sievers 1980

Fluid Phase Equilibria, 5 (1980) 35-54 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

35

AN EQUATION OF STATE FOR METHANE IN THE FORM OF BENDERS EQUATION FOR TEMPERATURES BETWEEN 91 K AND 625 K AND PRESSURES UP TO 500 BAR

U. SIEVERS and S. SCHULZ

Lehrstuhl Thermodynamik, Universitiit Dortmund, Postfach 500 500, D 4600 Dortmund 50 (F.R. G.)

(Received December 26th, 1979; accepted in revised form April 4th, 1980)

ABSTRACT

Sievers, U. and Schulz, S., 1980. An equation of state for methane in the form of Benders equation for temperatures between 91 K and 625 K and pressures up to 500 bar. Fluid Phase Equilibria, 5: 35-54.

The coefficients of Benders 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 Benders 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 ther- modynamical 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 mixtures 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. Benders 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; + Ep% + Fp$

+ (G + HP&,) p%ew(--a~ p&l (1)

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

F = al3

G = alhIT2 + aI5 /T3 + als/p

H= alTIp + aI8 IT3 + a19/p

(2)

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 inspec- tion 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 knowl- edge 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 Benders 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 (a 2Pl@9,m of the equation of state and commonly used in fitting the coefficients (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

S= ~~~ilP(T~~,i~~mcl~.~)-~~~~,~l~+~~~~ly(Br(Tr.~.~)-~~~~~,jI* 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 vapour- liquid 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 vapour- liquid 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 con- verted 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 repre- sent 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 of the equation of state (1) and (2). N is the number of data points.

Kind of data Author Range N

N-9

PP~T P&T PPItlT PP~T PP T

0 PIP T ft &pm T

& T % T & T PSP~ pVT m In

Douslin et al. (1964) 273-623 Pope (1972) 126-191 Roe (1972) 155-291 Rodosevich and Miller (1973) 91-115 Goodwin (1974) 92-300 Goodwin (1974) (corr. V) 92-186 Goodwin (1974) (corr. L) 92-188 Douslin et al. (1964) 273-623 Pope (1972) 126-159 Roe (1972) 168-263 Goodwin (1974) (corr.) 92-188

p(bar)

16-405 l- 47 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 shown by 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 equation of state (1) and (2) with coefficients 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. Measure- ments 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 analo- gously. For a good representation of the liquid properties and the vapour- liquid 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 Benders empirical correlation

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 J mole1 K-l a1 J dm3 moleV2 K-l

a2 J dm3 molew2

a3 J K dm3 molee2

a4 J K2 dm3 molem2

a5 J K3 dm3 molee2

'"6 J dm6 molee3 K-l

a7 J dms moleV3

aI3 J K dm6 moleF3

=9 J dmg moleV4 K-l

al0 J dmg molew4

=11 J dm12 molem5 K-l

(112 J dm12 molep5

013 J drn15 molew6

=14 J K2 dm6 molee3

a15 J K3 dm6 molem3

al6 J K4 dm6 moleM3

=17 J K2 dml* molem5

a18 J K3 dm12 molew5

a19 J K4 drn12 molee5

a20 dm6 molee2 M e mole-l

Tmin I< T p:z bKr pinax mole dmM3

Pmax Pmc /

8.3143 0.4531925285 0.2358287986 x lo3

-0.7173842009 x lo4 0.296.1509482 x 10

-0.8735505087 x 10 0.1052328359 x 10-l

-0.1694738647 x lo1 0.8637907281 x lo3 0.8533763036 x 1O-3

-0.4017208435 x 1O-2 0.3045691216 x 1O-4

-0.2563358654 x 10-l 0.8484753417 x 1O-3 0.2545831403 x lo6

-0.5522243593 x 108 0.2250857179 x lOlo

-0.3979182974 x 104 0.1268073106 x lo7

-0.5298037215 x lo* 0.9784545300 x 10-g

16.0430 90.68

623 500

28.6 2.83

TESTING THE NEW EQUATION 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 - PrnYPIn exp

are considered for this purpose.

(8)

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 500 600 - TIK

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.

the liquid 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

t

I

) l/2 -

A~rel = + 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

Authors -

Range Z &,,I &mrel

Michels and Nederbragt (1935)

Michels and Nederbragt (1936)

Schamp et al. (1958)

Douslin et al. (1964)

Epperly (1970)

Jansoone et al. (1970)

Cheng (1972)

Pope (1972)

Roe (1972)

Rodosevich and Miller (1973)

Goodwin (1974) pm < 10 mole dmW3

Pm = 10-14.5 mole dmm3

Pm = 14.5-20 mole dmm3

P,,, > 20 mole dme3

Goodwin (1974) (corr. L)

Goodwin (1974) (corr. V)

Gammon and Douslin (1976)

Gammon and Douslin (1976) (crit. reg.)

Mihara et al. (1977)

Nunes da Ponte et al. (1978)

273-423 18- 22 56

273-423 18- 389 119

273-423 18- 261 118

273-623 16- 405 319

458-610 3- 86 155

189-194 45- 50 62

111-309 218-1144 66

126-191 l- 47 138

155-291 3- 100 82

91-115 o- 2 11

176-300 28- 182 126

192-300 48- 329 85

190-290 27- 353 74

94-220 2- 353 268

92-186 o- 40 48

92-186 o- 40 48

143-323 6- 245 127,

189-191 44- 46 61

289-348 18- 86 46

110-120 14-l 281 86

Selected ppm T data of Table 1 gaseous region (pm < pmc) 92-623

critical region (T c 1.1 T,) ._ - 187-209 (9.6 Pmc -= Pm < 1.45 Pmc)

liquid region (pm > p,,) 91-398

l- 398

47- 93

o- 405

668

44

488

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 -

- -

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

0.17 0.18 0.19 0.27 0.20 0.73 0.48 1.36 - 0.13 - 0.24

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 part- icular 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~ ;;%

i

++++ 2x + ++ 0 ++

+ --8 -A

z 1

+++ 0 +++

+ ++ 0 +++

h=l xi

+

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 Renders 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 Benders 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 Benders 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 Benders 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 Benders 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 Goodwins 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 Goodwins 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 m 90.00 1 I I I I I I I I I t I I I I I I I 1 I 100.00 110.00 120.00 130.00 140.00 L50.00 160.00 170.00 180.00 190.0

TEMPERRTURE / K

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$l ei(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 deviation 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 Benders 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 ei and exponents ji of eqn. (14) for the ideal gas heat capacity C,, for methane

i ei ji i =i .ii

1 -0.7002279 x 10-S -3 5 -43.55974 1 2 0.3740558 x 10-l -2 6 110.7293 2 3 -0.7825957 -1 7 45.42033 3 4 11.13094 0 8 29.17310 5

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,.,,T data 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 remain- ing 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%. Benders 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 results Jones .age Faulkner and Katz 119631

0 10.36 bar x X.13 bar . 63.09 bar 0 55.16 bar

v 82.76 b.3~

Fig. 7. Comparison of experimental isobaric heat capacities C,, for methane with values calculated by the equation of state (1) and (2).

ia 0

:

l-

Fig. 8. Relative deviations AC Vm rel [eqn. (IS)] of experimental isochoric heat capacities Cv,,, exp for methane (Younglove, 1974) from heat capacities CVm calculated by the equation of state (1) and (2).

! j ! 100 200 300

- pl bar

Fig. 9. Comparison of experimental velocities of sound a for methane (Straty, 1974 and

1975) with values calculated by the equation of state (1) and (2).

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 Benders 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 pro- gram and of Mr. M. Koschowitz with some calculations is gratefully acknowl- edged.

LIST OF SYMBOLS

a

:, C, D, E, F, G, H & C Pm c VIXI

zm I J ji

L P PS R, T W

Pm

ACvrn rel

Aft

&Jrel

AP,, &An re1 AP m re1

velocity of sound coefficients of the equation of state (2) temperature functions of the equation of state (1) secondvirial coefficient molar isobaric heat capacity molar isochoric heat capacity coefficients in eqn. (14) molar enthalpy number of ppmT data number of second virial coefficients exponents in eqn. (14) number of vapour-liquid phase equilibrium data pressure vapour pressure universal gas constant temperature weighting factor molar density relative deviation in isochoric heat capacity molar enthalpy of evaporation relative deviation in pressure mean relative deviation in pressure relative deviation in density mean relative deviation in density

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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sievers 1980

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