ford and Phillips, a s w a s done by Herzog in h i s original article. Constants were a l s o derived for these equations using experimental parachors but no significant improvement in correlation w a s observed. Hence, it w a s decided t o u s e t h e calculated parachors throughout s i n c e calculated parachors a r e normally more convenient to use. Equations with t h e re-evaluated constants a r e shown in Tables I, 11, and 111, together with their per cent “reliabilities.” Reliabilities of t h e estimated T , values were calculated using the equation shown below. T h i s equation gives the maximum deviation to b e expected in 95% of the c a s e s .
The same approach w a s used in determining the p c and V, equation reliabilities. d=200
{-
1
n-1
T,(exptl.)
LITERATURE CITED (1) Herzog, R . , Znd. Eng. Chem. 36, 997-1001 (1944).
Received for review April 10, 1957. Accepted January 16, 1958.
Prediction o f Compressibility of Natural Gas Mixtures by Use of the Benedict-Webb-Rubin Equation of State JUI SHENG HSIEH’
and RICHARD H. ZIMMERMAN T h e O h i o State University, Columbus, Ohio
A
knowledge of t h e pressure-volume-temperature properties of natural g a s is of great importance in t h e transportation, storage, and u s e of t h e gas a s a fuel and a s a source of raw materials for chemical syntheses. Because of t h e variable composition of natural g a s and t h e time and skill required t o determine t h e P-V-T properties in t h e laboratory, i t h a s long been t h e goal of research engineers within t h e industry t o find a n accurate method for their calculation. Heretofore, t h e methods proposed have been of limited application. The tentative standards a s s e t forth by t h e Natural Gasoline Association of America (25) and t h e California Natural Gasoline Association (9, 10) a r e either not of sufficient accuracy or a r e limited t o relatively low pressures. Following t h e pseudo-critical concept of Kay (13), and assuming the law of corresponding s t a t e valid for any natural gas mixture, Dunkle (12) proposed a method using pure methane a s t h e b a s i s for evaluating t h e compressibility of natural g a s mixtures. Dunkle’s approach w a s later used by Zimmerman and Beitler (16, 17). Their
‘Present address, University of Bridgeport, Bridgeport, Conn.
Table
1.
method i s simple and accurate for most natural g a s e s , a n d h a s been adopted by t h e American G a s Association a s a new standard for t h e natural g a s industry (1). However when t h e ethane and/or nitrogen and carbon dioxide cont e n t s a r e high, t h i s method i s not so accurate a s i s generally desired. T h e present study was undertaken t o t e s t the application of the Benedict-Webb-Rubin equation of s t a t e (4, 5) to t h e prediction of t h e compressibility of natural g a s mixtures and particularly t o mixtures of high ethane and/or nitrogen and carbon dioxide content. T h e study was confined to t h e gaseous phase in the pressure and temperature ranges usually encountered in i t s transportation and storage. T h e data on natural g a s mixtures compiled’ by Zimmetman and Beitler (16, 17j were used for purposes of comparison in the study. CONSTANTS STATE
FOR
BENEDICT-WEBB-RUBIN
EQUATION OF
Benedict, Webb, and Rubin ( 5 ) have proposed t h e following empirical equation for the representation of the P-V-T relations of a pure gas:
Values of Constants i n Benedict-Webb-Rubin Equation o f State for Compounds Constituting Natural Gas Mixtures
Units. P = pounds per square inch absolute T = degrees Rankine (OF. + 459.63) d = pound-moles per cubic foot R = 10.7335 Ethane,
BO A0
c o x 10b a c
x 10-
a Y
BO A0
c o x 10b a c x 1v-
a Y
194
Methane, (4)
Propane, (4)
Isobutane, (4)
n-Butane, (4)
Isopentane,
Adjusted
0.682401 6,995.25 275.763 0.867325 2,984.12 498.106 0.511172 1.53961
1.00554 15,525.3 2,194.27 2.85393 20,850.2 6,279.39 1.00044 3.02790
1.55884 25,9 15.4 6,209.93 5.77355 57,248.0 25,247.8 2.49577 5.64524
220329 38,587.4 10,384.7 10.8890 117,047 55,977.7 4.4 1496 8.72447
1.99211 38,029.6 12,130.5 10.2636 113,705 61,925.6 4.52693 8.72447
2.56386 48,253.6 21,336.7 17.1441 226,902 136,025 6.98 77 7 11.8807
n-Pentane, (4)
n-Hexane, (4)
n-Heptane, (4)
Carbon Dioxide, Proposed
Nitrogen, ( 8)
2.51096 45,928.8 25,917.2 17.1441 246,148 161,306 7.43992 12.1886
2.84835 54,443.4 40,556.2 28.0032 429,901 296,077 11.5539 17.1115
3.18782 66,070.6 57,984.0 38.99 17 626,106 483,427 17.90 56 23.09 42
0.558255 8,192.54 1,864.48 1.36384 10,554.4 2,959.67 0.193192 1.07773
(4)
INDUSTRIAL AND ENGINEERING CHEMISTRY
(41
0.652648 3,973.31 98.4657 0.597292 1,516.34 142.564 0.522851 1.35999
VOL. 3, NO. 2
\
METHANE, MOLE PER CENT
(1)
\
\ \
\ \
(3) PIVOT N 0 . I
(5)
/
\
PIVOT N 0 . 2
( 7 ) PIVOT N p . 3
0
0
(12)
PIVOT N 0 . 5
Figure 1.
=
(
R T d + B,RT
- A,-
7
T1
Alignment chart for constant 19,
dl + (bRT - a)&+ Cd' T
Bad6+ 1(1
+ yd3e-Yd'
(1)
where P is absolute pressure, T is absolute temperature, d is molal density, R i s t h e universal g a s constant, and 1958
\
( 1 3 )CONSTANT B~ FOR MIXTURES CONTAINING CARBON DIOXIDE, NITROGEN, a HYDROCARBONS (VALUES OF Bo gBTAlNED FROM THIS SCALE ARE SUBJECTED TO ADJUSTMENT )
v
P
\
B o , A,, C,, b, a, c, a, and y a r e constants characteristic of the individual g a s e s and t h e dimensional units employed. T h e eleven chemical compounds most commonly appearing in natural g a s mixtures and the corresponding values of the eight characteristic constants required in Equation 1 a r e listed in Table I. ?he values of the constants for t h e hydrocarbons are principally those given by B e n e d i d , Webb, and Rubin (4), CHEMICAL AND ENGINEERING DATA SERIES
195
W 0
W -
W
N
W
w
P
W
V
W
I
r
W
W
-
J
W
W
l
g
~
\
\ (3)
\
\
PIVOT N0.I
/
(5)
/
fi
PIVOT N0.2
\
\
\
0
( 7 ) PIV T Np.3
-
D
METHANE, MOLE PER CENT
(I1
\
W
n
/
\
(9) CONSTANT A~ FOR M I X T ~ R E SCO,NTAINING HYDROCARBONS ONLY, OR PIVOT P
~0.4
VI
0 0
0
0
(2)
ETHAYE, MYLE PER'fENT
/ P V
0
V I I 0
0 0 0
0
0 0 0
0 0 0
0 0 0
f
W
' / I/ I
(12) PIVOT N0.5
Figure 2.
Alignment chart for constant A,
while those for nitrogen were evaluated by Bloomer and Rao (8). Constants A , and c for ethane were adjusted so that the calculated P-V-T relationships of gaseous ethane agree more closely with the experimental data in t h e temperature range 0 ' to 150' C. T a b l e I1 shows a comparison of average per cent deviations of pressure calculated by t h e equation of s t a t e using the original and the adjusted constants of gaseous ethane from the experimental data of
196
INDUSTRIAL AND ENGINEERING CHEMISTRY
Beattie, :ladlock, and Poffenberger (2) and Beattie, Su, and Simard (3). The constants of the equation of s t a t e for carbon dioxide given in Table I were evaluated according to the stepwise procedure described by Benedict, Webb, and Rubin (5), using the data on carbon dioxide of Michels and Michels (13). Table 111 shows the agreement between the experimental pressure and the pressure calculated by t h e equaVOL. 3, NO. 2
0
-
\
\
\
\
\
\
\
\\
,"\ /
( 3 ) PIVOT N0.I
\
(5)
PIVOT Iy0.2
\
\ ( 7 ) PIVOT N0.3,
1'1'
\
'I'I'I'I'I'I'I
/
(IO)
CARBON DIOXIDE, MOLE PER CENT
0
' / (12) PIVOT N0.5
0
-
r
.
J
W
P
V
I
0
)
-
4
Q
Figure 3.
1958
)
C
D
-O
-
-
Alignment chart for constant C,
CHEMICAL AND ENGINEERING DATA SERIES
197
W
W
0
-
W
~
W
m
m
W
m
m
u
)
METHANE, MOLE PER
(1)
w
w
w
w
~
u
)
u
-)
CENT
\
\
\
\
\
(3) PIVOT N9.I
/ n
(5) PIVOT N 0 . 2
\ \
\
( 7 ) PIVdT' N0.3 / \ (9) CONSTANT b FOR MIXTURES CONTAINING HYD6OCARBONS ONLY, OR PIVOT N0.4
n
E 0
2 -
N
-
,2
N
m
/w
4
0)
0
u) I,,,,,
(2)
\I
(12)
ETHANE, MOLE PER CENT
\
PIVOT
N0.5
Figure 4.
Alignment chart for constant b
tion of state. T h e carbon dioxide constants developed by Cullen and Kobe (11) were not available at the t i m e of this study. A comparison of the predicted compressibility factor of carbon dioxide using t h e s e constants and t h e constants presented i n Table I yields agreement to less than 0:1% for pressures under 2000 p.s.i.a. and within 0.2% for pressures up t o 3000 p.s.i.a.
198
INDUSTRIAL AND ENGINEERING CHEMISTRY
\
COMBINATION OF CONSTANTS I N EQUATION OF STATE FOR MIXTURES
In the application of the Benedict-Webb-Rubin equation of state to mixtures, the following equations were used for combining the constants of individual g a s e s to obtain t h e constants for the mixture: VOL. 3, NO. 2
~
w
(I)
\
METHANE, MOLE PER CENT
\
\
\ \
\ (3)
\
PIVOT NO.1,
/
/ '
\
\
(5) PIVOT, N0.2 (7)
\
P I V O T N0;3
\
(13) CONSTANT a FOR hIXTURES CdNTAINING CA
-
N
0 0 0
VI
0 0
V
I
9
0
0 0
0 0
11111111//111~11111~111
I 11
\
/
\
; z: in/ \g R \ 0 -
0
(4)
PROPAN?, MOLE PER
P
P
Ir
f\,
b
0
CENTO
A -
0
P
N
'
Q,
(x1
W
-
0
1
\ / \ / (12)
PIVOT
N0.5
I
I
I
I
0
-
Figure 5.
Bo = ( Z XB~o , ) A , = (Zx, A!,)2
- 16.0188 X M (0.007 XE - 0.0055 xN)
Alignment chart for constant a
(2)
b = (Zx, by)'
(3)
a=
(4 )
(Zx, a,Y),
c = (Zx,c,Y), CHEMICAL AND ENGINEERING DATA SERIES
(5)
(6)
(7) 199
\ \ \
\
\
\
\ (3) PIVOT N0.I
\ P ,\
1'
\
( 5 ) PIVOT N 0 . 2
\
\
/'\
\ y (7) PIVOT N 0 . 3
0
0 0
PJ
m
0
0
0 VI
-4
-
0)
P 0
0
VI
m
0
\ I
200
INDUSTRIAL AND ENGINEERING CHEMISTRY
VOL. 3, NO. 2
\
\ \
\
\
(3) PIVOT \NO. I
I
(5) PIVOT N0.2
\\ \
(7) PIV&\NO.3
\
\ 0
0
-
N
I (13) CONSTANT
d
I
0,
W
0
-
0
\
FOR MIXTURES CONTAINING CARBON DIOXIDE, NITROGEN, 8 HYDROCARBONS
in
0
0
-4
Q,
0
-
b
-
b
-
0
(6) BUTANES, MOLE \PER CENT I
(8) PENTANE 8 HIGHER HYDROCARBONS, MOLE PER CENT i’(l0) CARBON DIOXIDE, MOLE PER CENT
e
0
VI
0,
-4
0)
W
5
\
(12) PIVOT N0.5
I
I Figure 7. Alignmen( chart for constant a
1958
CHEMICAL AND ENGINEERING DATA SERIES
201
\ \ (3)
\
\
PIVOT\ v0.l
\
/ 0
\
/
O
0
c”
P
-
;
Y
w
b
i
i
/
(8) / I
\
i u L J b i n h 4 PENTANE & HIGHER HYDROCARBONS, MOLE PER CENT &
k
b
i
D
O
c
o
b
N V
0
-
(12) PIVOT N 0 . 5
I
\ Figure 8. Alignment chart for conatant y
202
INDUSTRIAL AND ENGINEERING CHEMISTRY
VQL. 3, NO. 2
VI
b,
-4
0
0
0
0
0
0
a
0 0
,
0 0
a
-
o
0 0
~
c
o
a
,
~
~
u
l
P
W
~
/
=
(2) T E M P E R A T U R E , DEGREES
(4) CONSTANT
P
8
a, 0
Bo
0
VI
(
D
O
)
-
4
b
,
V
I
P
W
N
FAHRENHEIT
OF MIXTURE
"\ \
0 -4
O
:
P
a, 0
VI
P
0
VI
-
(0
0
VI
0
\ w 0 0 0
01 VI
P 0
0
0 0
0
\ \
Figure 9.
Alignment chart for
CO
- and &RT in function F(t) T'
a = @xi a,'A )I
(8)
y = (Zx,y?)'
(9)
T h e s e equations with t h e exception of Equation 2 are given by Benedict, Webb, and Rubin (4), whose original
T a b l e 11. Comparison of Absolute Average P e r Cent D e v i a t i o n s of C a l c u l a t e d Pressures from Experimental D a t a of Gaseous Ethane for D e n s i t y Range 0.5 to 10 Gram-Moles per L i t e r
Temp. "C.
25
50
75
(2, 3)
100
125
150
By original constants 0.19 0.45 0.38 0.36 0.33 0.23 Byadjustedconstants 0.18 0.21 0.22 0.19 0.14 0.19
1958
Over-all Average 0.34 0.19
A0
+-CO - 60RT t'
equation for combining the constant E , is B o = X x , B o , . T h e modification of Equation 2 w a s found desirable on the b a s i s of a comparison of calculated P-V-T data with the T a b l e 111. Absolute Average P e r Cent D e v i a t i o n s of C a l c u l a t e d Pressures from Experimental D a t a of Gaseous Carbon D i o x i d e for Tempemture Range 0" to 150°C. (14)
Density, gram-moles/liter 1.05385 1.90565 3.10633 4.09273 5.03908 Abs. av. %deviation 0.05 0.04 0.08 0.14 0.21 Density, gram-moles/liter 5.99555 6.90826 0.23 Aba. av. %deviation 0.27 Over-all average % deviation 0.15
7.65599 8.46911 9.27348 0.11 0.30
0.16
CHEMICAL AND ENGINEERING DATA SERIES
203
-
VI
9
\ -
0
I
0
0
U
w
P
0
0
V
0
I
m 0
0
J
0
m
_
a 0
-
O 0
_N _
0
0
0
_
W
Figure 10. Alignment chart for bRT in function G ( t ) = bRT
0
L
-
-
V
0
0
0
I
-
~
0
_ 0
~
0
-o
-
0
m
c
CONSTANT
(1)
OF MIXTURE
I
I
I
(3) THE FUNCTION L ( t ) = C / T 2 0
0 0
0
O
0
a
0
0 0
VI
0
m
0
-
4
0
l
0
n
0
V
I 0
P
0
w
O 0
N
0
Figure 11.
204
INDUSTRIAL AND ENGINEERING CHEMISTRY
0 0 0
UI
0
-
J
o
0 . 0
E
a
0
w
0
-
4
0
Alignment chart for function
m
V
0
L ( t ) =-
0
0 0
0 0
0
0
I
P
0
0 0 0
w
0
N
0
0 0
0 0
0
0
-
0
0
C
T' VOL. 3, NO. 2
Q
)
i i \ \ \ 1 ,
I
I
,
,
//I I
I
,
\
I I I I Ill
1111
I
I
I
,
8
Jl,l,l,l,l,l#l,
I I l l I l l / IU
111 I
I
111 I
I
I I
I II I I
I
I1 I I I I I I I
I I 1 I Ill
(3) THE FUNCTION F ( t ) . d 2
I
\ 0 0
re
0 0
w
8 P
.
0
0
8
0 , 4 m ( D o
0 0
0 0
0 0
0 -
P w
:
0
p
P
0
0
w
e
2 0c n k0 0
0
0
Figure 13. Alignment chart for function F (t) d' 1958
CHEMICAL AND ENGINEERING DATA SERIES
205
w
0 0
Q,
-4
0
0 0
0
gg 00
0
0
0
0
-
Pul
N
a
c
0 0
0
0
W
p
-
o
0 0
-
o 0 0
-
VI
-
N
W
0
W
w
N
0
W
0
o
VI
0
0
0
0
0
\ I
0
I u w v , 0
0
0
0
0
0
0
0
o
w
0
0
w
0
0
0
- o
0 0
w
0 0
\
\ 0 -
I?
V
P
w
0 0
0 0
(3) THE FUNCTION G ( t ) , d 3
0
w
N
0 0 0
0 0
P
0
N
w
VI
R
0
0 w
0
x 0
g
8 2
0
0
0
Figure 14. Alignment chnri for functlon C (f) d 3
experimental data o n the methane-ethane and methanenitrogen systems. If either the ethane or nitrogen content is under 5 mole %, t h e modification made in Equation 2 for that component is small and may be neglected. If both ethane and nitrogen contents are under 5 mole %, Equation 2 reduces to the original form. Natural g a s is essentially a mixture of methane and ethane, commonly, with only minor quantities of the higher hydrocarbons, carbon dioxide and nitrogen. Accurate representation of the P-V-T relations of methane-ethane mixtures is essential t o the prediction of compressibility of Table
IV.
P e r Cent D e v i a t i o n s for Two Natural Gas Samples
% Deviation in Pressure
Temp. vange,
Pressure, Sample
P.S.L
A
3000 800
B
Benedict original
(4
F.
Present modified rules, (Equations 2 throu& 9)
Average Maximum Average Maximum
70 to 150 40 to 80
0.25 0.19
LO4 0.51
0.18 0.15
0.64 0.40
% Deviation in Compressibility Factor
Present modified rules, (Equations 2 through 9)
Zimmerman-BeitIer correlation (16, 1 7 )
Sample
Average
Maximum
Average
Maximum
A B
0.19 0.16
0.57 0.46
0.77 0.23
1.41 0.40
Note. Fractional analysis, mole per cent Sample A 85.36 methane, 6.14 ethane, 3.72 propane, 0.50 isobutane, 0.79 n-butane, 0.20 isopentane 0.07 n-pentane, 0.08 hexane, 0.04 heptanes, 3.10 carbon dioxide, 0.00 nitrogen Sample B. 71.25 methane, 5.69 ethane, 3.37 propane, 0.34 isobutane, 0.64 n-butane, 0.08 isopentane, 0.06 n-pentane, 0 . 1 3 hexane, 0.07 heptane, 0.10 carbon dioxide, 18.27 nitrogen
206
INDUSTRIAL AND ENGINEERING CHEMISTRY
natural gas. However, the equations given by Benedict, Webb, and Rubin for combining constants lead to results which give only a fair representation of t h e experimental data of methane-ethane mixtures. U s e of Equation 2 greatly reduces the average error of the calculated pressures. T h e average deviations of the calculated pressures from the experimental data of Bloomer, Gami, and Parent (7) for a mixture containing 85.06- mole % of methane and 14.94 mole % of ethane are 1.03 and 0.18% by the u s e of the equations of Benedict, Webb, and Rubin for combining constants and by Equations 2 through 9, respectively. A s most natural g a s e s contain considerably less than 15% of ethane, the present modification is considered satisfactory for natural g a s mixtures having a high ethane content. The expanding and ever-increasing u s e of natural gas, a s a fuel and in many industrial applications, forecasts the necessity of using g a s reserves of increasingly high nitrogen content. Again, in t h i s case, the equations of Benedict, Webb, and Rubin for combining constants lead to results which give only fair representation of actual data, T h e error in the calculated pressure is greatly reduced by the u s e of Equation 2. T h e average deviations of calculated pressures from the experimental data of Bloomer (6) for two methane-nitrogen mixtures, one containing 3 0 mole % and the other 50.06 mole % of nitrogen, a r e 0.84 and 1.04% respectively, by the u s e of t h e equations of Benedict, Webb, and Rubin for combining constants and 0.16 and 0.20% respectively, by the u s e of Equations 2 through 9. Table IV shows a sample of the average and maximum per cent deviations in pressure and in compressibility factor, P V / R T , a s predicted by three methods. Mixture A has a n ethane content which is higher than the average, and i t s propane and carbon dioxide contents a r e about the maximum amounts which appeared in a l l available samples. Mixture B is the only sample available with a high nitrogen content, but, unfortunately, data a t high pressures were not For a l l natural g a s mixtures studied in this obtained. work, t h e Zimmerman-Beitler method (16, 1Q gave a total over-all absolute average deviation of 0.32% in the comVOL. 3, NO. 2
VI
-
0
N
0 0
0 0
0 0
OF
a
w
0 0 0
0 0
CONSTANT
(I)
w
N VI
0
P
P
o
VI
0
r
0
0
111111 I I I I I I I I I I, I I I MIXTURE
0
u
l
n
0
0
0
0
\
ul 0 0 0
o
\
-
4 0 0 0
m
0 0 0
0 0 0
w
0 0 0
o
0
-
4
B
w
o
\
\
\
\
(3) P I V O T
0 -
0
-
0
0
0
N
0
VI
0
N VI
0
( 2 ) CONSTANT
q
P
w
O
0
0
OF MIXTURE
i i
0
0
i
i
i 0
0
0
0
0 -
0 0
0 N
0
U
0 0 -
0 0 N
0 0 V
0 -
-
N
0
O
N
V
0
0
I
-
N
0 0
1
c " 0
8:
0 0
I
Yllllllyrrrrrlli(lllllliTllTTTirnr'1'' ' ' I ' ' 0 0 0 0 0 0 0 0 0 - - - - - - -
-
-
u
1
0 0
0
0 N
N
cL1
P
0 N
V I m - 4 o ) c D o
0 N
VI
0
w
0
0 P 0
0
w VI
0 P
0
VI
O
V
0
I
V
0
V
I
I
O
0
m
V
0
o I
4
O
Figure 15. Alignment chart for function . a d 6
pressibility factor, P V / R T , a s compared t o a deviation of 0.18% by the method presented here.
T h e equation of s t a t e , 1, contains t h e following three temperature functions:
GRAPHICAL SOLUTION O F EQUATION OF S T A T E
Although the u s e of Equation 1 with Equations 2 through 9 t o predict the compressibility of natural gas is accurate and useful, the use of these equations by direct mathematical computations is very time-consuming. In order to reduce the t i m e required to handle these equations, a s y s tem of graphical solutions h a s been established. First, an alignment chart for each of the Equations 2 through 9 w a s constructed (Figures 1 through 8). The values of the cons t a n t s read from Figures 2 through 8 are final ones, ready for u s e in t h e equation of state, Equation 1, while the value of B o read from Figure 1 i s for the term Z x r B o , in Equation 2 and i s subjected t o adjustment according to Equation 2. 1958
L(t)=
2 TZ
Alignment charts f m these three temperature functions were constructed and are shown in Figures 9, 10, and 11. After the evaluation of the three temperature functions, the equation of s t a t e , 1, reduces t o the form: P = R T d - F ( t ) d a + G(t)d'+aad'+ L(t)d'(l CHEMICAL AND ENGINEERING
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(13)
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Figure 16. Alignment chart for function
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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VOL. 3, NO. 2
F i v e alignment charts for the evaluation of the five individual terms in t h e right-hand s i d e of Equation 13 have been constructed (Figures 12 through 16). T h e procedures which should be followed in using the charts are illustrated by t h e following example. The mole fractional a n a l y s i s for a natural g a s mixture c o n s i s t s of 84.0% of methane, 7.0% of ethane, 1.0% of propane, 1.5% of butanes, 1.0% of pentane and higher hydrocarbons, 1.5% of carbon dioxide, and 4.0% of nitrogen. Find t h e pressure of t h e mixture a t a temperature of 1 2 0 ° F . and a density of 0.3 pound-mole per cubic foot. In solving t h i s problem, Figures 1 through 8 are used to evaluate the eight constants for t h e mixture. In each of Figures 1 through 8, connect 84.0% of methane on No. 1 s c a l e and 7.0% of ethane on No. 2 s c a l e by means of an index line. Connect the intersection o L t h e f i r s t index line on No. 3 s c a l e and 1.0% of propane on No. 4 s c a l e by means of a second index line. Connect the intersection of the second index line on No. 5 scale and 1.5% of butanes on No. 6 s c a l e by means of a third index line. Connect the intersection of the third index line on No. 7 s c a l e and 1.0% of pentane and higher hydrocarbons on No. 8 s c a l e by means of a fourth index line. In the c a s e of pure hydrocarbon mixtures, read the value of the constant on No. 9 s c a l e a t its intersection with the fourth index line. Since the mixture i n this example contains carbon dioxide and nitrogen, connect the points defined by 1.5% of carbon dioxide on No. 10 s c a l e and 4.0% of nitrogen on No. 11 scale by means of a fifth index line. Connect the intersection of the fourth index line on No. 9 s c a l e and t h e intersection of the fifth index line on No. 12 s c a l e by means of a sixth index line. Read the value of the constant on No. 13 s c a l e a t its intersection with the sixth index line. Values of the constants for the mixture a s read from Figures 1 through 8 are a s follows:
= 0.7525 b = 1.095 a = 0.5980
A0 = 8040 a = 4610 Y = 1.777
80
Co = 506.5 X 10‘ c = 9.66 x 10’
Because the mixture contains more than 5% of ethane, the above value for BO should be modified to read, according to Equation 2 Bo = 0.7525
- 16.0188 x 0.007 X 0.84 X 0.07
3:
0.7459
In Figure 9, connect C 0 = 5 0 . 6 5 X 10’ on No. 1 s c a l e and t = 120°F. on No. 2 scale by means of an index line to obtain C o / T ’ = 1505 a t i t s intersection on No. 3 scale. Connect Be= 0.7459 on No. 4 s c a l e and t = 120°F. on No. 5 scale by means of a second index line to get B o R T = 4640 a t its intersection on No. 6 scale. Then F ( t ) = 8040 1505 - 4640 = 4905. In Figure 10, connect b = 1.095 on No. 1 s c a l e and t = 120°F. on No. 2 s c a l e by means of an index line to get W T = 6815 at i t s intersection on No. 3 scale. Then G ( t ) = 6815 4610 = 2205. In Figure 11, connect c = 9.66 x 10’ on No. 1 s c a l e and t = 120°F. on No. 2 scale by means of an index line to read L ( t ) = 2875 a t its intersection on No. 3 scale. In Figure U, connect t = 120°F. on No. 1 s c a l e and d = 0.30 pound-mole per cubic foot on No. 2 s c a l e by means of an index line to read RTd = 1870 a t i t s intersection on No. 3 scale. In F i g w e 13, connect F ( f ) = 4905 on No. 1 scale and d = 0 3 0 pound-mole per cubic foot on No. 2 s c a l e by m e a n s of an index Line to read F ( t ) d * = 440 at i t s intersection on No. 3 scale. In Figure 14, connect G ( t ) = 2205 on No. 1 s c a l e and d = 0.30 pound-mole per cubic foot on No. 2 scale by means of a n index line to read G ( t ) d’ = 59.5 a t its intersection on No. 3 scale. In Figure 15, connect a = 4610 on No. 1 s c a l e and a = 0.5980 on No. 2 s c a l e by means of an index line. Connect the intersection of the first index line on No. 3 s c a l e and d = 0.30 poundmole p e r cubic foot on No. 4 scale by means of a second index line. Read aadd a 2 . 0 1 on No. 5 s c a l e a t i t s intersection with the secund index line. In Figure 16, connect Y = 1.777 on No. 1 s c a l e and d = 0.30 pound-mole per cubic foot on No. 2 s c a l e by means of an index Line to 2btain Y d’= 0.159 a t i t s intersection on No. 3 scale. Connect d = Q30 pound-mole per cubic foot on No. 4 s c a l e and ( 1 + Y d ’ ) = 1.159 on No. 5 s c a l e by means of a second index line. Connect the intersection of the second index line on No. 6 scale and Y d ‘ = 0.159 on No. 7 s c a l e by m e a n s of a third index Line. Connect the intersection of the third index line on No. 8 s c a l e and L ( t ) = 2875 on No. 9 s c a l e by means of a fourth index line. Read L ( t ) d’ (1 Y d ’ ) e- Yd’ = 76 on No. 10 s c a l e a t i t s intersection with the fourth index line.
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Finally, by calculation, P = 1870 440 59.5 2.0 76 = 1567.5 pounds per square inch absolute. The compressibility factor may b e evaluated from P / R T d . In this example, R T d = 1870 and P = 1567.5. Hence, the compressibility factor is 0.8382.
T h e Eenedict-Webb-Rubin equation of state gives the pressure explicitly. Solving the equation for density for given values of pressure and temperature involves a trialand-error process. In the graphical solution only Figures 12 through 16 need b e repeatedly used in the trial process. ACKNOWLEDGMENT
The authors wish to express their sincere appteciation t o Samuel R. Beitler, Mechanical Engineering Department, The Ohio State University, for his guidance and criticism throughout the preparation of this work. They are greatly indebted, also, t o Salvatore M. Marco, chairman of the Department of Mechanical Engineering, and to Webster B. Kay, Chemical Engineering Department, T h e Ohio State University, who have critically read the manuscript and offered many valuable suggestions. NOMENCLATURE
Ao, BO,CO, a, b, c, a, Y = constants in equation of s t a t e m English units a s defined in Table I F ( t ) , G ( 0 , L ( t ) =temperature functions as defined by Fquations 10, 11, and 12, respectively P = absolute pressure, pounds per square inch R = 10.7335 (pounds) (cubic foot)/(square inch) (pound-mole) (degree Rankine) T = absolute temperature, degrees Rankine V = molal volume, cubic feet per pound-mole d = molal density, pound-moles per cubic foot e = b a s e of natural logarithms x = mole fraction of compoqent Subcripts
E = ethane M = methane N = nitrogen i = fth component L I T E RATU RE C I T E D (1) American Gas Association, “Supercompressibility Factols for Natural Gas,” v o l a I through VI, June 1955. (2) Beattie, J. A., Hadlock, C., Poffenberger, N., J. Chen. Phys. 3, 93 (1935). (3) Beattie, J. A., Su, G. J., Simard, G. L., J. Am. Chem. SOC. 61, 926(1939). (4) Benedict, M., Webb, G. E,Rubin, L. C., Chem. Eng. Pmgr. 47, 419 (1951). (5) Benedict, U, Webb, G. R , Rubin, L. C., 1. Chan. Phys. 8, 334 (1940). (6) Bloomer, 0. T., Ph.D. thesis, Illinois Institute of Technology, Chicago, IlL, 1953. (7) Bloomer, 0. T., Gami, D. C., Parent, J. D., Research Bull. 22, Institute of Gas Technology, Chicago, Ill., 1953. (8) Bloomer, 0. T., Rao, K. N., Ibid., 18, 1952. (9) California Natural Gasoline Association, Los Angeles, Calif., Bull. TS-354 (1936). (10) Ibid., TS-461 (1947). (11) Cullen, E. J., Kobe, K. A., A.Z. Ch. E. Journal 1, 452 (1955). (12) Dunkle, R V., Gas p. 41 (October, 1944). (13) Kay, W. B., Ind. Eng. Chem. 28, 1014 (1936). (14) Michels, A., Michels, C., Proc. Roy. SOC. London A-153, 201 (1935). (15) Natural Gasoline Association of America, Tulsa, Okla., Standard 4142, 1942. (16) Zimmennan, R. H., Beitler, S. R., Gas 28, 129 (October 1952). (17) Zimmennan, R. H., Beitler, S. R., Trans. Am. SOC. Mech. Engrs. 74, 945 (1952).
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Received for review April 1, 1957.
Accepted December 18, 1957.
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