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An equation of state for methane in the form of Bender's equation for temperatures between 91 K and 625 K and pressures up to 500 bar
Fluid Phase Equilibria, 5 (1980) 35-54 Elsevier Scientific Publishing Company,
35 Amsterdam
- Printed in The Netherlands
AN EQUATION OF STATE FOR METHANE IN THE FORM OF BENDER’S EQUATION FOR TEMPERATURES BETWEEN 91 K AND 625 K AND PRESSURES UP TO 500 BAR
U. SIEVERS
and S. SCHULZ
Lehrstuhl Thermodynamik, 50 (F.R. G.)
Universitiit
(Received December 26th, 1979;
Dortmund,
accepted
Postfach
500 500, D 4600
Dortmund
in revised form April 4th, 1980)
ABSTRACT Sievers, U. and Schulz, S., 1980. An equation of state for methane in the form of Bender’s equation for temperatures between 91 K and 625 K and pressures up to 500 bar. Fluid Phase Equilibria, 5: 35-54. The coefficients of Bender’s equation of state for pure fluid methane are refitted using recent, precise experimental data. This equation of state represents the thermodynamic properties of methane in the fluid region for temperatures between 90.68 K and 625 K, densities up to 28.6 mole dmm3 and pressures up to 500 bar with high accuracy. Only in the critical region is there a slightly lower accuracy. Compared with the results obtained with Bender’s original coefficients, improvements are achieved in wide regions, especially for the calculation of vapour--liquid phase equilibria and liquid properties.
INTRODUCTION
For the calculation and optimization of processes in the natural gas industry the thermodynamic properties of fluid mixtures of simple, non polar components must be known over a wide range of temperatures and pressures. The Benedict-Webb-Rubin (BWR) equation (Benedict et al., 1940) is used in different modifications for this purpose. All thermodynamic properties can be evaluated in combination with the temperature function of the isochoric heat capacity of the pure components in the ideal gas state. As the properties of the ideal gas can be calchlated highly accurately, using the statistical thermodynamical and quantum mechanical methods, the main problem is to find a good equation of state for the mixture. The quality of this equation depends on the accuracy of the equations of state for the pure components on the one hand and the mixing rules used in order to obtain the coefficients of the mixture’s equation from those of the pure fluids, on the other. This is discussed in detail by Sievers (1980). The purpose of this paper is to give a good equation of state for pure fluid methane, the main component of natural gas, which can 0378-3812/80/0000-0000/$02.60
01980
Elsevier Scientific Publishing Company
36
serve as a basis for an equation of state for fluid mixtures with methane as one component. EQUATION
OF STATE
AND
DETERMINATION
OF THE COEFFICIENTS
Bender (1971b) developed a modified BWR-type of equation of state with 20 coefficients for methane. McCarty (1974) refitted the coefficients of this equation and also determined the 33 coefficients of the Jacobsen equation (Jacobsen, 1972) for methane. In 1978 the more precise IUPAC equation for methane (Angus et al., 1978), a Jacobsen equation with 33 coefficients too, was published. When applying it to mixtures, the equation of state for the pure fluids should be of high accuracy but have a small number of coefficients, needing only a few mixing rules. Bender’s equation with its 20 coefficients is a good compromise between these two demands. It is successfully used for pure fluid argon, nitrogen, oxygen (Bender, 1971a), hydrogen (Bender, 1975a), ethylene and propylene (Bender, 1975b), ethane, propane, n-butane and n-pentane (Teja and Singh, 1977). The equation of state is p = R,Tp,
+ Bps
+ Cp& + Dp;
+ (G + HP&,) p%ew(--a~
+ Ep% + Fp$ (1)
p&l
where p and T are pressure and temperature, pm is the molar density and R, the universal gas constant. B, C, D, E, F, G and H are the following temperature functions for the pure fluids. B = al T -
a2 -
asIT -
ad/T2 -
as/T3
C = a6T + a7 + as/T D=aaT+alo E = alIT + al2
(2)
F = al3 G = alhIT2 + aI5 /T3 + als/p H=
alTIp
+ aI8 IT3 + a19/p
The coefficients a, to a2o are determined for each pure fluid by fitting the equation to experimental thermodynamic data. Bender (1971b) did this for methane, using data published up until 1969. Because of the great interest in the thermodynamic properties of natural gas, further precise measurements for its principal component methane have been published since then. An extensive experimental programme on the thermophysical properties of methane in the dense gas and liquid region was carried out by the National Bureau of Standards, Cryogenic Research Division (Goodwin (1974), Younglove (1974), Diller (1976), Haynes and Hiza (1977)). These and other new experimental results for methane required a refit of the coefficients a1 to a2o in
eqns. (1) and (2). Eubank (1972) came to the same conclusion after an inspection of the experimental data for methane. At the time when the coefficients of the IUPAC equation for methane were determined, there were no orthobaric densities thermodynamically consistent with density measurements of the homogenous region available (Angus et al., 1976). So the authors calculated the orthobaric densities from provisional equations of state for the gaseous and the liquid region by extrapolating the isochors to the vapour pressure. Meanwhile orthobaric densities, thermodynamically consistent with the densities of the homogeneous region, have been published (see Sievers (1980) for a detailed discussion). This improved knowledge of orthobaric densities, and the fact that there are further precise measurements for methane which have not been used in fitting the coefficients of the IUPAC equation, justify the refitting of the coefficients of Bender’s equation for methane even compared with the IUPAC equation. The new coefficients of the equation of state (1) and (2) are determined by a simultaneous, weighted least squares fit to selected experimental p&T data and second virial coefficients, to the thermodynamic conditions of equilibrium for the co-existing liquid and vapour phases and to the thermodaynamic conditions at the critical point. CvmpmT data, reflecting the derivation of the equation of state and commonly used in fitting the coeffi(a 2Pl@9,m cients (McCarty, 1974; Angus et al., 1978), have not been included in the fit in order to prevent the influence of another equation of state on the new coefficients, because the densities in these data for methane (Younglove, 1974) had been calculated using an equation of state by Goodwin and Prydz (1972). According to Schulz (1973) an equation of state is more accurate if the coefficients are fitted to ppmT data rather than to measurements of enthalpies or heat capacities. The sum of squares S of weighted residuals, eqn. (3), is minimized by using the techniques described by Hust and McCarty (1967) and Bender (1971a). J
~~~ilP(T~~,i~~mcl~.~)-~~~~,~l~+~~~~ly(Br(Tr.~.~)-~~~~~,jI*
S=
i=l
+
~c~b(Tc..., ~mc exp)-PC cd2
(3) The residuals are formed by the differences between the values calculated
38
from the equation of state with the experimental temperature TexP and the experimental density pm exp and the experimental pressure pexp, second virial coefficient B, exp, vapour pressure ps _,, and critical pressure pC _,, respectively. Each square of residual is individually weighted by,a factor w. Selected ppmT data of the gaseous and liquid regions and of the vapourliquid phase boundary are considered in the first sum. The number of these data is 1. The second sum includes J second virial coefficients. They are calculated from the equation of state by Bv = (al -
azlT -
a3/T2 -
ad/T3 -
a51T4)lR,
(4)
The thermodynamic conditions for vapour-liquid phase equilibrium are incorporated in the first and the third sum. The equality of the pressures for the saturated liquid and the saturated vapour at the same temperature is found in the first sum, as pp,,.,T data of the co-existing phases are also included. The criterion of equal fugacities for co-existing liquid and vapour phases is given in the third sum by the so called Maxwell-criterion, which is solved for the vapour pressure. Properties of the liquid phase are indicated by superscript L, those of the vapour phase by superscript V. The number of vapourliquid phase equilibrium data is L. The last three terms in eqn. (3) reflect the conditions at the critical point. As the critical density cannot be measured very accurately, the equation of state is not constrained to fulfil these conditions exactly. They are made part of the sum of squares like all the other data, according to a suggestion of Ahrendts (1977), and weighted by w,i, w,a and Wc3.
In an extensive study, all the available experimental thermodynamic data of methane have been thoroughly examined and reliable data have been selected for the determination of the coefficients (Sievers, 1980). All values were converted to SI units, temperatures corrected to the temperature scale ITPS-68. The data selected for fitting the coefficients of the equation of state are given in Table 1. Vapour pressure ps and orthobaric densities pk and pz must be known at the same temperature for the Maxwell-criterion. As these properties are measured independently as a function of temperature, data are not usually given at the same temperatures. Therefore correlations for the saturation properties are needed in order to obtain values at the same temperatures and also to smooth out experimental errors. The correlations of Good-win (1974) have been tested for this purpose (Sievers, 1980) and are used in this paper to represent the vapour pressure ps and the orthobaric densities pk and pz. Measurements with small experimental errors should have a higher influence on the determination of the new coefficients than those with greater experimental uncertainty. For the ppmT data this is achieved by choosing the weight Wpi proportional to the reciprocal absolute error of the i-th ppmT data point, which is calculated according to eqn. (5) from the mean absolute errors Ap, Apm and AT of the individual measurements, taken from the
39
TABLE
1
Selected thermodynamic data of methane * for fitting the coefficients state(1) and (2). N is the number of data points. Kind of data
Author
Douslin et al. (1964) Pope (1972) Roe (1972) Rodosevich and Miller (1973) Goodwin (1974) Goodwin (1974) (corr. V) Goodwin (1974) (corr. L) Douslin et al. (1964) Pope (1972) Roe (1972) Goodwin (1974) (corr.)
PP~T P&T
PPItlT PP~T PP T ‘0 PIP T ft &pm T & T % T & T PSP~m pVT In
of the equation of
Range
N
N-9
p(bar)
273-623 126-191 155-291 91-115 92-300 92-186 92-188 273-623 126-159 168-263 92-188
16-405 l47 3-100 o-2 3-351 o- 40 0- 42
0-
42
319 138 82 11 * 553 48 49 15 4 8 49
* A computer listing of the selected thermodynamic data for methane is available from the authors on request. The distribution of the selected ppmT data is shownby Sievers (1980) in a p-pm and a pT diagram.
authors
if possible
or estimated.
To start with, the derivatives @p/a&r
and @p/a Z&, are obtained from the determined from eqn. (3) with all weights set to 1.0, and corrected iteratively in the following steps with the coefficients of the preceding step until good results are achieved. Measurements of different authors and regions of the ppmT surface may be weighted by woP in a different way. The weights wei and wsI are calculated analogously. For a good representation of the liquid properties and the vapourliquid phase equilibria the weights w,,~ for the liquid region and w,, must be given large values. Empirical correlations for these weights, the absolute errors used in eqn. (5) and further details of the determination of the coefficients are given by Sievers (1980). The influence of different weights on the results of the equation of state is discussed by Koschowitz (1979). The properties of the critical point are chosen to be T, exp = 190.555 K, p. sxg = 45.988 bar and prnc exP = 10.1095 mole dm-s (Sievers, 1980) and the non-linear parameter azo according to Bender’s empirical correlation equation of state (1) and (2) with coefficients
asOAexp
= 1
The new coefficients for the equation of state (1) and (2) for methane are given in Table 2.
TABLE
2
Coefficients ci of the equation of state (1) and (2) for methane and the region of validity of the equation. The equation of state with the tabulated values for (I[ and the values of temperatures and density in the units K and mole dme3 respectively gives the pressure in J dme3. (Conversion of pressure units: 100 J dmw3 = 100 kPa = 1 bar)
%I a1 a2 a3 a4 a5 '"6 a7 aI3 =9 al0 =11 (112 013 =14 a15 al6 =17 a18 a19 a20
M
Tmin T
J mole1 K-l J dm3 moleV2 K-l J dm3 molew2 J K dm3 molee2 J K2 dm3 molem2 J K3 dm3 molee2 J dm6 molee3 K-l J dms moleV3 J K dm6 moleF3 J dmg moleV4 K-l J dmg molew4 J dm12 molem5 K-l J dm12 molep5 J drn15 molew6 J K2 dm6 molee3 J K3 dm6 molem3 J K4 dm6 moleM3 J K2 dml* molem5 J K3 dm12 molew5 J K4 drn12 molee5 dm6 molee2 e mole-l
I<
p:zbKr
pinax mole dmM3
Pmax / Pmc
TESTING
THE NEW EQUATION
8.3143 0.4531925285 0.2358287986 -0.7173842009 0.296.1509482 -0.8735505087 0.1052328359 -0.1694738647 0.8637907281 0.8533763036 -0.4017208435 0.3045691216 -0.2563358654 0.8484753417 0.2545831403 -0.5522243593 0.2250857179 -0.3979182974 0.1268073106 -0.5298037215 0.9784545300 16.0430 90.68 623 500 28.6 2.83
x lo3 x lo4 x 10’ x
10’
x 10-l x lo1 x lo3 x 1O-3 x 1O-2 x 1O-4 x 10-l x 1O-3 x lo6 x 108 x lOlo x 104 x lo7 x lo* x 10-g
OF STATE
For testing the accuracy of the new equation of state it is at first compared with experimental ppmT data of the homogeneous gaseous and liquid regions. The relative deviations between the experimental and the calculated pressures A~rel = (Pexp -
P)/P~
(7)
and the experimental and calculated densities AP m rel = (P, exp - PrnYPInexp
(8)
are considered for this purpose. The p-T diagram in Fig. 1 shows the relative deviations in density of the equation of state from selected ppmT data given in Table 1. Large parts of
43
0.5 100
200
300 -
LOO TIK
500
600
Fig. 1. Relative deviations Apm rel [eqn. (S)] of the equation of state (1) and (2) for methane from selected ppmT data given in Table 1.
and the gaseous regions are calculated with deviations of less than 0.1%. Near the critical point deviations in density increase up to 4.8%, while deviations in pressure do not exceed 0.22%. As discussed by Bender (1971a) a number of 20 coefficients is too small for a good representation of the critical region as well. For the critical temperature and the critical density, with the equation of state one calculates the critical pressure pc = 45.974 bar, which agrees with the chosen experimental value within the precision of the measurements. In the region of the critical isochore deviations between 0.2% and 0,5% occur. Except in the critical region the highest .deviations between 0.5% and 1.0% occur at pressures above 375 bar and high temperatures. Table 3 gives the mean relative deviations Aprel in pressure and A&, rel in density of the equation of state from the experimental pp,,,T data of several authors and also from the selected pp,,.,T data of Table 1 in various regions of the the ppmT surface. These deviations are defined by
the liquid
I
-
A~rel
=
t
+
l/2
c a~,‘., i )
i
1
TABLE
3
-
Mean relative deviations &&I [eqn. (9)] and Apmrel[eqn. (lo)] of the new equation of state (1) and (2) (first line) and the equation of state with the original coefficients of Bender (1971b) (second line) from p&, T measurements of several authors and from the selected ppm T data of Table 1 in different regions. Z is the number of data Z Range Authors &,,I &mrel
Goodwin
(1974)
(corr. L)
92-186
o-
40
48
-
Goodwin
(1974)
(corr. V)
92-186
o-
40
48
Gammon
and Douslin (1976)
143-323
6-
245
127,
Gammon and Douslin (1976) (crit. reg.) Mihara et al. (1977)
189-191
44-
46
61
289-348
18-
86
46
Nunes da Ponte et al. (1978)
110-120
14-l
281
86
0.20 0.29 0.16 0.20 0.18 0.16 0.02 0.10 4.45 5.49
0.04 0.20 0.17 0.24 0.12 0.16 0.20 0.19 0.45 0.45 3.50 3.71 0.62 0.46 0.09 0.21 0.16 0.36 0.02 0.19 0.31 0.77 0.44 0.54 0.13 0.27 0.03 0.24 0.04 0.28 0.33 0.38 3.29 5.50 5.01 10.62 0.02 0.11 0.22 0.19
92-623
l-
398
668
187-209
47-
93
44
91-398
o-
405
488
0.17 0.19 0.20 0.48 -
0.18 0.27 0.73 1.36 0.13 0.24
Michels and Nederbragt (1935)
273-423
18-
22
56
Michels and Nederbragt (1936)
273-423
18-
389
119
Schamp et al. (1958)
273-423
18-
261
118
Douslin et al. (1964)
273-623
16-
405
319
Epperly (1970)
458-610
3-
86
155
Jansoone et al. (1970)
189-194
45-
50
62
Cheng (1972)
111-309
Pope (1972)
126-191
l-
47
138
Roe (1972)
155-291
3-
100
82
91-115
o-
pm < 10 mole
176-300
28-
182
126
mole dmm3
192-300
48-
329
85
mole dmm3
190-290
27-
353
74
94-220
2-
353
268
Rodosevich
and Miller (1973)
Goodwin (1974) dmW3 Pm = 10-14.5 Pm = 14.5-20
P,,, > 20 mole dme3
Selected ppm T data of Table 1 gaseous region (pm < pmc) critical region (T c 1.1 T,) ._ (9.6 Pmc -= Pm < 1.45 Pmc) liquid region (pm > p,,)
218-1144
66
2
11
0.04 0.19 0.18 0.24 0.11 0.15 0.25 0.21 0.46 0.46 0.21 0.35 4.43 4.30 0.05 0.15 0.07 0.17 0.08 0.26 0.30 0.34 0.58 1.57 -
43
with Z as the number of ppmT data. In the gaseous region the relative deviations in pressure and density are about the same. Because of the very steep isotherms in the liquid region here small errors in density cause great deviations in pressure, although differences in density remain small. For comparison the mean relative deviations calculated using the coefficients of Bender (1971b) are also given in Table 3. In most cases the results of the new equation of state are closer to the experimental ‘data; this in particular applies to the liquid densities. Higher deviations only occur in the critical region and at pressures above 500 bar, for which the equation of state has not been fitted to experimental data. Summing up, one can say that the ppmT behavior of methane is very accurately represented by the new equation of state in the whole fluid region, for which ppmT measurements are available, with some restriction for the critical region. Second virial coefficients can be derived from ppmT data. Figure 2 shows the absolute deviations of the selected second virial coefficients B, exp given in Table 1 from the second virial coefficients B, calculated by eqn. (4). At low temperatures second virial coefficients change more rapidly with temperature; here deviations become greater. Over the whole temperature range they are somewhat greater than the accuracy of the measured values. For comparison deviations of the second virial coefficients of the IUPAC Table (Angus et al., 1978) from those calculated by eqn. (4) are shown too. Within a wide temperature range the selected experimental second virial coefficients are better represented by the new equation of state than by the IUPAC equation. The saturation properties are calculated from the equation of state for given temperatures by an iterative procedure (Bender, 1971a; Sievers, 1980). They are compared with measured values selected by Sievers (1980) and with the results of the IUPAC equation (Angus et al., 1978) and the Bender original equation (Bender, 1971b), which are treated further on as “experimental” data. The relative deviations AP, rel = h
exp - ps VP, exp
(11)
of the selected experimental vapour pressures ps exp from calculated vapour pressures ps are given in Fig. 3. For almost the whole temperature range from the triple point to the critical point the differences between the selected experimental data of Goodwin and Prydz (1972) and the results of the new equation of state are less than the experimental error of 0.1%. Vapour pressures in the IUPAC Table deviate by up to 0.17% from the experimental data of Prydz and Goodwin and up to 0.52% from the results of Bender.
44 0 =: .: d xz~ ;;% 2x 0 --8 i -A
++++ ++ + ++ + +++ 0 +++ + ++ 0
+++
h=l xi 1 z
+
Fig. 2. Absolute deviations of the selected experimental second virial coefficients B, exp for methane from second virial coefficients B, calculated by eqn. (4). For comparison the deviations of the second virial coefficients of the IUPAC Table from eqn. (4) are also shown.
For the density of the saturated liquid, relative deviations &J f;, rel = (Pk exp - Pkl)lPk exp
(12)
from the calculated values of pf;, of selected experimental data pk exp are
Fig. 3. Relative deviations Ap, rd [eqn. (ll)] of selected experimental vapour pressures ps exp for methane from the vapour pressure ps calculated by the equation of state (1) and (2). For comparison the relative deviations of the vapour pressures of the equation of state (1) and (2) with Render’s original coefficients and of the IUPAC equation from the new equation of state are also shown.
45
shown in Fig. 4. The measured values are represented with the experimental error of 0.1% by the equation of state in the temperature range between the triple point and 164 K. The saturated liquid densities for temperatures between 100 K and 150 K are used for calculations in custody transfer. The very precise experimental densities of Haynes and Hiza (1977) in this temperature range are calculated to within 0.05%. When approaching the critical point experimental errors and relative deviations increase. For temperatures from 164 K up to 187 K (0.98 Z’,) the differences between the selected experimental saturated liquid densities and the results of the new equation of state do not exceed 0.26%. The densities calculated with Bender’s coeffcients deviate by more than the experimental error from the measured values within a wide range. The relative deviations (13) from the calculated saturated vapour densities pz of selected experimental saturated vapour densities pz exp are presented in Fig. 5. The experimental error is estimated to be 0.3% increasing near the critical point. The equation of state represents the densities from the triple point up to 155 K within the experimental error, above 155 K the deviations exceed 0.3%. Compared with the densities calculated using Bender’s coefficients, the densities of the new equation are somewhat closer to the experimental results. The saturated vapour densities of the IUPAC Table (Angus et al., 1978) show even smaller
Fig. 4. Relative deviations A&!&l [eqn. (12)] of selected experimental saturated liquid densities & erp for methane from saturated liquid densities p&i calculated by the equation of state (1) and (2). For comparison the relative deviations of the saturated liquid densities of the equation of state (1) and (2) with Bender’s original coefficients from the new equation of state are also shown.
46
1
Fig. 5. Relative deviations ApxClrel [eqn. (13)] of selected experimental saturated vapour densities p$, exp for methane from saturated vapour densities calculated by the equation of state (1) and (2). For comparison the relative deviations of the saturated vapour densities of the equation of state (1) and (2) with Bender’s original coefficients and of the IUPAC equation from the new equation of state are also shown.
deviations. Considering the difficulties involved in measuring saturated. vapour densities the representation of these values by the new equation of state is quite satisfactory. The enthalpy of evaporation AH; can also be calculated from the equation of state (Bender, 1971a; Sievers, 1980). The absolute deviations of these values from the enthalpies of evaporation NE exp calculated with the Clausius-Clapeyron equation and Goodwin’s correlations for the vapour pressure and the orthobaric densities, from the values of a smooth curve (Sievers, 1980) through experimental data of Frank and Clusius (1937), Hestermans and White (1961) and Jones et al. (1963) and from the enthalpies of evaporation of the IUPAC Table (Angus et al., 1978) are shown in Fig. 6. In the temperature range from the triple point up to 182 K Goodwin’s data deviate from the equation of state by 38 J mole-’ at the most. The values of the smooth curve exhibit great differences at high temperatures caused by incorrect measurements made by Hestermans and White (1961) as
d
‘90.00
1
I
100.00
I
I
110.00
I
I
I
120.00
I
130.00
I
I
140.00
TEMPERRTURE
t
I
I
L50.00
/
I
160.00
K
I
I
170.00
m
I
I
180.00
1
I
190.0
Fig. 6. Absolute deviations of entbalpies of evaporation AH2 ex,, for methane from enthalpies of evaporation AlIz calculated by the equation of state (1) and (2).
pointed out by Bender (1970). The data of the IUPAC Table agree well with those of the new equation. One may say that saturation properties like vapour pressure, saturated liquid density, saturated vapour density and enthalpy of evaporation are reproduced very accurately with the new equation of state - over a wide range of measurements being within their margin of experimental error. Derived thermodynamic properties like enthalpies, entropies, heat capacities and velocities of sound can be calculated with the equation of state too, d the ideal gas heat capacity CG, is known. The thermodynamic functions for methane in the ideal gas state from McDowell and Kruse (1963) are fitted to the polynomial
Cv,/R, = i$lei(T/lOOO K)‘i
(14)
The coefficients ej are given in,Table 4. For the temperature range from 60 K to 650 K the mean relative de’viation is 0.04% and the maximum relative deviation 0.07%. Some of the derived thermodynamic properties are used here for testing thenew equation of state. Further examples and comparison with results calculated with Bender’s coefficients are given by Sievers (1980). Dawe and Snowdon (1974) published isothermal differences of enthalpies H, (1.01325 bar, T) - H,(T, p) for temperatures between 224 K and 367 K
48 TABLE
4
Coefficients methane
ei and exponents ji of eqn. (14) for the ideal gas heat capacity C”,,
i
ei
1 2 3 4
-0.7002279 0.3740558 -0.7825957 11.13094
x 10-S x 10-l
ji
i
=i
.ii
-3 -2 -1
5 6 7 8
-43.55974 110.7293 45.42033 29.17310
1 2 3 5
0
for
and pressures up to 100 bar. These data are represented by the equation of state with a maximum deviation of 33 J mole -‘. Differences of less than 10 J mole-l are shown by 40 out of 56 values. The IUPAC equation represents 49 of the 56 values to within 10 J mole-‘. Isobaric heat capacities have been measured by Jones et al. (1963) for temperatures from 116 K to 283 K and pressures up to 138 bar. Values calculated with the new equation of state differ - except in the critical region - by not more than 3% from the experimental ones. Agreement to within 1% is found for 296 of 400 data points and 164 are within 0.5%. Figure 7 shows a good agreement between calculated and experimental isobaric heat capacities for five isobars. The isochoric heat capacity is very sensitive to inaccuracies in the equation of state, because it is calculated from its second derivative with respect to temperature. Therefore it is especially useful for testing the equation of state. Younglove (1974) published experimental isochoric heat capacities for densities between 8.0 mole dme3 and- 28.1 mole dms, temperatures between 91 K and 300 K and pressures up to 330 bar. From these measurements Roder (1979) recalculated more precise C&p,.,,Tdata which are used here. (For further details see Sievers (1980)). For some measurements with nearly constant densities relative deviations A’&,
rep= (Cv, exp -
Cv,)/Cv,
exp
(15
)
of experimental isochoric heat capacities Cv, exp obtained from calculated values Cv, are presented in Fig. 8. The greatest differences occur at 28.1 mole dmW3 and very low temperatures and also in the critical region. As with the IUPAC equation the new equation of state is not able to reproduce the steep ascent of the isochoric heat capacity near the critical point. The remaining region above 100 K is accurately represented by the new equation of state. Deviation from the experimental values is less than 1% for 118 out of the 282 calculated values, 220 being less than 2%. Bender’s original coefficients produce results which are not as good: 91 values deviate by less than 1% and 187 by less than 2% from the experimental results. This test especially, using the very sensitive isochoric heat capacity, confirms the accuracy of the new equation of state. The velocity of sound calculated with the derivatives of the equation of
49
Erperfmentol Jones .“age
results Faulkner
and Katz 119631 0
10.36bar
x
X.13
.
63.09
bar
0
55.16
bar
v
82.76
b.3~
bar
Fig. 7. Comparison of experimental isobaric heat capacities calculated by the equation of state (1) and (2). ia 0 l-
C,,
for methane
with values
:
isochoric heat capacities Fig. 8. Relative deviations AC Vm rel [eqn. (IS)] of experimental Cv,,, exp for methane (Younglove, 1974) from heat capacities CVm calculated by the equation of state (1) and (2).
! 100 -
Fig. 9. Comparison 1975) with values
j
!
200
300
pl bar
of experimental velocities calculated by the equation
of sound a for methane of state (1) and (2).
(Straty,
1974
and
state for density and temperature (Sievers, 1980) also provides a very sensitive test for the equation of state. Measurements have been performed by Straty (1974 and 1975) for saturated liquid, liquid and gaseous methane in the temperature range from 91 K to 300 K and pressures up to 348 bar. Figure 9 illustrates that calculated and experimental data agree well. CONCLUSION
Bender (1971b) published an equation of state (1) and (2) for pure fluid methane, which can serve as a basis for an equation of state for fluid mixtures with methane as one component. Since Bender determined the 20 coefficients of his equation in 1971 using experimental thermodynamic data known at that time, several new and more precise measurements of the thermodynamic properties of methane have since been published. Therefore the coefficients of the equation of state (1) and (2) were newly evaluated in this work by a simultaneous, weighted least squares fit to ppmT data, second virial coefficients, thermodynamic conditions of equilibrium for co-existing liquid and vapour phases and thermodynamic conditions at the critical point. With the new coefficients of Table 2 this equation of state represents the thermo-
51
dynamic properties of methane in the whole fluid region at temperatures between 90.68 K and 625 K, densities up to 28.6 mole dmw3 and pressures up to 500 bar with a high degree of accuracy. Only some restrictions have to be made for the critical region. Within a wide range the fit is better than with Bender’s original coefficients, especially for vapour-liquid phase equilibria and the liquid region. ACKNOWLEDGMENTS
The helpf of Mr. F.-J. Vonnahme with the development of a computer program and of Mr. M. Koschowitz with some calculations is gratefully acknowledged. LIST OF SYMBOLS
velocity of sound coefficients of the equation of state (2) :, C, D, E, F, G, H temperature functions of the equation of state (1) secondvirial coefficient &’ C Pm molar isobaric heat capacity c VIXI molar isochoric heat capacity coefficients in eqn. (14) molar enthalpy I number of ppmT data J number of second virial coefficients exponents in eqn. (14) ji L number of vapour-liquid phase equilibrium data pressure P vapour pressure PS universal gas constant R, T temperature W weighting factor molar density Pm relative deviation in isochoric heat capacity ACvrn rel molar enthalpy of evaporation Aft relative deviation in pressure &Jrel mean relative deviation in pressure AP,, relative deviation in density &An re1 mean relative deviation in density AP m re1 a
zm
Indices C
exp L max
at the critical point experimental saturated liquid maximum
52
min V 0
minimum saturated vapour ideal gas state
REFERENCES Ahrendts, J., 1977. Die direkte Beriicksichtigung gemessener Siittigungsgrossen und der kritischen Temperatur bei der Aufstellung von Zustandsgleichungen. VDI-Thermodynamik-Kolloquium, Bad Diirkheim. Angus, S., Armstrong, B. and de Reuck, K.M., 1976. Final Report on IUPAC Methane Tables. Part I: Text. IUPAC Thermodynamic Tables Project Centre. PC/BB7 P, 123 pp. Angus, S., Armstrong, B. and de Reuck, K.M., 1978. International Thermodynamic Tables of the Fluid State - 5. Methane. IUPAC Thermodynamic Table. Pergamon Press, Oxford, 251 pp. Bender, E., 1970. Equations of state exactly representing the phase behavior of pure substances. Proc. Fifth Symposium on Thermophysical Properties. Amer. Sot. Mech. Engrs., New York: 227-235. Bender, E., 1971a. Die Berechnung von Phasengleichgewichten mit der thermischen Zustandsgleichung - dargestellt an den reinen Fluiden Argon, Stickstoff, Sauerstoff und an ihren Gemischen. Habilitationsschrift Ruhr-Universitiit Bochum. (English translation: The Calculation of Phase Equilibria from a Thermal Equation of State Applied to the Pure Fluids Argon, Nitrogen, Oxygen and their Mixtures. Verlag C.F. Miiller, Karslruhe, 1973,173 pp.) Bender, E., 1971b. Zur Aufstellung von Zustandsgleichungen, aus denen sich die Slittigungsgrossen exakt berechnen lassen - gezeigt am Beispiel des Methans. Kiiltetechnik-Klimatisierung, 23: 258-264. Bender, E., 1975a. Calculation of phase equilibria and thermodynamic properties of cryogenic mixtures with a new equation of state. Vth International Congress of Chemical Engineering,‘Chemical Equipment Design and Automation. CHISA 75, Prag. Section F. Vapour-Liquid Equilibrium F2.25, Bender, E., 1975b. Equations of state for ethylene and propylene. Cryogenics, 15: 667673. Benedict, M.. Webb, G.B. and Rubin, L.C., 1940. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures. J. Chem. Phys., 8: 334-345. Cheng, V.G., 1972. Measurements on the dense fluid equation of state and the melting parameters of argon, methane and nitrogen at high pressures. Ph.D. Thesis, Princeton University, Princeton, New Jersey. Dawe. R.A. and Snowdon, P.N., 1974. Enthalpy of gaseous methane in range 224.00366.70 K and l-100 bar. J. Chem. Engng. Data, 19: 220-223. Diller, D.E., 1976. Thermophysical properties data research on compressed and liquefied gases at the NBS cryogenic division. Adv. Cryog. Engng., 21: 522-531. Douslin, DR., Harrison, R.H.. Moore, R.T. and McCullough, J.P., 1964. P-V-T relations for methane. J. Chem. Engng. Data, 9: 358-363. Epperly, A.D.,, 1970. Determination of third virial coefficients from Burnett p-u-T measurements. Ph.D. Thesis, University of Missouri, Columbia. Eubank, P.T., 197 2. Interim Report on the Correlation of Thermodynamic Properties of Methane. IUPAC Thermodynamic Tables, Project Centre Report. PC/R23. Frank, A. and Clusius, K., 1937. Zur Entropie des Methans. Z. phys. Chem., B36: 291-300. Gammon, B.E. and Douslin, D.R., 1976. The velocity of sound and heat capacity in methane from near-critical to sub-critical conditions and equation of state implications. J. Chem. Phys., 64: 203-218. Goodwin, R.D. and Prydz, R., 1972. Densities of compressed liquid methane, and the equation of state J. Res. Nat. Bur. Stand., 76A: 81-101.
53
Goodwin, R.D., 1973. Dielectric constants and orthobaric densities of methane. Nat. Bur. Stand. Internal Report 73-300. Goodwin, R.D., 1974. The thermophysical properties of methane, from 90 to 500 K at pressures to 700 bar. Nat. Bur. Stand. Techn. Note 653. Haynes, W.M. and Hiza, M.J., 1977. Measurements of the orthobaric liquid densities of methane, ethane, propane, isobutane and normal butane. J. Chem. Thermodyn., 9: 179-187. Hcstermans, P. and White, D., 1961. The vapor pressure, heat of vaporization and heat capacity of methane from the boiling point to the critical temperature. J. Phys. Chem., 65: 362-366. Hust, J.G. and McCarty, R.D., 1967. Curve-fitting techniques and applications to thermodynamics. Cryogenics, 7: 200-206. Jacobsen, R.B., 1972. The thermodynamic properties of nitrogen from 65 K to 2000 K with pressures to 10 000 atmospheres. Ph.D. Thesis, Washington State University, Pullman, Washington. Jansoone, V., Gielen, H., de Boelpaep, J. and Verbeke, O.B., 1970. The pressure-temperature--volume relationship of methane near the critical point. Physica, 46: 213-221. Jones, M.L., Mage, D.T., Faulkner, R.C. and Katz, D.L., 1963. Measurement of the thermodynamic properties of gases at low temperatures and high pressure methane. Chem. Engng. Prog. Symp. Ser., 59: 52-60. Koschowitz, M., 1979. Eine therm&he Zustandsgleichung fiir das reine Fluid Methan. Studienarbeit Lehrstuhl Thermodynamik. Abt. Chemietechnik. UniversitPt Dortmund. McCarty, R.D., 1974. A modified Benedict-Webb-Rubin equation of state for methane using recent experimental data. Cryogenics, 14: 276-280. McClune, C.R., 1976. Measurement of the densities of liquefied hydrocarbons from 93 to 173 K. Cryogenics, 10: 289-295. McDowell, R.S. and Kruse, F.H., 1963. Thermodynamic functions of methane. J. Chem. Engng. Data, 8: 547-548. Michels, A. and Nederbragt, G.W., 1935. Isotherms of methane between 0 and 150% and densities of 19 and 53 amagat (Pressures between 20 and 80 atm). Physica, 2: IOOO1002. Michels, A. and Nederbragt, G.W., 1936. Isotherms of methane between 0 and 150% for densities up to 225 amagat. Calculated specific heat, energy and entropy in the same region. Physica, 3: 7, 569-577. Mihara, S., Sagara, H., Arai, Y. and Saito, S., 1977. The compressibility factors of hydrogen-methane, hydrogen-ethane and hydrogen-propane gaseous mixtures. J. Chem. Engng. Jpn., 10: 395-399. Nunes da Ponte, M., Streett, W.B. and Staveley, L.A.K., 1978. An experimental study of the equation of state of liquid mixtures of nitrogen and methane, and the effect of pressure on their excess thermodynamic functions. J. Chem. Thermodyn., 10: 151168. Pope, G.A., 1972. Calculation of argon, methane and ethane virial coefficients at low reduced temperatures based on data obtained by isochorically coupled Burnett experiments. Ph.D. Thesis, Rice University, Houston, Texas. Prydz, R. and Goodwin, R.D., 1972. Experimental melting and vapour pressures of methane. J. Chem. Thermodyn., 4: 127-133. Roder, H.M., personal communication, 1979. Rodosevich, J.B. and Miller, R.C., 1973. Experimental liquid mixture densities for testing and improving correlations for liquefied natural gas. AIChE J., 19: 729-735. Roe, D.R., 1972. Thermodynamic properties of gases and gas mixtures at low temperatures and high pressures. Ph.D. Thesis, Imperial College of Science and Technology, London. Schamp, H.W., Mason, E.A., Richardson, A.C.B. and Altmann, A., 1958. Compressibility and intermolecular forces in gases: methane. Phys. Fluids, 1: 329-337.
54 Schulz, S., 1973. Thermische oder kalorische Messungen; Wege zur Bestimmung des thermodynamischen Zustandsverhaltens. Arch. Techn. Messen, Blatt V2-1, Lfg. 453: 181184. Sievers, U., 1980. Berechnung thermodynamischer Stoffeigenschaften mit einer thermischen Zustandsgleichung fur fluide binare Gemische mit der Komponenten Methan, Argon, Stickstoff und Wasserstoff bis au Driicken van 500 bar. Dissertation Universitat Dortmund, Abt. Chemietechnik. Straty, G.C., 1974..Velocity of sound in the dense fluid methane. Cryogenics, 14: 367370. Straty, G.C., 1975. Hypersonic velocities in saturated and compressed fluid methane. Cryogenics, 15: 729-731. Teja, A.S. and Singh, A., 1977. Equations of state for ethane, propane, n-butane and n-pentane. Cryogenics, 17: 591-596. Younglove, B.A., 1974. The specific heats, C,, C,, of compressed and liquefied methane. J. Res. Nat. Bur. Stand., ‘78A: 401-410.
35 Amsterdam
- Printed in The Netherlands
AN EQUATION OF STATE FOR METHANE IN THE FORM OF BENDER’S EQUATION FOR TEMPERATURES BETWEEN 91 K AND 625 K AND PRESSURES UP TO 500 BAR
U. SIEVERS
and S. SCHULZ
Lehrstuhl Thermodynamik, 50 (F.R. G.)
Universitiit
(Received December 26th, 1979;
Dortmund,
accepted
Postfach
500 500, D 4600
Dortmund
in revised form April 4th, 1980)
ABSTRACT Sievers, U. and Schulz, S., 1980. An equation of state for methane in the form of Bender’s equation for temperatures between 91 K and 625 K and pressures up to 500 bar. Fluid Phase Equilibria, 5: 35-54. The coefficients of Bender’s equation of state for pure fluid methane are refitted using recent, precise experimental data. This equation of state represents the thermodynamic properties of methane in the fluid region for temperatures between 90.68 K and 625 K, densities up to 28.6 mole dmm3 and pressures up to 500 bar with high accuracy. Only in the critical region is there a slightly lower accuracy. Compared with the results obtained with Bender’s original coefficients, improvements are achieved in wide regions, especially for the calculation of vapour--liquid phase equilibria and liquid properties.
INTRODUCTION
For the calculation and optimization of processes in the natural gas industry the thermodynamic properties of fluid mixtures of simple, non polar components must be known over a wide range of temperatures and pressures. The Benedict-Webb-Rubin (BWR) equation (Benedict et al., 1940) is used in different modifications for this purpose. All thermodynamic properties can be evaluated in combination with the temperature function of the isochoric heat capacity of the pure components in the ideal gas state. As the properties of the ideal gas can be calchlated highly accurately, using the statistical thermodynamical and quantum mechanical methods, the main problem is to find a good equation of state for the mixture. The quality of this equation depends on the accuracy of the equations of state for the pure components on the one hand and the mixing rules used in order to obtain the coefficients of the mixture’s equation from those of the pure fluids, on the other. This is discussed in detail by Sievers (1980). The purpose of this paper is to give a good equation of state for pure fluid methane, the main component of natural gas, which can 0378-3812/80/0000-0000/$02.60
01980
Elsevier Scientific Publishing Company
36
serve as a basis for an equation of state for fluid mixtures with methane as one component. EQUATION
OF STATE
AND
DETERMINATION
OF THE COEFFICIENTS
Bender (1971b) developed a modified BWR-type of equation of state with 20 coefficients for methane. McCarty (1974) refitted the coefficients of this equation and also determined the 33 coefficients of the Jacobsen equation (Jacobsen, 1972) for methane. In 1978 the more precise IUPAC equation for methane (Angus et al., 1978), a Jacobsen equation with 33 coefficients too, was published. When applying it to mixtures, the equation of state for the pure fluids should be of high accuracy but have a small number of coefficients, needing only a few mixing rules. Bender’s equation with its 20 coefficients is a good compromise between these two demands. It is successfully used for pure fluid argon, nitrogen, oxygen (Bender, 1971a), hydrogen (Bender, 1975a), ethylene and propylene (Bender, 1975b), ethane, propane, n-butane and n-pentane (Teja and Singh, 1977). The equation of state is p = R,Tp,
+ Bps
+ Cp& + Dp;
+ (G + HP&,) p%ew(--a~
+ Ep% + Fp$ (1)
p&l
where p and T are pressure and temperature, pm is the molar density and R, the universal gas constant. B, C, D, E, F, G and H are the following temperature functions for the pure fluids. B = al T -
a2 -
asIT -
ad/T2 -
as/T3
C = a6T + a7 + as/T D=aaT+alo E = alIT + al2
(2)
F = al3 G = alhIT2 + aI5 /T3 + als/p H=
alTIp
+ aI8 IT3 + a19/p
The coefficients a, to a2o are determined for each pure fluid by fitting the equation to experimental thermodynamic data. Bender (1971b) did this for methane, using data published up until 1969. Because of the great interest in the thermodynamic properties of natural gas, further precise measurements for its principal component methane have been published since then. An extensive experimental programme on the thermophysical properties of methane in the dense gas and liquid region was carried out by the National Bureau of Standards, Cryogenic Research Division (Goodwin (1974), Younglove (1974), Diller (1976), Haynes and Hiza (1977)). These and other new experimental results for methane required a refit of the coefficients a1 to a2o in
eqns. (1) and (2). Eubank (1972) came to the same conclusion after an inspection of the experimental data for methane. At the time when the coefficients of the IUPAC equation for methane were determined, there were no orthobaric densities thermodynamically consistent with density measurements of the homogenous region available (Angus et al., 1976). So the authors calculated the orthobaric densities from provisional equations of state for the gaseous and the liquid region by extrapolating the isochors to the vapour pressure. Meanwhile orthobaric densities, thermodynamically consistent with the densities of the homogeneous region, have been published (see Sievers (1980) for a detailed discussion). This improved knowledge of orthobaric densities, and the fact that there are further precise measurements for methane which have not been used in fitting the coefficients of the IUPAC equation, justify the refitting of the coefficients of Bender’s equation for methane even compared with the IUPAC equation. The new coefficients of the equation of state (1) and (2) are determined by a simultaneous, weighted least squares fit to selected experimental p&T data and second virial coefficients, to the thermodynamic conditions of equilibrium for the co-existing liquid and vapour phases and to the thermodaynamic conditions at the critical point. CvmpmT data, reflecting the derivation of the equation of state and commonly used in fitting the coeffi(a 2Pl@9,m cients (McCarty, 1974; Angus et al., 1978), have not been included in the fit in order to prevent the influence of another equation of state on the new coefficients, because the densities in these data for methane (Younglove, 1974) had been calculated using an equation of state by Goodwin and Prydz (1972). According to Schulz (1973) an equation of state is more accurate if the coefficients are fitted to ppmT data rather than to measurements of enthalpies or heat capacities. The sum of squares S of weighted residuals, eqn. (3), is minimized by using the techniques described by Hust and McCarty (1967) and Bender (1971a). J
~~~ilP(T~~,i~~mcl~.~)-~~~~,~l~+~~~~ly(Br(Tr.~.~)-~~~~~,jI*
S=
i=l
+
~c~b(Tc..., ~mc exp)-PC cd2
(3) The residuals are formed by the differences between the values calculated
38
from the equation of state with the experimental temperature TexP and the experimental density pm exp and the experimental pressure pexp, second virial coefficient B, exp, vapour pressure ps _,, and critical pressure pC _,, respectively. Each square of residual is individually weighted by,a factor w. Selected ppmT data of the gaseous and liquid regions and of the vapourliquid phase boundary are considered in the first sum. The number of these data is 1. The second sum includes J second virial coefficients. They are calculated from the equation of state by Bv = (al -
azlT -
a3/T2 -
ad/T3 -
a51T4)lR,
(4)
The thermodynamic conditions for vapour-liquid phase equilibrium are incorporated in the first and the third sum. The equality of the pressures for the saturated liquid and the saturated vapour at the same temperature is found in the first sum, as pp,,.,T data of the co-existing phases are also included. The criterion of equal fugacities for co-existing liquid and vapour phases is given in the third sum by the so called Maxwell-criterion, which is solved for the vapour pressure. Properties of the liquid phase are indicated by superscript L, those of the vapour phase by superscript V. The number of vapourliquid phase equilibrium data is L. The last three terms in eqn. (3) reflect the conditions at the critical point. As the critical density cannot be measured very accurately, the equation of state is not constrained to fulfil these conditions exactly. They are made part of the sum of squares like all the other data, according to a suggestion of Ahrendts (1977), and weighted by w,i, w,a and Wc3.
In an extensive study, all the available experimental thermodynamic data of methane have been thoroughly examined and reliable data have been selected for the determination of the coefficients (Sievers, 1980). All values were converted to SI units, temperatures corrected to the temperature scale ITPS-68. The data selected for fitting the coefficients of the equation of state are given in Table 1. Vapour pressure ps and orthobaric densities pk and pz must be known at the same temperature for the Maxwell-criterion. As these properties are measured independently as a function of temperature, data are not usually given at the same temperatures. Therefore correlations for the saturation properties are needed in order to obtain values at the same temperatures and also to smooth out experimental errors. The correlations of Good-win (1974) have been tested for this purpose (Sievers, 1980) and are used in this paper to represent the vapour pressure ps and the orthobaric densities pk and pz. Measurements with small experimental errors should have a higher influence on the determination of the new coefficients than those with greater experimental uncertainty. For the ppmT data this is achieved by choosing the weight Wpi proportional to the reciprocal absolute error of the i-th ppmT data point, which is calculated according to eqn. (5) from the mean absolute errors Ap, Apm and AT of the individual measurements, taken from the
39
TABLE
1
Selected thermodynamic data of methane * for fitting the coefficients state(1) and (2). N is the number of data points. Kind of data
Author
Douslin et al. (1964) Pope (1972) Roe (1972) Rodosevich and Miller (1973) Goodwin (1974) Goodwin (1974) (corr. V) Goodwin (1974) (corr. L) Douslin et al. (1964) Pope (1972) Roe (1972) Goodwin (1974) (corr.)
PP~T P&T
PPItlT PP~T PP T ‘0 PIP T ft &pm T & T % T & T PSP~m pVT In
of the equation of
Range
N
N-9
p(bar)
273-623 126-191 155-291 91-115 92-300 92-186 92-188 273-623 126-159 168-263 92-188
16-405 l47 3-100 o-2 3-351 o- 40 0- 42
0-
42
319 138 82 11 * 553 48 49 15 4 8 49
* A computer listing of the selected thermodynamic data for methane is available from the authors on request. The distribution of the selected ppmT data is shownby Sievers (1980) in a p-pm and a pT diagram.
authors
if possible
or estimated.
To start with, the derivatives @p/a&r
and @p/a Z&, are obtained from the determined from eqn. (3) with all weights set to 1.0, and corrected iteratively in the following steps with the coefficients of the preceding step until good results are achieved. Measurements of different authors and regions of the ppmT surface may be weighted by woP in a different way. The weights wei and wsI are calculated analogously. For a good representation of the liquid properties and the vapourliquid phase equilibria the weights w,,~ for the liquid region and w,, must be given large values. Empirical correlations for these weights, the absolute errors used in eqn. (5) and further details of the determination of the coefficients are given by Sievers (1980). The influence of different weights on the results of the equation of state is discussed by Koschowitz (1979). The properties of the critical point are chosen to be T, exp = 190.555 K, p. sxg = 45.988 bar and prnc exP = 10.1095 mole dm-s (Sievers, 1980) and the non-linear parameter azo according to Bender’s empirical correlation equation of state (1) and (2) with coefficients
asOAexp
= 1
The new coefficients for the equation of state (1) and (2) for methane are given in Table 2.
TABLE
2
Coefficients ci of the equation of state (1) and (2) for methane and the region of validity of the equation. The equation of state with the tabulated values for (I[ and the values of temperatures and density in the units K and mole dme3 respectively gives the pressure in J dme3. (Conversion of pressure units: 100 J dmw3 = 100 kPa = 1 bar)
%I a1 a2 a3 a4 a5 '"6 a7 aI3 =9 al0 =11 (112 013 =14 a15 al6 =17 a18 a19 a20
M
Tmin T
J mole1 K-l J dm3 moleV2 K-l J dm3 molew2 J K dm3 molee2 J K2 dm3 molem2 J K3 dm3 molee2 J dm6 molee3 K-l J dms moleV3 J K dm6 moleF3 J dmg moleV4 K-l J dmg molew4 J dm12 molem5 K-l J dm12 molep5 J drn15 molew6 J K2 dm6 molee3 J K3 dm6 molem3 J K4 dm6 moleM3 J K2 dml* molem5 J K3 dm12 molew5 J K4 drn12 molee5 dm6 molee2 e mole-l
I<
p:zbKr
pinax mole dmM3
Pmax / Pmc
TESTING
THE NEW EQUATION
8.3143 0.4531925285 0.2358287986 -0.7173842009 0.296.1509482 -0.8735505087 0.1052328359 -0.1694738647 0.8637907281 0.8533763036 -0.4017208435 0.3045691216 -0.2563358654 0.8484753417 0.2545831403 -0.5522243593 0.2250857179 -0.3979182974 0.1268073106 -0.5298037215 0.9784545300 16.0430 90.68 623 500 28.6 2.83
x lo3 x lo4 x 10’ x
10’
x 10-l x lo1 x lo3 x 1O-3 x 1O-2 x 1O-4 x 10-l x 1O-3 x lo6 x 108 x lOlo x 104 x lo7 x lo* x 10-g
OF STATE
For testing the accuracy of the new equation of state it is at first compared with experimental ppmT data of the homogeneous gaseous and liquid regions. The relative deviations between the experimental and the calculated pressures A~rel = (Pexp -
P)/P~
(7)
and the experimental and calculated densities AP m rel = (P, exp - PrnYPInexp
(8)
are considered for this purpose. The p-T diagram in Fig. 1 shows the relative deviations in density of the equation of state from selected ppmT data given in Table 1. Large parts of
43
0.5 100
200
300 -
LOO TIK
500
600
Fig. 1. Relative deviations Apm rel [eqn. (S)] of the equation of state (1) and (2) for methane from selected ppmT data given in Table 1.
and the gaseous regions are calculated with deviations of less than 0.1%. Near the critical point deviations in density increase up to 4.8%, while deviations in pressure do not exceed 0.22%. As discussed by Bender (1971a) a number of 20 coefficients is too small for a good representation of the critical region as well. For the critical temperature and the critical density, with the equation of state one calculates the critical pressure pc = 45.974 bar, which agrees with the chosen experimental value within the precision of the measurements. In the region of the critical isochore deviations between 0.2% and 0,5% occur. Except in the critical region the highest .deviations between 0.5% and 1.0% occur at pressures above 375 bar and high temperatures. Table 3 gives the mean relative deviations Aprel in pressure and A&, rel in density of the equation of state from the experimental pp,,,T data of several authors and also from the selected pp,,.,T data of Table 1 in various regions of the the ppmT surface. These deviations are defined by
the liquid
I
-
A~rel
=
t
+
l/2
c a~,‘., i )
i
1
TABLE
3
-
Mean relative deviations &&I [eqn. (9)] and Apmrel[eqn. (lo)] of the new equation of state (1) and (2) (first line) and the equation of state with the original coefficients of Bender (1971b) (second line) from p&, T measurements of several authors and from the selected ppm T data of Table 1 in different regions. Z is the number of data Z Range Authors &,,I &mrel
Goodwin
(1974)
(corr. L)
92-186
o-
40
48
-
Goodwin
(1974)
(corr. V)
92-186
o-
40
48
Gammon
and Douslin (1976)
143-323
6-
245
127,
Gammon and Douslin (1976) (crit. reg.) Mihara et al. (1977)
189-191
44-
46
61
289-348
18-
86
46
Nunes da Ponte et al. (1978)
110-120
14-l
281
86
0.20 0.29 0.16 0.20 0.18 0.16 0.02 0.10 4.45 5.49
0.04 0.20 0.17 0.24 0.12 0.16 0.20 0.19 0.45 0.45 3.50 3.71 0.62 0.46 0.09 0.21 0.16 0.36 0.02 0.19 0.31 0.77 0.44 0.54 0.13 0.27 0.03 0.24 0.04 0.28 0.33 0.38 3.29 5.50 5.01 10.62 0.02 0.11 0.22 0.19
92-623
l-
398
668
187-209
47-
93
44
91-398
o-
405
488
0.17 0.19 0.20 0.48 -
0.18 0.27 0.73 1.36 0.13 0.24
Michels and Nederbragt (1935)
273-423
18-
22
56
Michels and Nederbragt (1936)
273-423
18-
389
119
Schamp et al. (1958)
273-423
18-
261
118
Douslin et al. (1964)
273-623
16-
405
319
Epperly (1970)
458-610
3-
86
155
Jansoone et al. (1970)
189-194
45-
50
62
Cheng (1972)
111-309
Pope (1972)
126-191
l-
47
138
Roe (1972)
155-291
3-
100
82
91-115
o-
pm < 10 mole
176-300
28-
182
126
mole dmm3
192-300
48-
329
85
mole dmm3
190-290
27-
353
74
94-220
2-
353
268
Rodosevich
and Miller (1973)
Goodwin (1974) dmW3 Pm = 10-14.5 Pm = 14.5-20
P,,, > 20 mole dme3
Selected ppm T data of Table 1 gaseous region (pm < pmc) critical region (T c 1.1 T,) ._ (9.6 Pmc -= Pm < 1.45 Pmc) liquid region (pm > p,,)
218-1144
66
2
11
0.04 0.19 0.18 0.24 0.11 0.15 0.25 0.21 0.46 0.46 0.21 0.35 4.43 4.30 0.05 0.15 0.07 0.17 0.08 0.26 0.30 0.34 0.58 1.57 -
43
with Z as the number of ppmT data. In the gaseous region the relative deviations in pressure and density are about the same. Because of the very steep isotherms in the liquid region here small errors in density cause great deviations in pressure, although differences in density remain small. For comparison the mean relative deviations calculated using the coefficients of Bender (1971b) are also given in Table 3. In most cases the results of the new equation of state are closer to the experimental ‘data; this in particular applies to the liquid densities. Higher deviations only occur in the critical region and at pressures above 500 bar, for which the equation of state has not been fitted to experimental data. Summing up, one can say that the ppmT behavior of methane is very accurately represented by the new equation of state in the whole fluid region, for which ppmT measurements are available, with some restriction for the critical region. Second virial coefficients can be derived from ppmT data. Figure 2 shows the absolute deviations of the selected second virial coefficients B, exp given in Table 1 from the second virial coefficients B, calculated by eqn. (4). At low temperatures second virial coefficients change more rapidly with temperature; here deviations become greater. Over the whole temperature range they are somewhat greater than the accuracy of the measured values. For comparison deviations of the second virial coefficients of the IUPAC Table (Angus et al., 1978) from those calculated by eqn. (4) are shown too. Within a wide temperature range the selected experimental second virial coefficients are better represented by the new equation of state than by the IUPAC equation. The saturation properties are calculated from the equation of state for given temperatures by an iterative procedure (Bender, 1971a; Sievers, 1980). They are compared with measured values selected by Sievers (1980) and with the results of the IUPAC equation (Angus et al., 1978) and the Bender original equation (Bender, 1971b), which are treated further on as “experimental” data. The relative deviations AP, rel = h
exp - ps VP, exp
(11)
of the selected experimental vapour pressures ps exp from calculated vapour pressures ps are given in Fig. 3. For almost the whole temperature range from the triple point to the critical point the differences between the selected experimental data of Goodwin and Prydz (1972) and the results of the new equation of state are less than the experimental error of 0.1%. Vapour pressures in the IUPAC Table deviate by up to 0.17% from the experimental data of Prydz and Goodwin and up to 0.52% from the results of Bender.
44 0 =: .: d xz~ ;;% 2x 0 --8 i -A
++++ ++ + ++ + +++ 0 +++ + ++ 0
+++
h=l xi 1 z
+
Fig. 2. Absolute deviations of the selected experimental second virial coefficients B, exp for methane from second virial coefficients B, calculated by eqn. (4). For comparison the deviations of the second virial coefficients of the IUPAC Table from eqn. (4) are also shown.
For the density of the saturated liquid, relative deviations &J f;, rel = (Pk exp - Pkl)lPk exp
(12)
from the calculated values of pf;, of selected experimental data pk exp are
Fig. 3. Relative deviations Ap, rd [eqn. (ll)] of selected experimental vapour pressures ps exp for methane from the vapour pressure ps calculated by the equation of state (1) and (2). For comparison the relative deviations of the vapour pressures of the equation of state (1) and (2) with Render’s original coefficients and of the IUPAC equation from the new equation of state are also shown.
45
shown in Fig. 4. The measured values are represented with the experimental error of 0.1% by the equation of state in the temperature range between the triple point and 164 K. The saturated liquid densities for temperatures between 100 K and 150 K are used for calculations in custody transfer. The very precise experimental densities of Haynes and Hiza (1977) in this temperature range are calculated to within 0.05%. When approaching the critical point experimental errors and relative deviations increase. For temperatures from 164 K up to 187 K (0.98 Z’,) the differences between the selected experimental saturated liquid densities and the results of the new equation of state do not exceed 0.26%. The densities calculated with Bender’s coeffcients deviate by more than the experimental error from the measured values within a wide range. The relative deviations (13) from the calculated saturated vapour densities pz of selected experimental saturated vapour densities pz exp are presented in Fig. 5. The experimental error is estimated to be 0.3% increasing near the critical point. The equation of state represents the densities from the triple point up to 155 K within the experimental error, above 155 K the deviations exceed 0.3%. Compared with the densities calculated using Bender’s coefficients, the densities of the new equation are somewhat closer to the experimental results. The saturated vapour densities of the IUPAC Table (Angus et al., 1978) show even smaller
Fig. 4. Relative deviations A&!&l [eqn. (12)] of selected experimental saturated liquid densities & erp for methane from saturated liquid densities p&i calculated by the equation of state (1) and (2). For comparison the relative deviations of the saturated liquid densities of the equation of state (1) and (2) with Bender’s original coefficients from the new equation of state are also shown.
46
1
Fig. 5. Relative deviations ApxClrel [eqn. (13)] of selected experimental saturated vapour densities p$, exp for methane from saturated vapour densities calculated by the equation of state (1) and (2). For comparison the relative deviations of the saturated vapour densities of the equation of state (1) and (2) with Bender’s original coefficients and of the IUPAC equation from the new equation of state are also shown.
deviations. Considering the difficulties involved in measuring saturated. vapour densities the representation of these values by the new equation of state is quite satisfactory. The enthalpy of evaporation AH; can also be calculated from the equation of state (Bender, 1971a; Sievers, 1980). The absolute deviations of these values from the enthalpies of evaporation NE exp calculated with the Clausius-Clapeyron equation and Goodwin’s correlations for the vapour pressure and the orthobaric densities, from the values of a smooth curve (Sievers, 1980) through experimental data of Frank and Clusius (1937), Hestermans and White (1961) and Jones et al. (1963) and from the enthalpies of evaporation of the IUPAC Table (Angus et al., 1978) are shown in Fig. 6. In the temperature range from the triple point up to 182 K Goodwin’s data deviate from the equation of state by 38 J mole-’ at the most. The values of the smooth curve exhibit great differences at high temperatures caused by incorrect measurements made by Hestermans and White (1961) as
d
‘90.00
1
I
100.00
I
I
110.00
I
I
I
120.00
I
130.00
I
I
140.00
TEMPERRTURE
t
I
I
L50.00
/
I
160.00
K
I
I
170.00
m
I
I
180.00
1
I
190.0
Fig. 6. Absolute deviations of entbalpies of evaporation AH2 ex,, for methane from enthalpies of evaporation AlIz calculated by the equation of state (1) and (2).
pointed out by Bender (1970). The data of the IUPAC Table agree well with those of the new equation. One may say that saturation properties like vapour pressure, saturated liquid density, saturated vapour density and enthalpy of evaporation are reproduced very accurately with the new equation of state - over a wide range of measurements being within their margin of experimental error. Derived thermodynamic properties like enthalpies, entropies, heat capacities and velocities of sound can be calculated with the equation of state too, d the ideal gas heat capacity CG, is known. The thermodynamic functions for methane in the ideal gas state from McDowell and Kruse (1963) are fitted to the polynomial
Cv,/R, = i$lei(T/lOOO K)‘i
(14)
The coefficients ej are given in,Table 4. For the temperature range from 60 K to 650 K the mean relative de’viation is 0.04% and the maximum relative deviation 0.07%. Some of the derived thermodynamic properties are used here for testing thenew equation of state. Further examples and comparison with results calculated with Bender’s coefficients are given by Sievers (1980). Dawe and Snowdon (1974) published isothermal differences of enthalpies H, (1.01325 bar, T) - H,(T, p) for temperatures between 224 K and 367 K
48 TABLE
4
Coefficients methane
ei and exponents ji of eqn. (14) for the ideal gas heat capacity C”,,
i
ei
1 2 3 4
-0.7002279 0.3740558 -0.7825957 11.13094
x 10-S x 10-l
ji
i
=i
.ii
-3 -2 -1
5 6 7 8
-43.55974 110.7293 45.42033 29.17310
1 2 3 5
0
for
and pressures up to 100 bar. These data are represented by the equation of state with a maximum deviation of 33 J mole -‘. Differences of less than 10 J mole-l are shown by 40 out of 56 values. The IUPAC equation represents 49 of the 56 values to within 10 J mole-‘. Isobaric heat capacities have been measured by Jones et al. (1963) for temperatures from 116 K to 283 K and pressures up to 138 bar. Values calculated with the new equation of state differ - except in the critical region - by not more than 3% from the experimental ones. Agreement to within 1% is found for 296 of 400 data points and 164 are within 0.5%. Figure 7 shows a good agreement between calculated and experimental isobaric heat capacities for five isobars. The isochoric heat capacity is very sensitive to inaccuracies in the equation of state, because it is calculated from its second derivative with respect to temperature. Therefore it is especially useful for testing the equation of state. Younglove (1974) published experimental isochoric heat capacities for densities between 8.0 mole dme3 and- 28.1 mole dms, temperatures between 91 K and 300 K and pressures up to 330 bar. From these measurements Roder (1979) recalculated more precise C&p,.,,Tdata which are used here. (For further details see Sievers (1980)). For some measurements with nearly constant densities relative deviations A’&,
rep= (Cv, exp -
Cv,)/Cv,
exp
(15
)
of experimental isochoric heat capacities Cv, exp obtained from calculated values Cv, are presented in Fig. 8. The greatest differences occur at 28.1 mole dmW3 and very low temperatures and also in the critical region. As with the IUPAC equation the new equation of state is not able to reproduce the steep ascent of the isochoric heat capacity near the critical point. The remaining region above 100 K is accurately represented by the new equation of state. Deviation from the experimental values is less than 1% for 118 out of the 282 calculated values, 220 being less than 2%. Bender’s original coefficients produce results which are not as good: 91 values deviate by less than 1% and 187 by less than 2% from the experimental results. This test especially, using the very sensitive isochoric heat capacity, confirms the accuracy of the new equation of state. The velocity of sound calculated with the derivatives of the equation of
49
Erperfmentol Jones .“age
results Faulkner
and Katz 119631 0
10.36bar
x
X.13
.
63.09
bar
0
55.16
bar
v
82.76
b.3~
bar
Fig. 7. Comparison of experimental isobaric heat capacities calculated by the equation of state (1) and (2). ia 0 l-
C,,
for methane
with values
:
isochoric heat capacities Fig. 8. Relative deviations AC Vm rel [eqn. (IS)] of experimental Cv,,, exp for methane (Younglove, 1974) from heat capacities CVm calculated by the equation of state (1) and (2).
! 100 -
Fig. 9. Comparison 1975) with values
j
!
200
300
pl bar
of experimental velocities calculated by the equation
of sound a for methane of state (1) and (2).
(Straty,
1974
and
state for density and temperature (Sievers, 1980) also provides a very sensitive test for the equation of state. Measurements have been performed by Straty (1974 and 1975) for saturated liquid, liquid and gaseous methane in the temperature range from 91 K to 300 K and pressures up to 348 bar. Figure 9 illustrates that calculated and experimental data agree well. CONCLUSION
Bender (1971b) published an equation of state (1) and (2) for pure fluid methane, which can serve as a basis for an equation of state for fluid mixtures with methane as one component. Since Bender determined the 20 coefficients of his equation in 1971 using experimental thermodynamic data known at that time, several new and more precise measurements of the thermodynamic properties of methane have since been published. Therefore the coefficients of the equation of state (1) and (2) were newly evaluated in this work by a simultaneous, weighted least squares fit to ppmT data, second virial coefficients, thermodynamic conditions of equilibrium for co-existing liquid and vapour phases and thermodynamic conditions at the critical point. With the new coefficients of Table 2 this equation of state represents the thermo-
51
dynamic properties of methane in the whole fluid region at temperatures between 90.68 K and 625 K, densities up to 28.6 mole dmw3 and pressures up to 500 bar with a high degree of accuracy. Only some restrictions have to be made for the critical region. Within a wide range the fit is better than with Bender’s original coefficients, especially for vapour-liquid phase equilibria and the liquid region. ACKNOWLEDGMENTS
The helpf of Mr. F.-J. Vonnahme with the development of a computer program and of Mr. M. Koschowitz with some calculations is gratefully acknowledged. LIST OF SYMBOLS
velocity of sound coefficients of the equation of state (2) :, C, D, E, F, G, H temperature functions of the equation of state (1) secondvirial coefficient &’ C Pm molar isobaric heat capacity c VIXI molar isochoric heat capacity coefficients in eqn. (14) molar enthalpy I number of ppmT data J number of second virial coefficients exponents in eqn. (14) ji L number of vapour-liquid phase equilibrium data pressure P vapour pressure PS universal gas constant R, T temperature W weighting factor molar density Pm relative deviation in isochoric heat capacity ACvrn rel molar enthalpy of evaporation Aft relative deviation in pressure &Jrel mean relative deviation in pressure AP,, relative deviation in density &An re1 mean relative deviation in density AP m re1 a
zm
Indices C
exp L max
at the critical point experimental saturated liquid maximum
52
min V 0
minimum saturated vapour ideal gas state
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53
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54 Schulz, S., 1973. Thermische oder kalorische Messungen; Wege zur Bestimmung des thermodynamischen Zustandsverhaltens. Arch. Techn. Messen, Blatt V2-1, Lfg. 453: 181184. Sievers, U., 1980. Berechnung thermodynamischer Stoffeigenschaften mit einer thermischen Zustandsgleichung fur fluide binare Gemische mit der Komponenten Methan, Argon, Stickstoff und Wasserstoff bis au Driicken van 500 bar. Dissertation Universitat Dortmund, Abt. Chemietechnik. Straty, G.C., 1974..Velocity of sound in the dense fluid methane. Cryogenics, 14: 367370. Straty, G.C., 1975. Hypersonic velocities in saturated and compressed fluid methane. Cryogenics, 15: 729-731. Teja, A.S. and Singh, A., 1977. Equations of state for ethane, propane, n-butane and n-pentane. Cryogenics, 17: 591-596. Younglove, B.A., 1974. The specific heats, C,, C,, of compressed and liquefied methane. J. Res. Nat. Bur. Stand., ‘78A: 401-410.