Cap 15 McCain
Transcript of Cap 15 McCain
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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(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
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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
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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
-
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(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
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(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
-
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(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
-
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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
-
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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
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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