Benedict Equation of State APPLICATION TO METHANE, ETHANE, n-BUTANE, AND IIPBENTANE J. B. OPFELL, W. 6. SCHLINGEW, ANDB.H. S l G E California Institute of Technology, Pasadena. Calif.
T
HE equation of state developed by Benedict, Webb, and Rubin (9, 10) is a satisfactory relation to describe the volu-
metric behavior of the lighter hydrocarbons. Except for methane, the early work of Benedict and coworkers was limited to a maximum pressure of approximately 5000 pounds per square inch. The extension of the range of pressures of active industrial interest has made it desirable to explore the accuracy of the application of this equation of state a t high pressures. In addition, its suitability for describing the behavior in the liquid phase is of inteiest. Selleck and cowoikers ( 2 1 ) extended the application of this equation for propane to pressures of 10,000 pounds per square inch in the temperature interval between 100' and 460" F. They included a number of sets of coefficients and recommended a set which yielded an average standard error of estimate for the compressibility factor from the molal volume and temperature of 0.0094, treating pressure as a dependent variable. (Subsequently the abbreviation "standard error of estimate" is used in place of "standard error of estimate for the compressibility factor from the molal volume and temperature.") This work illustrated the feasibility of extending the application of the Benedict equation to much higher pressures than originally used by Benedict, but also showed that the equation was not well suited to extrapolation beyond the range of conditions for which experimental data were employed in evaluating the coefficients. Brough (12) proposed a statistical method for the evaluation of the coefficients of this equation which followed the original proposals of Benedict (6). Matrix techniques ( I S ) used in conjunction with available automatic digital computing equipment serve as an effective means of establishing the coefficients of the Benedict equation of state. Coefficients for the lighter hydrocarbons published by Benedict and coworkers (7-11) describe with accuracy the volumetric behavior of the gas phase and the vapor pressure of the pure substances, and for mixtures of the thermodynamic properties of the gas phase and the composition of the coexisting phases. These coefficients are of primary utility at pressures be lo^ 4000 pounds per square inch. It is the purpose of the present discussion to present values for each of the coefficients of the Benedict equation for methane, ethane, nbutane, and n-pentane which describe the volumetric behavior of the liquid and the gas phases for pressures up to 10,000 pounds per square inch in the temperature interval between 100' and 460' F. These coefficients do not describe the vapor pressure as well as the coefficients proposed by Benedict. For present purposes it is convenient to write the Benedict equation in the following forms:
The expression for the compressibility factor is convenient in the zpplication of the equation to gases, whereas Equation 2 permits an iterative solution for volume at a specified pressure and ternpcrature. The methods proposed by Brough ( 1 2 )were extended in a more recent application of the equation of state to propane ( 2 1 ) and n-cre employed in the present evaluation of the coefficients for a number of the lighter hydrocarbons. Thc details of the methods employed in these studies arc available (21, 2 2 ) and no discussion of the extended calculations required in order to obtain the results presented is included here. In each case iavestigations were made of the agreement of the equation of state with the experimental data for several different values of the nonlinear coefficient y shown in Equations 1 and 2. I t was necessary to determine the optimum value of this coefficient by trial (21). For each chosen value of y , the corresponding values of each of t,he other coefficients were obtained by least squares techniques ( 2 1 , 2 2 ) . In addition, the etandard error of estimate of the equation of state was computed. From these calculations the set of coefficients yielding the smallest standard error of estimate was chosen by inspection. METHANE
Methane has been experimentally investigated in detail and at present there is an abundance of data for predicting the appropriate coefficients. The measurements of Michels and Kedcrbragt (15) represent an extended study of this compound, which was supplcmented by more recent measurements ( 1 7 ) a t pressures up to 10,000 pounds per square inch in the temperature interval between 100' and 460' F. These two sets of data in turn were supplemented at lower temperatures by the measurements of Kvalnes and Gaddy ( 1 4 ) . Figure 1 depicts the experimental data employed in establishing the coefficients. The envelope of the region in which Benedict fitted the coefficients of the equation is indicated by the hachured lines. The points corresponding to the three sets of data employed have been differentiated in the figure. Figure 2 illustrates the variation in the standard error of estimate with values of y. A minimum standard error of estimate for the data recorded in Figure 1 was obtained with a value of y of 1.2. The values of the coeffirients corresponding to a y of 1.2 are recorded in a part of Table I. For temperatures from -100' to 460" F. and for pressures up to 10,000 pounds per square inch the standard error of estimate is 0.0156. Figure 3 shoas the differrnce between the computed and experimental values of the compressibility factor for a number of the states depicted in Figure 1. The figure presents the information both with volume and with preesure as the dependent variable. The variation in the distribution of the differences is evident. In a part of Table I is recorded the root-mean-square deviation, calculated from the deviations shown in Figure 3, with pressure and temperature as the independent variables. These values are not greatly different from the root-mean-square deviation obtained with volume and temperature as the independent variables. It may be seen from Figure 3 that the disper1286
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1954
0 MICHELS A N D NEDERBRAGT(l3)
p
study. The data from each of the investigations were given the same weight and no smoothing of the experimental measurements was involved. From the data shown in Figure 4 several sets of values for the coefficients of the Benedict equation as found in Equations 1 and 2 were determined by statistical methods already described (21). The r lationship of the standard error of estimate to the coefficient y is depicted in Figure 2. The minimum standard error of estimate corresponds to a value of y of approximately 2.5 and the coefficients recorded for ethane in a part of Table I correspond to this value. These coefficients yielded a standard error of estimate of 0.0138, which is slightly less than the standard error of estimate found for methane. However, the ranges of pressures and temperatures were somewhat smaller, and it is not surprising that the equation permitted a better description of the behavior of this compound.
-
REAMER SAGE A N 0 L A C E Y ( I 6 ) A N ~G A D (14) ~ ~ ~
0-OLOS
KVAL'NES
1287
0
ZOO
0
100
300
200
TEMPEUTURE
400
Of
Figure 1. Experimental Data Used i n Evaluating Coefficients for Methane sion, especially a t the higher pressures, is smaller when molal volume is used as the dependent variable than when pressure is so used.
:-I $
1 1 1 1 1 2.5
5.0
7
(cu.
7.5 10.0 FT P E R LB.YOLE)*
J
12.5
Figure 2. Dependence of Standard Error of Estimate upon Y TABLE
I.
COEFFICIEXTS FOR EQU.4TIOh' PREDICTIONa
FOR
VOLUMETRIC
Coefficient
Methane Ethane Propane n-Butane n-Pentane R 10,73147b 10.73147 10.73147 10.73147 10.73147 7001 .40 15913.1 22784,O 60215.5 A0 4910.53 0.237507 0.560703 1.00517 BO 0.455158 3,69003 334,26 15491.4 6315.35 19628.9 x 10-8 448,753 26547. 9 67141.1 a 4551.18 253507 0 203941 . O 3.43107 7.41650 19.7116 b 1 ,03508 16.0875 6476.86 19262.1 93708.5 C X 10-6 619,147 116061 .O 2.39047 U 0.332260 0,742830 1 ,60300 6.67703 2.50000 4.24021 6 40000 1 .20000 IO. 5000 44.094 16.042 30,068 72.146 58.120 0.0094 Standard error of 0 . 0 1 5 0 0.0138 0.0258 0.0166 estimate 88 0.0065 0,0165 . 0.0571 0.136 av 0.0053 0.0126 . . .. 0.0015 0.0103 a These coefficients are recommended only for prediction of the behavior in the homogeneous regions. b T h e values recorded are dimensionally consistent when used in t h e equation of state wit% pressure expressed in pounds per square inch, temperature in R., and volume in cubic feet per pound mole.
c.
L
.. .
ETHANE
The volumetric behavior of ethane has been investigated in detail. For the present study the measurements of Beattie and coworkers (1, 5 ) were employed along with data ( 1 8 ) which included measurements a t higher pressures. The experimental data of Beattie extended from 77' to 527' F. for pressures up to 5000 pounds per square inch. The more recent data (18) were limited t o temperatures between 80" and 460' F. for pressures up t o 10,000 pounds per square inch. Figure 4 depicts the number and distribution of experimental points used in t,his
The difference between the compressibility factor as predicted by thc equation of state and as determined by experiment is shown for a number of states in Figure 5 . In this instance there is a greater difference between the predicted and actual behavior with prrssure as the dependent variable than with molal volume. At the low temperatures and high pressures the use of pressure as the dependent variable places an unusually severe requirement upon the equation of state, since a small uncertainty in the molal volume results in an unusually large change in the predicted equilibrium pressure. At the higher presurcs, thc cocffirients of Table I yield a more accurate description of the volumetiic behavior of ethane than was obtained from the oi iginal coc?fficients suggested by Benedict. Such a difference is not unexpected, as the coefficients suggested by Rcnedict ( 7 ) wcre bawd upon expcrimental data extending only to 5000 pounds per square inch It has becn found (21) that the Benedict equation does not rxtrapolate accurately the behavior of pure substances to statcs beyond the field of experiment. PROPANE
For the convenience of the reader the coefficients for propane developed earlier ( 2 1 ) are included in Table I. These data yield a standard error of estimate of 0.0094, which is smaller than that found for any of the other compounds so far investigated. No graphical presentation of the behavior of propane with the coefficients listed in Table I is given, as such information is already available (21).
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 46, No. 8
n-BUT4NE
0.01 5 ~
FiYF4T 0
TEMPERATURE F I ooo 4Q0°
0
0.010
r
0
N
0.005
z
0 9
b
o
2,
n
-0.005
8
-0.010
I 5000 2. 0 PRESURE,
Figure 3.
l
l
9500
10,000
The volumetric behavior of n-butane has been studied in sufficient detail to permit the coefficients of the Bencdict equat,ion to be established with some accuracy. The experimental background (4,17, 20) employed t o evaluate the coefficients is shown in Figure 6. The three sets of experimental data involved are indicated on this diagram. The data from the two sources were Lyeighted equally. The measurements of Beattie and coworkere (4)n-ere used along with measurements a t higher pressures (19) in order to establish the coefficients in the same fashion as has been employed for t,he lighter hydrocarbons. The hachured lines enclose the states employed by Benedict. I n t,liis instance the normal matrix of the least, squares niet,hod was somewhat ill conditioned ( 1 4 ) and it was not possible to obtain precise values for the standard errors of estimate. Figure 2 presents the effect of y upon the standard error of estimate. A value of y of 6.4 corresponded closely to the minimum value of the st,andard error of estimate. Figure 7 presents the differences between the experimental and predicted compressibility fa,ct,or for a number of the states investigated. The influence of the choice of dependent variable upon the agreement of the equation of state wit,h the experimenta,l data is pronounced in this instance. Throughout the liquid region th? deviations Yith pressure are much larger than with molal volume as the dependent, variable. The standard error of estimate as determined from the data of Figure 6 is reported in a part of Table I and xas 0.0166, or ainiost twice the value for propane.
Le. PER S0.IN.
Deviations of Compressibility Factor for Jlethane
n-PENTANE
The application of the Benedict equation to n-pentane a t pressures up to 10,000 pounds per square inch involves a large area upon the pressure-temperature plane that lies within the liquid
l0,000
0.01
5000
0
-0.01 f
200c
2 U z 9
d
'u-0.02
W
I
O F
z
B.
3
!0-0.03 -
u)
4
e
l . l
U
a
I
N
loot
50C
2,- z = P( ye - ~ ) / R T
0
2,-
Z -(P,
DEPENDENT VARIABLE
~
0
P
Y
0
0
- P) Y / R T TEMPoERATURE F I ooo
0-e-
-0.04
20c
B)
4ooo
-0.05
o.REAMER'OLDS SAGE IO(
8 lo0
AND LACEY (18) R E A M E R ' OLD; SAG;, A N D LACEY (18) BEATTIE' HADLbCK. A N 0 POFFENBERGER(I
200
300
TEMPERATURE
Figure 4.
400
I
2500
500
PRESSURE,
OF
Experimental Data Used in Evaluating Coefficients for Ethane
5000
Figure 5.
I
I
7500 l0,OOO LO.. PER SQ. IN.
Deviations of Compressibility Factor for Ethane
1289
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1954
region. Figure 8 shows the distribution of the experimental data (2, 19, 90) which were employed. The computing work was carried out before the recent measurements of Beattie and coaorkers (5)became available. However, as good agreement was experienced among the three sources, it does not appear that the coefficients x-ould be modified significantly by the inclusion of the more recent data. The minimum standard error of estimate of 0.0258 was found for a value of y of 10.5, which represents the largest value of this coefficient so far employed for the Benedict equation when applied to high pressures. The large value of this standard error of estimate reflects the preponderance of experimental points falling in the liquid region. The hachured line encloses the region of temperature and pressure uscd by Benedict in predicting coefficients for this compound. A number of values of the molal volume were computed iteratively for the experimental values of pressure and temperature. Likewise, values of the pressure corresponding to the experimental values of molal volume and temperature were computed from the coefficients for n-pentane recorded in Table I. Deviation of the predicted and experimental compressibility factor is recorded in Figure 9 as a function of state. The deviations with pressure as the dependent variable are extremely large a t the low temperatures and high pressures. Corresponding root-mean-square errors with volume and with pressure as the dependent variable are recorded in a part of Tabltx I.
the terms of Equations 1 and 2, especially for small values of ?, Figures 12 and 13 show the compressibility factor for npentane predicted from the coefficients of Table I and from those proposed by Benedict ( 7 , 8, 11). Only data for the liquid phase are involved a t 100" F. Smoothed experimental data have been shown for comp'arison with the predictions of the equation of state.
a
0.02
0.050
0.01
0.025
\
\
>e
?* >".
-
0.
I
0
0
e'
1 I
a
N
N I
1
0
N
N
-0.01
-0.025
z
0
+-
0
5
2n
->
-__
-0.02
.0.050
IEPENDEI JARIABLE
OF
:L
100'
CO\lPARISON OF COEFFICIENTS
400'
0.
0.03
The effect of the molecular weight of the compound upon each of the coefficients is presented in Figures 10 and 11. For most of the coefficients there is a rapid increase with molecular weight. The exception in the case of 13, between methane and ethane is not understood. It is desirable t o carry the same number of significant figures for all coefficients because rather small differences between large numbers are encountered in summing u p
h u
.0.075
I
5 3 PRESSURE,
2500
Figure 7.
7500
LB. PER
I
IOPOO SO.IN.
Deviations of Compressibility Factor for n-Butane
10,000
SO00
2000
m
i
200 W'
8: 3
100
VI W
c--0-
n
a
c-05c
20
10
5
0-0-
o-c-cc-c-c-
0.0-
0-0-c-
0-0-
c-c-c-
c-0-
c-c-c-
0-e
0-0-0-
0 C-0
100
'
BEATTIE, S I M A R D , A N D S U ( 4 ) S A G E A N D L A C E Y (20) OLOS, R E A M E R , SAGE, A N D L A , C E Y (17)
200
300
TEMPERATURE
Figure 6.
400
500
10
5
100
OF.
Experimental Data Used in Evaluating Coefficients for n-Butane
200
300
TEMPERATURE
Figure 8.
400
500
OF
Experimental Data Used in Evaluating Coefficients for n-Pentane
*-INDUSTRIAL AND ENGINEERING CHEMISTRY
1290
Vol. 46, No. 6
I I Ze-Z:P(ye-Y)/RT
o z,
0.02
- z = ( - P) !/R
T
0 2
DEPENDENT
TEMPERATURE
VA R l ABLE
10,000
2
0.01
\ h
>.
0 1
I
,
I
>a:
0.
00
v
a
0
R
I
I
I
r4
\
>.
a
8
0
e-
0
c
a2
, I &
L
IOOC
W Y
U
1
N "
-0.01
z 0
-0 I
0
t-
5 >
I-
4
2
N I
N "
>
2
-0.2 0"
-0.02
o-
-0.3
-0.03
w
?3 I
c
IOC
z 4
I
IC
0-
0-
PRESSURE,
k
LB. PER SO.IN.
Figure 9. Deviations of Compressibility Factor for n-Pentane
--
_-____
1 40 UOLECULAR
Figure 11.
10,000
I 20
I
eo
I
WEIGHT
Effect of IIlolecular Weight upon 1alt:ec of Coefficients Proposed hj Benedict
pressures, excellent agreement was obtained FTith both sets of coefficients. For this reason it is recommended that the coefficients of Table I not be used outside the range of pressures and F---*--+ tc'niperatures discussed here. The coefficients suggested by Benedict ( 7 ) are suitable for the prediction of vapor pressures of the several compounds in question. The restraints imposed upon the coefficients in Table I t o describe the volumetric behavior of the liquid and to extend their applicability to higher pressures do not permit them to be used a s effectively as Benedict's in predicting vapor pressure. In addition, t,hey should not be applied (8, 11) with existing interaction coefficients to the prediction of the phase behavior of mixtures. X o adjustment of these volumetric coefficients n-as made to adapt them t o the prediction of phase behavior. ACKNOW LEDSMENT
This paper is a contribution from American Petroleum ltcsearch Project 37 located at the California Institute of Technology. F. T. Selleck aided with the numerical calculations. W. S. Lacey reviewed the manuscript, and Olgs Strandvold assisted in its preparat,ion. NOMENCLATURE
MOLECULAR
WEIGHT
Figure 10. Effect of Molecular Weight upon Magnitude
of Coefficients of Table I These figures indicate that a sct of coefficients for the Benedict equation should not be relied upon to predict behavior accurately beyond the ranges of temperature and pressure covered by the experimental data employed in determining these coefficients. At 400" F. good agreement was obtained with both sets of coefficients a t pressures up t o 5000 pounds per square inch. HOTever, at 100' F. significant departures were experienced, particularly with pressure as the dependent variable. In the case of methane, for which Benedict (9) fitted his coefficients at the higher
do.Bo,C,,u, h, e, 01, y = coefficients for the Benedict equation of state exp ( ) = exponential function SI = molecular weight = pressure, lb. per sq. inch absolute P = universal gas constant, (Ib./sq. inchj(cu. f t . ) per (Ib. R mole) ( ' R.) = root-mean-square deviation of experimental comSP pressibility factor from the predicted isotherm when the compressibility factor is computed from the molal volume and the temperature = root-mean-square deviation of experimental ?omSa pressibility factor from the predicted isotherm when the compressibility factor is computed from the pressure and the temperature T = absolut'e temperature, R. = molal volume, cu. feet per Ib. mole Y = compressibility factor z
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1954
Pnissuw.
LB. P i a 50
1291
IY
Figure 12. Predicted and Experimental Compressibility Factor for n-Pentane with Molal Volume as the Dependent Variable Subscript e = value of property calculated using the Benedict equation of state LITERATURE CITED (1) Beattie, J. A,, Hadlock, C., and Poffenberger, N., J . Chem. Phys., 3, 93 (1935). (2) Beattie, J. A., Levine, S. W., and Douslin, D. R., J . Am. Chem. Sac., 73, 4431 (1951). (3) I b i d . , 74,4778 (1952). (4) Beattie, J. A,, Simard, G. L., and Su, G.-J., Ibid., 61,26 (1939). (5) Ibid., p. 924. (6) Benedict, M., Ibid., 59, 2224 (1937). (7) Benedict, M., Webb, G. B., and Rubin, L. C., Chem. Eng. Progr., 47,419 (1951). ( 8 ) Ibid., p. 449. (9) Benedict, M., Webb, G. B., and Rubin, L. C., J . Chem. P h y s , 8 , 334 (1940). (10) Ibid., 10, 747 (1942). (11) Benedict, M., Webb, G. B., Rubin, L. C., and Friend, L., Chem. Eng. Progr., 47,571 (1951). (12) Brough, H. W., Schlinger, W. G., and Sage, B. H., IND. ENG. CHEM.,43, 2442 (1951).
2ow
4000 PICSURE.
eo00
LB P C R sa
eo00 IN
Figure 13. Predicted and Experimental Compressibility Factor for n-Pentane with Pressure as the Dependent Variable (13) Hartree. D. R., “Numerical Analysis,” London, Oxford University Press, 1952. (14) Kvalnes, M., and Gaddy, V. L., J . Am. Chem. Sac., 53, 394 (1931). (15) Michels, A,, and Nederbragt, G. W., Physiea, 3, 569 (1936). (16) Olds, R. H., Reamer, H. H., Sage, B. H., and Lacey, W. S . , IND.ENG.CHEM.,35, 922 (1943). (17) Ibid., 36, 282 (1944). (18) Reamer, H. H., Olds, R. H., Sage, B. H , and Lacey, W. N., Ibid., 36, 956 (1944). (19) Sage, B. H., and Lacey, W. N., Ibid., 34,730 (1942). (20) Sage, B. H., and Lacey, W . N., “Thermodynamic Properties of the Lighter Hydrocarbons and Nitrogen,” New York, American Petroleum Institute, 1950. (21) Selleck, F. T., Opfell, J. B., and Sage, B. H., IND.ENG.CHEM., 45, 1350 (1953). (22) Selleck, F. T., and Sage, B. H., Washington, D. C., Am. Documentation Inst., Doc. 3914 (1953). REChIVED
for review January 14, 1954.
ACCEPTEDFebruary 3, 1954
Residual Viscosity of Paint Systems at Infinite Shear Velocity J
W. K. ASBECK‘ AND M. VAN LOO The Sherwin- Williams Go., 115th St. and Cottage Grove Ave., Chirago, Ill.
T
HE critical pigment volume concentration of a paint ( 4 )
depends on the dispersive capacity of the vehicle used. Application qualities such as relative ease of brushing and flow also vary according t o the solid/solid and solid/liquid interactions occurring in the paint. Such relationships have been the subject of numerous rheological studies. These studies have been carried out almost exclusively a t low and intermediate shear velocities. Few high shear velocity measurements above 1000 reciprocal seconds (1000 sec. -1) have been attempted, apparently chiefly because of the lack of suitable instrumentation. 1
Present address, Carbide & Carbon Chemicals Co., 5. Charleston, W. Va.
BAND VISCOMETER
The band viscometer has been described elsewhere ( 1 1 , 1 4 ) . It is a simple high shear apparatus in which a thin flexible band is pulled between two thermostatted jaws, carrying paint with i t from a small well in the jaws. The apparent viscosity may be calculated from the equation
dK = 2WHV
where q is the viscosity at the given shear velocity, d is the clearance between the jaws, K is the wei h t pulling the band, W is the width of the band, H is the heigft of the active surface of
