 2

    -10

    The calculated isotherm

    of

    Figure 15 for temperaturebelow critica}

    temperature exhibits the van der Waals

    loop

    At certain temperatures,

    the calculated pressures in the loop are

    neg

    ative, as shown. This loop

    does not appear experirnentally. For a

    p

    ure substance a horizontal line

    connects equilibriurn gas and

    lqu

    id.

    Ag

    ain see Fi

    g

    ure 2-10

    .

    The van der Waals loop is used

    t

    o

    determine

    the molar volumes

    of

    the

    equilibrium gas and liquid and

    th

    en is

    replac

    ed by a tie-line between

    these two volumes. Toe

    basi

    s

    of

    this calculation is that the

    of the equilibrium

    ga

    s e

    qual

    s the of the

    equilibrium liquid.

    Equations

    of

    state can be

    use

    d to calculate gas-liquid equi

    l ib

    ria a

    s

    an

    altemative to using K-factor correlations . Toe assumption must be made

    that the equations

    of

    state presented in

    C

    hapters 3 and 4 predict pressure-

    volume-temperature relationships fo

    r

    liquids as well as for gases.

    This chapter

    i

    s an

    t

    o

    utili

    zing

    e

    quation

    s

    of

    state in

    g

    as-

    liquid equilibria calculations. conveys a general understanding of the

    subject and is as simple and short as po

    ss

    ible

    . Man

    y o

    f

    the complexities

    of

    this subject are not

    di

    scu

    ss

    ed. Thus, the study of

    thi

    s chapter

    w

    ill not

    result in the ablity to apply

    equation

    s

    of

    state to complex petroleum

    mixtures.

    G a s - L i q u i d E q u i l i b r i a

    C a lc u la t io n s W i t h

    E q u a t i o n s o f S t a t e

    15

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    Chemical potential also is called

    F

    i

    gu

    re 15-

    2

    A

    gives a

    ca

    lcula

    t

    ed i

    sot

    h

    e

    r

    m

    below the

    criti

    ca

    te

    mpera

    -

    ture. Point a is selected arbi

    t :

    ra

    ril

    y along the liquid part of the i

    so

    therm.

    Po

    in

    t d is s elec ted arbit:rarily along the gas part of the isotherm. Points b

    and e are mnimum and maxim

    u

    m

    points

    on the van

    de

    r W

    aa

    ls

    loo

    p

    .

    Points e

    re p

    resent the points along

    th

    e

    l

    oop for which

    th

    e ch

    emic

    aJ

    potentials are

    eq

    ual. Point f is the point along line be which has the same

    press

    ure as

    poi

    nts e.

    Figure 1 5

    -

    28 gives the

    c

    orresponding chemical potentials

    calcu

    l

    at

    ed

    as in Equat

    i

    on 1 5-1. A loop also appears on this

    figu

    re. The loop is

    nonexiste

    nt physically but can be used analytically. The point of

    inte

    rsecti

    o

    n

    , e,

    m

    eets the

    requireme

    nts of equilibria for the g

    a

    s and liquid

    of a pure substance. At point e, the pressure of the gas equals t:he

    pressure of the liquid, and the chemical potentials of the two phases are

    equ

    al. Point

    f h

    as the same

    p

    ressure as

    po

    ints e but is not

    a

    n equili

    br

    ium

    point becau s e

    i

    ts chemical potential

    i

    s h

    i

    gher than that of points e.

    The G

    ,

    of a

    pur

    e flu

    i

    d at constant temperature

    ma

    y

    be calcu

    la

    ted as

    Chemical

    Potential

    Fig. 15-1.

    t

    . . .

    415

    a

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    @

    Chemlcal potentlal,

    G

    e

    e

    a

    a

    a.

    . .

    e

    a.

    d

    e

    . . .

    Figure 15-3 indicates the situation a

    s

    it would be measured e

    x

    peri-

    mentall

    y The nonphysical loop

    ebfc

    ein Figure 15-2Ahas been repl

    ac e

    d

    with the

    tie-

    line ee. Figure

    15-3B shows

    the elimination

    of th

    e

    corresp

    o

    nding loop ebfce. an equation of state could be de

    v i

    sed tha

    t

    would

    e

    xactly reproduce isotherm aeed

    o

    n Figure 15-3A, the c

    a

    lc ulated

    chemical potentials of all states along tie-Iine ee would be

    identi

    c al.

    Toe point e to the left on Figure

    l5-

    3

    A is

    the position of the bubble

    point at the temperature of the

    isotherm

    , and the point e to the right

    i

    s the

    position of the dew point. the above analysis were performed at

    various temperatures below the

    criti

    c

    a }

    temperature, the phase en

    velo

    pe

    would defined. Figure 15-4 shows the position of the phase e

    nvel

    ope

    along w ith

    thr

    ee isotherms.

    . . .

    .

    a

    416

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    (1 5

    -4)

    Sin

    ce Equation 15-3 defines fu g

    a

    city in differential fonn, a

    r

    e

    fere

    nce

    value is required.

    (15-3)

    The

    c

    h

    e

    m i

    cal potential of a real

    fl

    uid can be expressed by

    r

    e

    pl

    acing

    pressu

    r

    e Equation 15-2 with a property called

    T h e u g a c i t y

    (

    1

    5

    2)T d(ln p).

    RT

    /

    p

    )

    dp

    Thus

    fo

    r

    a

    pure ideal gas

    (3-13)

    T

    The calculation of the

    proper

    ties

    o

    f

    equilibr

    ium gas and Iiquid

    r

    esol

    v

    es

    in

    to the calculation of the chemi

    c

    al potentials of the two

    ph

    ases.

    A n equation of state is required to evaluate Equation 15-1. For an i

    d

    eal

    ga

    s

    o :

    I

    1

    o

    417

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    The Peng-Robinson equation of state, Equation 4-35, will

    u

    sed

    with Equ

    ati

    on 15-7 to develop a

    proc

    edure for calculating

    of

    the vapor

    pressure of a pure substance. Toe vapor pres

    s

    ure is simply the pressure ,

    points e on Figure 15-2, for which the fugacity of the liquid equals the

    fugacity of the gas.

    The application here of the

    Peng-Robinso

    n equation of state doe

    s

    not

    mean that it is the best equation of state. This equation was

    m

    ere l

    y

    selec

    ted to illustrate the application

    of

    a typical

    three-constant,

    cubic

    equation of

    s

    tate to gas-Iiquid equilibria calculations.

    Example of State alculation for a Pure Substance

    where z is the z-factor as defined Equations 3-39 and 3-4

    0

    .

    Fo

    r

    a pure

    s

    ubstance, the ratio

    of

    fugacit

    y to pressure,

    f

    /

    p, is called

    The following expression can

    b

    e derived from Equation 15-5 under

    the con

    s

    traint

    of

    Equation

    15-4.

    In

    . . _

    z - 1 - In z _l RT - dV

    (

    15-7)

    p RT

    00

    Fugacity oefficient

    (15-6)

    Remember that chemical potential for

    th

    e liquid must equal chemi

    c

    al

    potential for the gas at equilibrium.

    F

    or a pure

    s

    ubstance this mean

    s

    that

    at any point along the vapor pressure

    li n

    e, the chemical potential of the

    liquid must equal the chemical potential

    of

    the gas. Thus Equation

    15-3

    shows that the fugacity of the liquid

    mu

    st equal the fugacity of the gas at

    equtiorium

    tf1'Clhe" Vaporr

    press1ii 'e 'lin

    e

    :S o g m ; . . :} i q t l"

    q

    ilibria c an be

    calculated under the condition that

    (

    1 5-5)

    ) dp

    Note that fugacity simply replaces pr

    es

    sure in an ideal gas equation to

    form a real gas equation. Fugacity has pressure units. Equation 15-4

    merely states that at low pressures the fluid acts Iike an ideal fluid.

    Combination

    of

    Equation

    15-1

    w

    ith

    15-3

    results in the defining

    equation for the fugacity of a pure substance.

    PETROLEUM FLUIDS

    18

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    (15-13)

    . b 0.07780 and O .

    a

    = 0.45724

    The acentric

    factor

    ,

    is a constant. Each pure

    s

    ubstance has a different

    value

    of

    acentric

    factor

    .

    The coefficients in Bquations

    15-9

    and

    15-10

    often are assigned

    symbols as follows.

    C l Y i ; : : ; 1 (0.37464

    l.542

    26w

    - 0.26992c . 1 > 2 )

    1

    -

    ) (15-12)

    where

    e x

    is determined as

    (15-11)

    The term b is a constant. Toe term aT vares with temperature; a, is its

    value at the critica

    temperature

    . The temperature variation of term aT

    resides in

    e x ,

    that is

    0.07

    780

    RT

    c

    (15-9)

    P e

    and

    ac

    0.45724 R

    T

    / .

    (15-10)

    P e

    Application of

    Equation

    s

    4-8

    at the critical point results in

    3 -

    ~ T

    b ) v

    M 2

    2

    b:T

    -

    3b

    2

    ) VM

    -

    b (

    ; -

    b

    -

    b o .

    (15-8)

    can be arranged into cubic form,

    (4-35)

    The Peng-Robinson equation

    of

    state

    419

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    Equation

    15-14

    is a cubic equation w ith real coefficients. Thus, three

    values of z-factor cause the equation to equal zero. These three roots"

    are all real when pressure and temperature are on the vapor pressure

    line-that

    is, when liquid and gas are pre

    se

    nt. One real root and two

    complex roots exist when the temperature is above the critical tempera-

    ture.

    Figure

    15-5

    gives the

    shap

    e

    of

    an

    i

    sotherm calculated with Equation

    15-14 at a temperature below the critical temperature. Points a through

    are equivalent to points a through

    fon

    Figures 15-2 and 15-3. Points e

    are the values

    of

    z-factor that would

    b

    e mea

    s

    ured experimentally. Point

    f

    is a nonphysical

    solution

    .

    As before, curve ebfce is eliminated. See Figure

    15-6.

    The upper

    point e is the z-factor of the equilibrium g

    a

    s, and the

    lo w

    e

    r

    point e is the

    z-factor of the equilibrium

    liquid,

    Toe dotted line

    c

    onnecting these two

    points has no physical

    meaning

    .

    The dashed curve represents the

    complete phase

    envelope.

    Notice the

    similarit

    y

    of

    the isotherm on Figure

    15-6

    to the experimental 104F

    is

    otherm of

    Figur

    e

    3-4

    .

    Thus, Equation

    15-14

    is

    solv

    ed for its three

    roots

    . th

    e

    re is only one

    real root, temperature

    i

    s

    above

    th

    e critica temperature. there are three

    real roots, the largest is the

    z

    -factor of the equilibrium gas and the

    (15-16)

    p

    RT

    and

    (15-15)

    where

    (15-14)

    z

    3

    - (1 - B

    )z2 (A

    -

    2B

    - 3B

    2)z

    -

    (AB

    -

    B2

    -

    B3

    )

    O ,

    into the Peng-Robinson equation of

    sta

    te g

    iv

    es

    (3-39)

    RT

    /

    p

    Substitution

    of

    PETROLEUM FLUIDS

    20

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    a

    421

    Fig .

    d

    a

    \ Crltlcal polnt

    . . . .

    \/

    c a

    -

    I

    1

    N

    /

    e

    .

    Pressure, p

    Pressure,

    . . . .

    -

    a

    -

    N

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    (

    15-9)

    007780) (10.73

    2

    )

    (734 .

    13)

    . (5 27 .

    9)

    (3-41)

    _ _

    190 459.67 0

    .

    88495

    r Te 274.46 459.67

    First,

    calculate those coefficients

    wh ic

    h are not pressure dependent

    EXAMPLE 15-1:

    Equation 15-17 is applied twice: on

    c

    e with the liquid z-factor to

    calculate the fugacity of the liquid and again with the gas

    z-facto

    r to

    calculate the fugacity of the gas.

    Toe

    pro c

    edure to calculate the vapor pressure of a pure

    sub

    s

    tan

    ce

    invol

    ves Equations 15-9 through 15-17 . Once temperature is

    sele

    cted

    ,

    the results of Equations 15-9 through 15-12 are fixed. The problem

    then

    is

    to find a pressure for use in Equations 15-14 through 1

    5

    -16

    which will give values of z-factors for gas and liquid which will result in

    equal values of fugacities of gas and liquid from Equation 15-17.

    I n (~

    )

    z - 1 - ln(z -

    8)

    - I n

    [ Z

    L

    (2 1 1 2

    1)8

    ]

    .

    (1 5-17

    )

    p L L 21.58

    Z

    L

    -

    (2112

    -

    1)8

    and

    { 1

    5

    -17

    )

    (

    fg ) - _ _ _ _

    A

    [

    Zg (2 1 1 2 l)B

    ]

    I n p - g

    l

    ln(zg 8)

    2 1 . 5 8

    I n Z g _ (2 1 1 2

    _

    l)B

    smallest is the z-factor of the equilibrium liquid. These are points e on

    Figure 15-5. The middle root, represented by point fin Figure 15-5, is

    discarded .

    Equation

    15-14

    can be used to

    compl

    e

    te the integration of Equation

    15-7, re

    s

    ulting in

    4

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    fg 1 76 . 79 psia

    1 n ( . . ) = 1 ~ ln(z - B) -~

    Z

    g 21 1 2 l}B

    \

    (

    1

    5-1

    7)

    p

    -; g 2 1 . 5 8 Zg

    -

    (2

    1 1 2 - l)B

    ]

    Toen

    ZL

    0.06

    7258 and Zg 0.70786

    ar

    e

    0067258

    ,

    0.18678, and 0.70786.

    (1 5

    -1

    4)

    3 - 2 2

    -

    (AB

    -

    B

    2

    -

    B3

    )

    O

    The

    th

    ree

    ro

    ots which salve

    (1 5

    -1

    6)

    (1 5-15)

    57995) (22

    8 .

    79)

    o .

    27295

    (10.

    732)

    2

    (649.67

    )

    2

    =_.EE._ =

    (1.1611

    ) (228

    .79

    ) =

    0 038101

    RT

    (1 0

    .

    7

    3

    2

    )(649.67)

    S

    eco

    nd ,

    by trial and error, fi nd a

    p

    ress

    ur

    e

    w

    hich causes the fuga

    c

    ity of

    the liquid calculated with Equations

    1

    5-14, 15-15, 15-16, an

    d

    1

    5-17

    to

    equ

    a

    l

    t

    he fugacity of

    the

    ga

    s

    c

    a

    lc

    ula

    t

    ed

    w

    ith the same

    equa

    t

    i

    ons

    .

    Onl

    y

    the

    last calc

    u

    lation with a final

    tri

    al

    valu

    e

    of

    p 228. 79

    p

    s

    ia

    will

    be

    shown .

    (

    1 5-11)

    T 8c U =

    (53,765)(1.0

    787) = 57,995

    = n R 2 T\ = (0 .45724) (1 0 .732)2 (734

    .13)2

    = 53,765 (1 5-10

    )

    Pe (5 27

    .

    9)

    o : 1 2

    1 (0.37464

    l.54

    226w

    -

    0.2

    6992w

    (1

    - T/

    h) (15

    12)

    u =

    1.0787 where

    c o

    =

    0.1

    852

    Caiculatkm

    s

    423

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    Toe results do not always agree

    th i

    s closely with experimental

    observation

    s

    .

    Toe trial-and-error process illustrated in Example 15-1 is rather

    tedious . Severa methods have been proposed to speed convergence to

    the correct solution. These

    method

    s can be into two general

    Vapor pressure, psia

    228.8

    22

    8.3

    Saturated liquid molar volume,

    2.050

    2 .035

    cu f

    U

    lb mole

    Saturated vapor Molar volume,

    2

    1.57

    21.68

    mole cu ft/lb

    Liquid z-factor

    0.0673 00666

    Ga

    s z-fa

    c

    tor

    0.7078

    0.

    7

    101

    Compar

    e results with experimental data

    (3-39)

    .050 cu ft/lb mole

    (0.06725 8) (10

    .

    732) (649.67)

    (2

    2

    8.79)

    (3-39)

    1 . 5 7 cu fUib mole

    (0.70786) (1 0 .732) (649.67)

    g p (2

    2

    8.79)

    f L

    f

    g,

    thus the trial value of p, 228.79 psia, is the vapor pre

    ss

    ure of

    iso-butane at 190F.

    Notice that the molar

    volum

    es can calculated easily.

    fL 1

    76

    .

    79 psia

    I

    n(

    ~)

    =

    z L

    - 1 -

    ln(zL

    - B

    )

    - _A__

    I n

    (

    Z L

    2 1 1 2

    l)B \

    (15-1

    7)

    p 2L

    S

    B Z L -

    ( 2 1 1 2

    - l)B

    l

    and

    PETROLEUM FLUIDS

    2

    4

  • 7/25/2019 Cap 15 McCain

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    for ali components

    j.

    This is analogous to the development

    of

    the

    equations for ideal solutions , early in Chapter 1 2, in

    w

    hich the partial

    pressure

    of

    the Iiquid (Dalt

    on's equ

    ation

    ) w

    as se

    t

    equal to the

    parti

    al

    pressure of the gas (Raou

    lt's

    equation).

    (

    15-20)

    Tha

    t is,

    a

    s pressure approa

    c

    hes zero the fluid a

    pproac

    hes ideal behavior

    and the fugacity

    of

    a

    com

    ponen

    t ap

    p

    roa

    ches the parta pre

    ss

    ure

    of

    that

    component.

    Remember

    t

    ha

    t

    t

    he Gj, for a

    c

    omponent

    of

    a

    mixture a

    t equili

    b

    riu

    m

    m u

    st be the

    sam

    e in

    bo

    th

    t

    he gas and the liquid.

    Thus Equation 15-18 shows

    that

    a

    t

    eq

    uilibri

    um the

    of

    a

    component must be equal in

    bo

    th

    th

    e gas

    an

    d the liquid

    .

    So gas-Iiqud

    equilibra can

    b

    e c

    alc

    ulated

    und

    e

    r

    the

    conditio

    n

    tha

    t

    (15-19)

    im

    fj

    p

    O

    The

    ref

    erence

    v

    al

    u

    e

    fo

    r

    fug

    acity

    th

    is

    eq

    uatio

    n

    is

    (15-1

    8 )

    T

    d(ln

    a s

    The chemi

    c

    al potential o

    f

    a o

    f

    a may be calculated

    Chemical

    Potential

    Mixtures

    The situation

    w

    ith

    regar

    d to

    mixtu

    res is so

    m

    ew

    ha

    t

    mo

    re difficult to

    visuali

    z

    e.

    Howeve

    r,

    e

    qui libriu

    m

    i

    s

    attained wh

    en

    th

    e

    che

    mi

    c

    al

    poten

    tial

    of each component in the liqud equals the chemical potential

    of

    that

    component in the gas .

    t

    ypes: succe

    ss iv

    e s

    ub

    st

    tut

    ons and

    Newton-Raph

    son

    .

    These

    t

    e

    chnque

    s

    will not

    be

    dscu

    ss

    ed in this te xt.

    425

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    The Peng-Robinson equation of state is

    Exampleof State Ca cu ation tor Mixtures

    where f L i f g j at equilibrium.

    f i . .

    j

    K j

    X j P

    Y j

    < p g

    f g

    X j

    Y j P

    where z is z-factor as defined in Chapter 3 .

    Further, the ratio of the fugacity coefficients can used to calculate

    K-factor.

    ln

    Z,

    [

    RT

    -

    ( c t p

    ]

    dV

    -

    dilj

    00

    Fugacity coefficient may be calculated as2

    ,k _J

    t J .

    YjJ)

    Another useful term is Fugacity coefficient far

    each component of a mixture is defined as the ratio of fugacity to partial

    pressure.

    Fugacity oefficient

    Values of the fugacity far each component are calculated with an

    equation of state. Any equation of state can be used far these calcula-

    tions. Later,

    asan

    example of the

    procedure

    ,

    we develop equations using

    the Peng-Robinson equation of state.

    PETROLEUM FLUIDS

    26

  • 7/25/2019 Cap 15 McCain

    14/24

    and

    1 (0.3

    7464 1.542260 0 ;

    -

    0

    .26992

    00/) (1

    - T

    r j Y i ) .

    (1 5-1

    2)

    (1

    5-10

    )

    cj

    0

    .45724

    R

    Tc/

    Pc i

    where

    (

    1 5-11)

    and

    (1

    5

    9)

    ; 0.07780

    RTc ;

    P

    e ;

    a

    s

    V

    alue

    s

    of the

    c

    o

    efficien

    ts

    far

    th

    e

    ind

    i

    v

    idual components

    ar

    e

    calc

    ul

    a

    t

    e

    d

    (

    1 5

    25)

    and

    (

    1 524)

    w

    he

    re su

    bscri

    pts and ref

    er

    to componen

    ts.

    Also

    I

    J

    I

    l

    J

    and

    (4

    38)

    Mix

    ture

    ru le

    s are

    427

  • 7/25/2019 Cap 15 McCain

    15/24

    l < j > j

    - ln (z -

    l)B

    'j

    A (A

    '

    -

    B 'j)

    - 21.58

    ln

    [

    z

    (

    2

    Jn

    (1 5

    26

    )

    z

    (2

    1 2

    l)B

    w

    here

    B'

    -

    (1527)

    J

    b

    E quation 1514 is cubic in

    z-f

    actor, Thus

    ,

    three values of z-factor

    cause t

    h

    e

    e

    qu

    a

    tio

    n to equal

    ze

    ro

    .

    Th

    e

    s

    e

    thre

    e

    roots

    are ali

    r

    e

    al w

    h

    e

    n

    pres

    s

    ure and temperature are such that the m

    i

    xture is two phase. There

    will one real root and two

    compl

    ex roots when the mixtu re is s ingle

    phase

    .

    When three

    r

    oots are obtained, the

    l

    owest root is the z-factor of the

    liquid. Toe

    hi

    ghest root is the z-fac

    to

    r of the

    ga

    s , and the middle root is

    discard

    ed

    . This is analogous to eliminating point f on Figure

    1

    5

    .

    Combi

    nin

    g the

    Peng-Robin

    son

    equ

    at

    i

    on of state and

    Equat

    ion 1

    5

    22

    res

    ult

    s in an equation for

    fu

    gacity

    coefficie

    nt of each component.

    3

    (15-16)

    p

    nd

    (15-

    1

    5

    )

    w

    he

    r

    e

    (1 5-1

    4

    )

    z3 - (1

    - B)z2 (A - 2B - 3B

    2

    )z

    - (AB - B2 - B3)

    O

    ,

    A

    s before, the Peng-Robinson

    equ

    ation

    ca

    n be written

    a

    s

    PETROLEUM

    F

    LU

    ID

    S

    28

  • 7/25/2019 Cap 15 McCain

    16/24

    (15-

    2

    1 )

    Equations 15-26 through 15-28 are written twice: once far values of z,

    b ,

    and aT o

    f

    the gas and again for

    va1ue

    s

    of

    z

    , b,

    and ay

    of th

    e

    l iq

    u

    id

    .

    Toe proc edure far calculating gas-liquid equilibria at given

    temp

    era-

    ture and pressure is as

    follows

    .

    V

    alue

    s o

    f

    aT

    j

    and

    bj

    for each component

    of

    the mixture are obta

    in

    ed

    w ith Equation

    s

    15-9 through

    15-1

    2 from a knowledge of the c

    ritica

    }

    properties and acentric

    factor

    s of

    t

    he pure components.

    A first trial set of K-factors is obt

    a

    ined. For instance, the K-fa

    c

    tor

    equation

    g

    iven in Appendix B can

    b

    e applied

    t

    o get the first trial set of K-

    factors. These are used in a gas-

    I

    iquid equilibria calculation, as described

    in Chapter 12, to determine the compositions of the gas and liquid. Toe

    rem

    aining equations are solved tw ic

    e,

    onc

    e

    for the liquid and once

    far

    the

    gas .

    V

    al

    u

    e

    s

    o

    f

    aT

    and b are calculated from Equations

    4-38

    and

    4-40

    w

    ith

    th

    e c

    om pos

    itions determined above. When the

    com

    position o

    f t

    he

    l

    iquid

    is used, the values are aTL and bL. When the composition of the gas is

    used, tbe values are

    a T g

    and bg.

    Valu

    es of binary interaction coeffici

    e

    nts ,

    6

    i j

    ,

    c

    a

    n

    b

    e

    in

    c luded in Equation 4-

    40

    they are known.

    unknown,

    the

    values

    o

    f 6

    i j

    can be set equal to zero. Values of A and B far the Iiquid, A L

    and

    BL ,

    are calculated with Equations 1 5-15 and 15-16

    u

    s ing aTL and

    bL.

    Wben

    a T g

    and bg are used

    Equation

    s 15-15 and 15-16, and B

    8

    res

    ult.

    Also , B\ and A'i must be calculated for each componentj. B 'j

    L

    results

    when b i . is used in Equation 15-27 , and B jg results when b

    8

    is u

    s

    ed. A

    '

    iL

    and

    A

    'j g

    r

    esult

    s

    imilarly from

    Equatio

    n 15-28.

    The smallest root of Equation 15-14 is zL when and B L

    ar

    e used.

    The

    lar

    gest root of Equation 1514 is Zg when Ag and B

    g

    are used.

    Equati

    o

    n 15-26 is solved for the

    fu

    gacity coefficients of the compo-

    nents of the liquid, < > L j , using v

    alue

    s of AL, B L , A'jL , and B'jL

    Values of q >

    g

    result when the c orr

    e

    sponding

    ga

    s coefficients and z-factor

    ar

    e used in Equation

    15-26.

    TI1

    en v

    alue

    s of liquid fugacity and gas fugacity far each componen are

    obtained from

    (

    15-2 8)

    and

    429

  • 7/25/2019 Cap 15 McCain

    17/24

    where K{ are the K-factors just

    calculat

    ed and the K/ are the trial values

    of K-factors .

    Conve

    rg

    en

    c

    e on a correct solution is obtained when the sum

    o

    f the

    error

    function

    s is less than sorne selected tolerance. the sum of the

    error fu

    n

    ctions is greater the tolerance, the

    K{

    are used as

    ne

    w trial

    values of Ki , and the proces

    s

    is repeated.

    (15-3

    1)

    is less than sorne selected toleran

    c

    e.

    Another error function used in

    c

    onverging on

    a

    correct solution

    b

    y a

    rnethod of successive substitution

    inv

    olves K-factors. Toe for the

    mixture are determined frorn the

    fug

    acity coefficients with Equation

    15-23

    .

    To

    e

    n

    (15-30)

    wh

    er

    e a solution is obtained when the Euclidean norrn

    of

    the

    E

    i

    (1

    5

    9)

    There are as many Equations 15-20 as there are components. All

    these equations cannot be satisfied simultaneously. Thus sorne sort o

    f

    err

    or

    func

    tion based on Equation

    15

    -20 mu

    s

    t be devised, One approach

    is

    (15-

    20

    )

    Equilibrium is obtained and the calculation is complete when ali

    (15

    -21)

    a n d

    PETROLEUM FLU IDS

    30

  • 7/25/2019 Cap 15 McCain

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    (159)

    j 0.0778

    0 RT

    c i

    P c i

    (1 5

    0 )

    2

    T .2

    a

    c j

    cJ_

    P

    c j

    1 (0.37464 l .54226(1)j - 0.26992w/) (1 -

    T

    1

    (1

    512)

    First, calculate the coefficients of

    t

    he components of the m

    i

    xture.

    1

    .0000

    M ethane 0.5301

    n-

    Butane 0.1055

    n-Dec ane 0.3644

    fracton

    EXAMPLE 15-2:

    4 3

  • 7/25/2019 Cap 15 McCain

    19/24

    (4

    -

    38)

    (4

    40)

    Third, calc

    ul

    ate the compo

    s

    ition de

    p

    endent c

    oefficients

    n

    ecessar

    y

    fo

    r

    z-factor c alculations for both I iqu

    i

    d and

    g

    a

    s

    .

    Z

    j

    Xj

    C

    3.992 0 .5301 0.2408 0 .9613

    n-

    C4

    0.2413

    0.1055 0.1517

    0

    .0366

    n-

    C1 0

    0.00340 0.3644 0.6075 0 .0021

    1

    .0000 1.0000

    1.

    0000

    Thi

    s

    c a

    lcula

    tion requires

    t

    rial and error;

    o

    nly the final trial

    w

    ith ff

    g

    0 .40 1

    5

    is s

    how

    n.

    kX k Zj

    ; J J n

    g

    Second ,

    se

    l

    ec

    t tria values of K-factors and calculate tri

    a

    l com

    posi

    tio

    n

    s

    of e

    q

    uilibriumgas and liquid.

    On

    ly

    th

    e fi

    na l trial ,

    with

    K-fact

    ors as given

    belo

    w, is

    s

    hown.

    343.0 666.4

    0 .0 1 0

    4

    0

    .

    748

    1 9,297 6,956 0.4297

    765.3 5 5 0 .

    6 0.199

    5 1.1394 56,017 63,827 1.1604

    1111.7 305 .2 0.4898 1.6139 213,240 344,149 3.0411

    Tcj

    PETROLE

    UM

    FLUID

    S

    32

  • 7/25/2019 Cap 15 McCain

    20/24

    (15-27)

    J b

    (

    1

    5-28)

    Fifth , calculate the co

    m

    positi on

    depend

    ent coefficie

    nts neces

    sary for

    calculating

    fugacit

    y coeffic i

    en

    ts

    for

    l

    iq

    uid and

    gas

    .

    0

    90

    51

    L

    0 .3922 and Z

    g

    (

    15-14)

    z3

    - (1 -

    B )

    z

    2 (A

    -

    2B

    - 3

    B

    2

    )z

    - (AB -

    B

    2 -

    B

    3)

    O

    ,

    Fourth,

    calc

    ulate

    z-fa

    c

    t

    o

    rs

    o

    f li

    quid a

    n

    d

    gas.

    0

    .

    3198

    3

    0

    .

    06945

    3.876

    6

    0 . 1

    8 4

    9

    2 .

    1 270

    0 .4

    619

    1

    7

    1

    ,

    446

    8,17

    7

    Liquid

    Gas

    A

    hase

    (15-16)

    (15-15)

    433

  • 7/25/2019 Cap 15 McCain

    21/24

    (1 5

    -31)

    (

    15-23)

    Seventh,

    calcu

    la

    te the

    K

    -

    fa

    c

    tors o

    f

    th

    e

    c om

    pon

    e

    nts and

    t

    he error

    functions.

    0.92065

    0 .53373

    0.20600

    3

    .

    67552

    0 .

    1 2

    878

    0.000699

    < l >

    L j

    (15-26)

    In < l > j

    {

    z

    -

    B) (z

    -

    2

    1

    ~

    B

    Ai - B 'i)

    In

    [

    z

    (2 1 : l)

    B J

    z -

    (2

    l

    )B

    Sixth, calculate the fugacity coefficients o

    f

    th e components of liquid

    and gas.

    434

    PETR

    OLEUM FLUIDS

    B

    B J

    J

    J

    Cl

    0. 38893 0.20204

    1.8440 0.93042

    1

    .2

    2127

    0. 54559 5 .5014

    2 .

    51

    253

    n-C

    10

    2.83309 1 .42979

    1 2.5445

    6.5 8435

  • 7/25/2019 Cap 15 McCain

    22/24

    zRT

    (3-39)

    p

    _pM

    (3

    39)

    zRT

    z

    VM

    Ma

    ft

    L iq u

    i

    d

    0.3922

    2.61

    99.12 38.00

    G as 0.9051

    6

    .02 17.84

    2.96

    A l

    s

    o, the molar volumes and den

    s

    itie

    s

    can be calculated.

    Xj

    Y J

    Xj

    Y J

    C

    0 .

    24

    08

    0 .9613

    0.242

    0.963

    n-C4

    0 .1517 0 .0366

    0.152 0.036

    n-C

    10

    0.607

    5

    0.0021

    0.606

    0.

    002 1

    1.0000

    1 .0000

    1.000

    1 .0

    011

    Tbe sum of the error functions is le

    ss

    than a tolerance of O.001, so the

    s

    et

    of

    trial

    values of

    Kfactors

    was

    corr

    ec

    t and the calculated values o

    f

    Iiquid

    an

    d gas comp

    o

    sitions are corre

    c

    t.

    Compare results with experimental

    data

    .

    0.000

    0.000

    0.000

    3.992

    0.2413

    0.00340

    435

  • 7/25/2019 Cap 15 McCain

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    1.000 0

    . 99

    93

    0

    .

    8

    26

    0.16

    7

    0 .

    0 0 6

    3

    0.485

    0

    .412

    0.103

    M ethane

    n-Butane

    n-Decane

    gas

    C

    ompare your answer with

    ex

    perimental results shown b

    e

    low.

    5

    1 .0

    000

    0 .5 532

    0 .

    3630

    0 .

    0

    8

    38

    Methane

    n-Butane

    n-De

    c

    ane

    Exercises

    15-1. Use the Peng-Robinson equ

    a

    tion of state to calculate

    th

    e vapor

    pre

    ssure of ethane at 32F. Al

    so

    , calcul

    a

    te the

    den

    sities of the

    liquid and gas at 32

    F.

    Compar

    e your answers with v

    alues

    from

    Figure

    s

    2-7, 2-12

    ,

    and

    3-3

    .

    15-2.

    U s

    e the Peng-Robin

    s

    on equati on of

    s

    tate to calculate the v

    apo

    r

    pr

    ess

    ure of propane at i04 F . " Ais , calcuiate the' 'densities of the

    liquid and gas at 104"F . Compare your answers with valu es from

    Figure

    s

    2-7, 2-12, and

    3

    4

    .

    15-

    3.

    U

    se the Peng-Robinson equ

    a

    tion of

    s

    tate to calculate the c om

    po -

    sition

    s

    and densities of the e

    q

    uilibrium liquid and gas of the

    mixtur

    e

    given below at 1 60F and 2000 psia. U se

    bin

    ary

    int

    er

    a

    c

    tion coefficients

    of

    0

    .0

    2 for

    methane-n-but

    a

    ne

    ,

    0

    .

    035

    fo

    r

    methane-n-decane, and O

    .

    O for n-butane-n-decane.

    Much more sophisticated and

    po

    werful methods of converging on a

    solution are available. They will not di

    s

    cussed here.

    These equations can be used

    al

    s

    o

    t

    o

    c

    alculate the bubble points and

    de

    w points of mixtures. The solution

    t

    echniques in these

    applic ations

    differ from those used in Example 1 52.

    PETROLEUM FLU IDS

    36

  • 7/25/2019 Cap 15 McCain

    24/24

    Chao , K.

    C

    and

    Robinson

    ,

    R

    .L., Jr. (eds.):

    Ad

    van

    c e

    s

    Chemistry Series

    A CS

    ,

    Washingt

    o

    n (1979).

    General References

    Pha

    s

    e

    SPE Reprint Serie

    s

    SPE, Dallas

    (1981

    ).

    References

    Fussell, L.T.: A Technique for Calc ulating Multiphase Equilib-

    ria

    ,

    J.

    (Aug. 19

    79

    ) 203-210.

    2. Edmister, W.C. and Lee, B .I.:

    2nd ed., Gulf Publishing Co., Houston (1984).

    3 . Peng,

    D

    Y. and Robinson,

    D.B

    . : A

    N

    ew Two-Constant Equation

    of State, (1 976) No. 1 , 59-64.

    4. Fuss

    ell

    , DD. and Yanosik, J.L.:

    "

    AJl lterative Sequence far Phase

    Equilibria Calculations Incorporating the Redlich-Kwong Equation

    of State, J (June 19

    7

    8) 173-182.

    5. S ag

    e

    , B.H. and Lacey, W.N.: Thermodynamic

    Properti

    e

    s of the

    Lighter Paraffin

    Hydrocarbon

    s and Nitrogen,"

    API, New York (1950).

    Compare your answer with the results of Example

    2-8.

    1.000

    0.500

    0.150

    0.350

    Methane

    Propane

    n-Pentane

    fraction

    15-4. Use the Peng-Robinson equation of state to calculate the compo-

    s

    itions, densities, and quantities (lb moles) of the

    equilibri

    um

    liquid and gas of the mixture

    given

    below at 160F and 500

    ps

    ia.

    U se binary interaction coeffici

    e

    nt

    s of O . O for methane-propane,

    0 .02 for methane-n-pentane,

    a

    nd

    0.01

    for propane-n-pentane

    .

    43

    7