A Simple Cubic Equation of State for Hydrocarbons and ...hydrocarbons, hydrocarbon mixtures, and...

SPE Paper # 26667

A Simple Cubic Equation of State for Hydrocarbons and other Compounds Mohsen Mohsen-Nia 1, U. of Illinois at Chicago; Hamid Moddaress, Amir-Kabir U.; and G.A. Mansoori, * U. of llllnols at Chicago

* SPE Member

ABSTRACf

A new simple two-constant cubic equations of state for hydrocarbons, hydrocarbon mixtures, and other nonassociating compounds is introduced. This equation of state is based on the statistical mechanical information available for the repulsive thermodynamic functions and the phenomenological knowledge of the attractive intermolecular potential tail contributions to the thermodynamic properties. This new two-constantparameter cubic equation is in the following form:

<I Ia11)/RT Zin ::oh: V + 1.3191.tgi � _1 "'°'I __ _

RT v -.tm T'12(v + I $1)I

whera .tm = (1/4)(31: I XiXj $i; + I X1 .tu),

o _ (o 1/3 + I, 1/3)3/8.

c,ij - c,11 jj ,

I I I

and a11

= 0.48748 R2T al 512 /P cii and I, 11 = 0.064662 RT cit IP al

It is shown that this equation of state is more accurate than the Redlich-Kwong equation, which has been considered to be the best two-constant-parameter cubic equation of state. This new equation is applied for calculation of thermodynamic properties of hydrocarbons and other non-associating fluids and fluid mixtures of interest in the oil and gas industries.

651

INTRODUCTION

The success of oil and gas recovery, production, and processing operations is related to the availability of accurate and simple equations of state for prediction of thermodynamic properties of fluids. Recent developments in modeling the phase behavior of reservoir fluids indicate that more serious considerations must be given to fundamental approaches to produce simple and accurate equations of state1•9• Although, there has been considerable progress made in the development of equations of state based on the statistical mechanics, the cubic equations of state are still widely preferred in industrial calculations. Equations such as the van der Waals, Redlich-Kwong, Heyen and their modifications are extensively used in petroleum andnatural gas engineering calculations1•7 •

In this report, we first present a simplified, but accurate, equation for the positive (repulsive) term of equations of state. Then we introduce a new version of cubic equations of state using the simplified repulsive term developed in part 1. A new cubic equation of state which contains two constant parameters is then introduced. The resulting equation of state is used for property prediction of a variety of hydrocarbons, other fluids, and mixtures of practical interest. This equation of state is shown to be a simple cubic equation with superior accuracy than the RI< equation which has been considered to be the most accurate two-constant-parameter cubic equations of state.

DOI: http://dx.doi.org/10.2118/26667-MS

This paper appeared inProceedings of the 68th Annual SPE Meeting, Pages 651-659, held in Houston, TX, 3-8

October 1993

Authors email addresses: M. Mohsen-Nia ([email protected] ); H. Modarress ([email protected] ); G.A. Mansoori ([email protected])

(1). Present address: Kashan University, Kashan, Iran

2

ntE NEW REPULSIVE TERM OF EQUATIONS OF STATE

The importance of short-range repulsive forces in an equation of state for determining the thermodynamic properties of fluids is well established by the works of numerous investigators.2•6,7·11•16• This is because moreaccurate expression for the repulsive term of an equation of state results in a better agreement with the experimental data. However, the major weakness of the industrial equations of state is due to their inaccurate repulsive term,

(1)

An accurate equation for representing the repulsive effect is the MCSL equation which reduces to the following form for pure fluids 11:

This equation can replace the repulsive term, v /(v-b), in the cubic equations. However, due to the fact that after this modification the resulting equation of state was not going to be cubic any more Scott14 introduced the following equation for the repulsive term of pure fluid equations of state.

v min appearing in Table 1 is the smallest molar volume for fluids which can be calculated based on different equations of state (approximately the molar volume at the triple-point) when the MCSL repulsive equation of state is used to modify the above equations (replacing v /(v0b) with eq. 2). Of course the MCSL equation of state is valid for v /b values as low as 0.25. Considering the fact that the v /b in the cubic equations of state (excluding the van der Waals equation which, because of its inaccuracy, is not used for practical calculations) do not go below 0.7 we propose L'le following repulsive equation of state for v /b > 0.7:

Z = Pv /RT= (v + 0.62b)/(v-0.47b) rep (5)

This equation reproduces the MCSL equation for the repulsive fluid with good accuracy for v /b > 0.7 as it is demonstrated in Figures 1 and 2. In Figure 1 the percentage deviation of calculating the repulsive compressibility factor from the exact MCSL equation as calculated by eq. (5) is compared with the Scott and Kim, et.al models. According to this figure eq. (5) is in better agreement with the MCSL equation than the other simplified repulsive equations for v /b > 0.7.

In Figure 2 the repulsive excess Helmholtz free energy (over the ideal gas) ,

Z<�> = Pv /RT= (v + 0.5 b)/(v -0.5 b) (3) (A-A18)/RT = - 2.319Ln(l - 0.47b/v) (6)

Similarly, Kim, et al. (1986) presented another equation for repulsive term of equations of state:

Z<�> = Pv /RT= (v + 0.77 b)/(v - 0.42 b). (4)

Neither of the above two equations are accurate enough to replace eq.(2) as it is demonstrated in Figure 1. In order to produce a simple, but accurate, repulsive term for cubic equations of state, we consider a number of representative cubic equations of state in a generalized from as shown in Table 1.

Table 1: Parameters of a generalized equation of state E. 2

Z = Pv /RT= Z(rep) -(av /R1) / [T (v +/!Jv-&bc)]according to different cubic equations

Equation of State

Van der Waals Heyen Peng-Robinson Redlich-K wong

E

0 0 0 1/2

0 1>H: :.b b

0 1 b/c 0

0.581 0.806 0.833 0.893

652

as derived from eq. (5) is compared with the similarcalculations using the Scott and Kim, et.al models.According to this figure eq. (6) is in better agreementwith the accurate MCSL equation of state than the othersimplified repulsive models.

We have extended the applicability of equations (5) and(6) to repulsive mixtures by writing them in thefollowing forms

zm(rep) = (v + 0.62bm)/(v - 0.47bm) (7)

(A - A 1

)/RT= - 2.319 Ln(l - 0.47bm

/v) (8) m mg

and using the following mixing and combining rules,

bm= (1/4)(3L I, XiXjbij + L Xi bu)

where;

I I

3 b,. = Cbu t/3 + b..113) /8

I II

(9)

Eqs. (7) and (8), joined with eq. (9), are tested extensively for discrete as well as polydisperse repulsive mixtures and it is shown to be in very good agreement with the

(2)

M. Mohsen-Nia, H. Modarress, G.A. MansooriA Simple Cubic Equation of State for Hydrocarbons and Other Compounds

SPE Paper # 26667, Proceed. 68th SPE Ann. Tech. Conf. & Exhib., Houston, TX, USA, 3-8 Oct. 1993 ____________________________________________

3

accurate, but complicated, MCSL11 repulsive mixture equation for v /b > 0.7. We have calculated the thermodynamic properties of equimolar binary mixtures (x1=x2=1/2) with (b22/b11)113 = 1.5, 2.0, 2.5, and 3.0. Theresults of these calculations are compared with the accurate MCSL repulsive mixture equation of state as reported in Figures 3 and 4. The maximum percentage deviation of the binary repulsive mixture compressibility factor and excess Helmholtz free energy for molecular diameter ratios of (b2ib11)113 = 1.5 to 3 is calculated to be 3% for v /b > 0.7, which is in the range of accuracy of MCSL equation.

In order to test further the accuracy of the proposed repulsive term of the mixture equation for multicomponent mixtures we have used it to calculate thermodynamic properties of polydisperse (continuous) repulsive systems through different distribution functions f(r). Following the method of Salacuse and Stell16 we can express eq.(9) for the polydisperse systems as the following:

bill

= (1/16) (7� + 9Jltil'lli) (10)

Where, m1, ffii and m3 are the first, second and third

moments of distribution function, respectively, defined as:

II\ = [ r• f(r) dr

For the normal distribution:

the moments are given by

II\= r + (2'1 /n)1n.

Mz = f 2 + 11 + 2(211 /n)112

II\= r 3 + 2(2/n)11'Lri312 + 311 r + 3 r 2(211 ht )112

(11)

(12)

(13) (14) (15)

where 11 and r are the variance and mean, respectively. Therefore, the mixture bm/v, eq. (10) for the normal distribution is as the following:

653

We can calculate the compressibility factor and Helmholtz free energy of this polydisperse repulsive mixture by joining eq. (16) with eq's (7) and (8).

For the Schultz distribution:

f s(r) = _l (S !. 1 )s + 1 r8 [-(S : 1 ) r]

sl r r (17)

where r is the first moment of the distribution, and s is related to variance of f

8(r) by,

-2T\ (r) = var(r) = ..1...-Js+l

(18)

we can derive the following expressions for the bu/v of the Schultz polydisperse repulsive mixture:

b /v=4 ( pNo? )((s + 2)(8s + 15))m 6v B(s + 1)2

(19)

Then by replacing this equation in eq's (7) and (8) we can calculate the compressibility factor and Helmholtz free energy of the Schultz polydisperse repulsive fluids.

We have calculated the thermodynamic properties of polydisperse repulsive mixtures obeying the normal and schultz distributions. The results of these calculations are compared with the accurate polydisperse MCSL repulsive equation of state11 in Figures 5 to 8. According to these figures the deviations of the multi-component repulsive mixture compressibility factor and Helmholtz free energy fo.!:_ Shultz (with f=l & T)=l/3,1/6,1/11) and normal (with f=l & h=0.1,0.3,0.5) distributions are quite low and acceptable for v /b > 0.7 range of interest in industrial equations of state.

We have also made calculations for values of r other than unity with similar results as to Figures 5-8 which are not be reported here due to space limitation.

At this stage we propose the following modified equation of state:

Z=Pv/RT

= (v+0.62b)/(v-0.47b)-(av /RT)/T \v2+tv-dbc) (20)

The difference between this equation and the generalized equation reported in Table 1 is the replacement of the new repulsive compressibility factor given by eq (5). Provided the proper numerical values of E., I, , and & are replaced in this equation one can reproduce the new versions of the equations of state



4

reported in Table 1. It should be pointed out that this change of the repulsive term will not keep the PengRobinson and Heyen equations in cubic forms any more.

By a pplying the critical point constraints: (oP /ov)T =(o2P /ov2)r =0 (for all equations); and Z = Z

c C C

(for the three-parameter equations like Heyen) we can derive expressions for parameters of the new version of equations of state with respect to the critical properties as reported in Table 2.

Table 2: Parameters of eq. (20) with respect to the critical properties.

Equation of State Original

Van der Waals 1.429 u.542 0 0.364 o.375

Heyen t.945' 0.371" 0.991 0.27 0.27 Peng-Robin,on 1.951 0.370 0 0.272 0.307

Redlich-Kwong 1.513 0.341 0 0.314 0.333

"These values are based onZc=0.27.

According to this table, except for the Heyen equation for which Zc is an input parameter, the Z

c values calculated

based on the hew version of the other equations of state are closer to the experimental z, values than the Z/s calculated by the original form of the equations of state.

THE NEW TWO-CONSTANT-PARAMETER CUBIC EQUATION OF STATE

By replacing ci. =0 in eq. (20) it will become a cubic equation in the following form

Z = (v + 0.62b)/(v - 0.471>) - (a/RT)/'JC(v + £.). (21)

The choices of E, and I, are still in our disposal. For the sake of simplicity we choose f. =1/2, as proposed by Redlich and Kwong10, and t=0.4i'b as appears in the denominator of the repulsive term of eq. (21). As a result we introduce a new cubic equation of state in the following form:

Z = (v + 1.13191)/(v -t)-(a/RT)/T 112(v +£.) (22)

Equation (22) is a cubic equation of state in terms of

volume and contains only two constant parameters. By applying the critical point constrains, we determine parameters a and I, as the following:

a = 0.48748 R2Tc

512 /Pc

and I,= 0.064662 RTc /Pc (23)

The critical compressibility factor Zc=0.333 will remain the same as the RK equation of state. In order to demonstrate the validity of the repulsive term of this equation of state we have calculated v

m10/b as reported

in Table 3 for different fluids. According to this table for all the fluids studied v mln/b values are in the allowable range of v /b > 0.7 where the repulsive part of eq.(22) accurately represents the MCSL repulsive equation of state.

TableJ: Calculation of the minimum v /b ratio for different fluids according to Eq. (23)

Fluid Vmln/b

Ethane o.76n-Butane 0.76n-Propane 0.73Oxygen 0.80lsobutane 0.73Ethylene 0.80Np, 0.89H

iS 0.86

N 3 0.73

Extension of equation (22) to mixtures will take the following form:

(! l:a11)/RT Z = h. = v + 1.319ltm _ 1 i (24)

RT v - tm T112(v + I, 8-i)

where

and

tm = (l/4){3l: l: XiXj t11 + I Xi tu) I J I

and

In eq. (24) we have applied the new mixing rule for repulsive term (eq. 9). For the attractive term, we have used the usual cubic equation mixing rules in order to be able to observe the corrections in the repulsive term of this equation for mixture calculations. It should be pointed out that there are also other possibilities for the mixing rules of the attractive term of this equation of state11• This new equation of state is used to predict properties of pure fluids and mixtures as it is reported below.

654



5

RESULTS AND DISCUSSION

Cubic equations of state may be classified into two categories: (i) Those with two constant parameters like the van der Waals, Berthelot, Dieterici, and RedlichKwong (RK) equations. (ii) Those with more than two parameters (or two or more temperature-dependent parameters) and they include the Peng-Robinson, Heyen and Soave equations1•7• Among the first category of cubicequations of state the RK equation is still considered to be the most accurate equation of all. As a result in order to test the accuracy of eq. (22), which is also of the first category of equations, for prediction of the properties of pure fluids and mixtures it is compared here with the RK equation. Table 4 shows the comparisons of the result of density calculations using eq. (22) with the RK equation and the experimental data. According to this table the result of eq. (22) are generally superior to the RK equation for 17 out of 19 pure fluids tested.

In order to test the applicability of eq. (24) for mixtures, we have used it for calculation of the properties of a number of systems. In all the calculations reported here for eq. (24), and for the RK equation, we have chosen the value of the unlike-interaction parameters, k

i/s, equal to zero. This choice of k

1i=O is made in order not tocomplicate the comparisons of different equations of state with the fitting of their k

ii's. However, for moreaccurate calculations of mixture properties this parameter may be fitted to some mixture data.

On Figures 9 and 10 the density calculations for two different binary high-pressure (500 bar) liquid mixtures (n-octane + n-dodecane and n-decane + n-tetradecane) are reported and are compared with the results of the RK equation. According to these figures predictions by eq. (24) are better than the RK equation.

We also have performed solubility and VLE calculations for mixtures. On Figure 11 the solubility of methane in nhexane is reported versus temperature at two different pressures of 40.5 and 81.1 bars. According to this figure also the predictions by eq. (24) are closer to the experimental data than the RK equation.

In Figures 12 to 14 equilibrium pressure-composition diagrams for methane + n-decane (at 344.26 K}, carbon dioxide + n-decane (at 344.26 K), and for nitrogen + carbon dioxide system (at T = 270.0 K) are reported. According to these three figures the predictions by eq. (24) are far better than the predictions by the RKequation.

655

As it was mentioned earlier , in all the calculations reported here, the unlike interaction parameters, k

1/s, are assumed to be zero. By choosing appropriate non-zero k

11 values the proposed equation of state would

accurately correlate and predict properties of the mixtures reported in this paper. The k

11 data bank for

this equation is not reported here due to space limitation.

The proposed equation of state may be made even more accurate by choosing its parameters (a & b) to be dependent on temperature as it Is usually done in other industrial equations of state1•7•

ACKNOWLEDGEMENTS

This research Is supported in part by the National Science Foundation, Grant No. ClS-9108595.

NOMENCLATURE

A a b C f,(R) KiiNop

R

r

r

T X

z s

= = = = =

= =

= = = =

=

= =

=

Greek letters p

1t

Tl

= =

=

Subscripts C = i,j = ij = ig = m = N = s =

Helmholtz free energy attractive constant in equation of state repulsive constant in equation of state third parameter in equations of state distribution function binary interaction parameter Avogadro number pressure Universal gas constant random variable the mean value and first moment in normal and Schultz distribution function respectively absolute temperature mole fraction compressibility factor a parameter in Schultz distribution function, defined in eq.(19)

density 3.1415927 variance of normal distribution function

critica.l property component indices property of i-j interaction ideal gas property mixture property normal distribution Schultz distribution



6

REFERENCES

1, Edmister, W.C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics, Gulf Pub. Co., Houston, TX, 1986.

2. Elliot, J. and Daubert, T.: "Revised Procedure forPhase Equilibrium Calculations with Soave Equation ofState," I & EC Proc. Design & Develop., 1985, vol 23, pp.743-748 .

3. Du, P.C. and Mansoori, G.A.: "Continuous MixtureComputational Algorithm of Reservoir Fluid PhaseBehavior Applicable for Compositional ReservoirSimulation," SPE Paper# 15953, Proceedings of the 1986Eastern Regional Meeting of SPE and Proceedings of the9th SPE Symposium on Reservoir Simulation, Society ofPetroleum Engineers, Richardson, TX, 1986.

4. Benmekki, E.H. and Mansoori, G.A.: "MinimumMiscibility Pressure Prediction with Equations of State,"SPE Reservoir Engineering,, May 1988, 559°564.

5� Chom, L.G. and Mansoori, G.A. (Editors),: C7+ Fraction

Characterization Advances in Thermodynamics, Vol. I, Taylor & Francis Pub. Co., New York, N.Y., 1988.

6. Mansoori, G.A. and Savidge, J.L.: "PredictingRetrograde Phenomena and Miscibility using Equationsof State," SPE Paper No. 19809, Proceedings of the 1989Annual SPE Meeting, Society of Petroleum Engineers,Richardson, Texas, 1989.

7. Ahmed, T.: "Hydrocarbon Phase Behavior", Gulf Pub.Co., Houston, TX, 1989.

8. Kawanaka, S. Park, S.J. and Mansoori, G.A.: "OrganicDeposition from Reservoir Fluids" SPE ReservoirEngineering Journal, May 1991, 185-192

9. Escobedo, J. and Mansoori, G.A. "Heavy OrganicDeposition and Plugging of Wells (Analysis of Mexico'sExperience),: " Proceedings of the JI LAPEC, SPE Paper #23696, Society of Petroleum Engineers, Richardson, TXMarch, 1992.

10. Redlich, 0., and Kwong, J.N.S.: "On Thermodynamicof Solutions V," Chem. Rev., 1949, 44, 233.

11. Mansoori, G.A., Carnahan, N.F., Starling, K.E., andLeland, T.W.: "Equilibrium Thermodynamic Propertiesof the Mixture of Hard Sphere," J. Chem. Phys., 1971, 54,1523; see also Carnahan, N.F., and Starling, K.E.:"Equation of state for nonattracting spheres," J. Chem.Phys., 1969, 51, 635o636.

12. De Santis, R., Gironi P., and Marrelli, L.: "VaporLiquid Equilibrium from a Hard Sphere Equation ofState," Ind. Eng. Chem. Fundamentals., 1976, 15, 183.

13. Kim, H., Lin, H.M., and Chao, K.C.: "Cubic Chain ofRotators Equation of State," Ind. Eng. Chem.Fundamentals., 1976, Vol. 25, No. 1, 75-83.

14. Scott, R.L.: Physical Chemistry, an AdvancedTreatise, Henderson, D., ed., Vol. BA, Chapt. 1,Academic Press, New York, 1971.

15. Haile, J.M. and Mansoori, G. A.: Molecular-BasedStudy of Fluids, Advances in Chemistry Series, Vol. 204,ACS, Washington, D.C., 1983.

16. Salacus, J.J. and Stell, G.: "Polydisperse System:Statistical Thermodynamic, with Applications toSeveral Models Including Hard and Permeable Spheres,"J. Chem. Phys., Oct. 1982, 77(7), 3714-3725.

17. Mansoori, G.A.: "Mixing Rules for Cubic Equation ofState," ACS Symposium Series 300, 1986, Part 15, 314-330, , Washington, D. C.

18. Takagi, T., and Teranishi, H.: " Ultrasonic Speed andThermodynamics for Binary Solutions of n-AlkanesUnder High Pressures," Fluid Phase Equilibria, 1985, 20,315.

19. Shim, J. and Kohn, J.P.: "Multiphase and VolumetricEquilibria of Methane-n-Hexane Binary System atTemperatures Between 110° and 150°C," J. Chem. & Eng.Data, 1962, 7, 3-8.

20. Reamer, H.H., and Sage, B.H.: "Phase Equilibrium inHydrocarbon Systems, Volumetric and Phase Behaviorof the n-Decane+Carbon dioxide System," J. Chem. Eng.Data, 1963, 8, 508-S13.

21. Reamer, H.H., Fiskin, J.M., and Sage, B.H.: "PhaseEquilibria in Hydrocarbon Systems, Phase Behavior inthe Methane-n-Butane-n-Decane System," Ind. Eng.Chem., 1949, 41, 2871-2875.

22. Somalt, F.A. and Kidnay, A.: "Liquid-VaporEquilibria at 270 K for Systems containing Nitrogen,Methane and Carbon dioxide," J. Chem. Eng. & Data, .1978, 23, 301-305.

656



'l'al)le 4 s Capareion of this work and 11K eqw,tions ot stllte tor d•nsity pr.diotion ot diff•r•nt coapounds,

Coapound 'l' .. por•ture bnqe, CJ!)

CO2 aoo-1000 H28 300-700 so2 a85°500 11a 200-1000 02 200-1000 !!ethane 91-500 Propane 200-700 i-sut•n• aoo-100 n-P•ntane 300-650 llaopetan• 300-600 n-H•x•n• 298-1000 n·H•ptan• 300-1000 n-octan• 320-1000 n-Nonan• 300-600 n-oec:an• 300-500 n-Undea .. no 300-570 n•OOdeaan• 300-570 n-Tridaoan•l00-570 n-c15 310-410

Overall

Reduced Oensltv

flpn S • The percentage deviation of compressibility factor from the exact mlxlur1 MCSL bard sphere cq11atlon vcra111 th• mixture reduced density, lb11•bullav), and

for dlffmnt 111olecvlar dlNlltlU nlloa for l>lrtasy 1111xlure1 of hard sphere fluldf.

Preaure Ranqe, (&ta)

40-400 5-400 1-315 10-700 5•500 10-700 5-700 10-600 10-600 5-400 1"500 1-500 1-1100 15-1000 15•500 50-1000 1-600 50-1050 1-600

657

No, of MDI l>&ta

143 101 115 183 120 170 140 130 160 60 100 130 80 90 130 40 20 50 30

1992

iiJr" THIS HORK

3,9 2,4 1,8 2,2 5,0 2,6 3,4 2,8 1,9 l,7 3,0 3,4 3,1 2,6

3,3 3,1 4,6 3,9 3,1 2,8 5,7 3,3 5,8 2,8 6,3 2,2 7,9 2,6 8,8 2,5 10.3 3,2 11,3 3,7 13,3 4,7 15,G 6 ,4

6,2 3,1

Figure,· The er<cess Helmhollz free energy according to Eq. (91 and the exact MCSL hard aphere mixture equation of stale versus the reduced density, (b11+bJ/(8vl, for

the binary mlxlure hard sphere fluid wllh (bJb1J111 "3, For (b1/b1�111 ratios !us than 3 lhe two theorlu would match even bettor than this reported cue.



N

v., .• 0m D 0011 01 0111 02 0211 03 0116 O<I

fledU* DanllV Flp• I• 'lbt pt,.nlqt dnlallon al complltllblD!y l1<1or 1rGm lht dHI mlllort MCSL hold ophtrt ,._111on ol olalf lot lht polydlopo11t hold ophtrt a,I...,. wllh no .... l dllbt1,ollol\,•t11_, lht lllht•N nd_, dtntlly, WWN,Pm, old for 1111n .. 1., al r • l tad dllffftlll ••rltncu. '11tt lhlrd momeal. "'r al lht normal dlMrlbllllonhddlatdbyc�.IJII.

41..---------------------,

:t

.,. .,. -

Vlr.•1/3

·3' •••• Var.=1/11

D 11011 o1 0111 02 11211 Reduoed Denllty

. .. ,, ...... 03 0311 ...

flpn 7 • TIit ,. ........ dmolhla of .................. compnnlbRlty , ..... , f,011 thc _11...,.,. MaL hud ophtN ...,11loa al- forlht polydlopcnt h1NI 1pht1W mbda1t wllh lhlilla dllbl .... lloa, -• lht mlllurt rtduttd dt•lly, �MHIP, ind !Of 11,cn •II• of i • I ind 1111- .,rlcnott.

i O .. ·-·--.. -·--·· .. -1·

'o ·10 g

i ·20

.. ., ..... ,._

NEW

ff(

....... ______ _,,_._._ ..... --1 o.o 0,2 o... 0.6 0.8 1.0

fl&,,rt t • TIit fA!cullltd pttttnt,p dt•llllon ol ln«t- ♦ n-dodtcoat) ll••ld mblurt dtnllly lnlm lht aptrlDltflltl doll IIIL occonllna to lht p- and RK .,.,,._ ...,..iloM al llllt llc,fGI. .,,.., ..-.... , 11alr lndlon II T-.u K ••• ......

2.w

I r·

,·

i 1,6· , .. ,

, .. ,...,,·,.,;

�

a&

,., •• Jt•' , ..

,,,.,,,... -�'

.;e•',,,,.

0o 0011 01 Q 111 02 Qati Reduced Del'ICl!y

HtUtll

MCSL

QB 1136 0.4

Flaon 6. Tho -llllmholla Inc,...., occOftll,. .. � I'll ond 1111 no,1111athold ophctt mlll•" �uall•a al llolt .,,.., tu nd•td dtntlty, IJIIINtP•• i., tlM polydllftnt hard ,,.,... """"'" with aomol dl,tdb.llon with - ..... of ; � I ... ,..tua • O.l Por .-... .... lhon IU lht ..... lhcodts woold m11Ch -llctlftthaathloNpONd-

2

MCSl..(vlr,•1/�_,, MCSI., (var •.. 111,, •

1.11 Eq(SO,(vlr.•1� D

Eq(lill,(V•.•1/11)

1

006 01 0111 02 11215 Rllcllnd 01nlly

03 0311 Q◄

Flgurtl • n,.-,h ..... phtrt ml••lldmholll 1 ....... 'IY ..... ldla910 '4.t 1nd lht Oll(I MCSL hotd 1phttt ma.i.n �•adon al lhlt ¥111UO lht ...... dtntlly, llllllylt+lllMIMotll', for lht polydl.,.IN hw ophm 11blun wllh lhuldllll'lbulloa wll• a,w, •1IH of;• l anf •lrl-■ Ill • tilt.

10

l

'Ii &

·10

IN&W

-ao A(

� .00+----.,_. ...... -.---,.------1

o.o D.2 0.4 0.8XI 0.8 1.0

fls-10. ,'h, colwltltd pt•llllp ... lalloaol 111.._ ♦ -◄-tradocanc) n'l"kl mlwlurt dtntlty ln,a, lht taptd-1 ,a1o IIIL occotdi. lo lht pnotnl aod U mlalun .. ,lloo1 of olalt I.._.CIL .,,.., ll•lllndt<- malt l11C1loa II T�ll It -·---


SPE Paper # 26667, Proceed. 68th SPE Ann. Tech. Conf. & Exhib., Houston, TX, USA, 3-8 Oct. 1993____________________________________________

l I s

I i

I

1.0

0.8

0.8

0,4

P • 40,5 blr: - NEW , ... , ........ , ... F 0

p • $1.1 bar. - NBW '"'"'''"' .... F •

, ..... ,, .. .........,

.,.....,""""' ........... " ... ..............

.... "" ....._

�.._,-........,

Exp.

�.

-·-............ , ............ , ....... i_,_.,..,..0.2

0.0 200 250 Temperature k aoo

FIi•" II , Prt41ctlON of Iii, 19Jublll7 of 11tth,ne lo 1,hn1nt accor41aJ to !htpmtnl Ind IIK 11lt1u11 1111Uo11t 11 11.111 lkirOI, """' ltmp111ture at IWOdlllcrtnt prtt1u11� n, 11J11rl111nt,i data rtponcd on 1h11 111ph 111 11ktn from 11,,11,�

400---------_______ .., - N!W

0

,: soo J

0

i 200

100

0

o+-__________ ,_..,..__,,_--4

0.0 o., 0.8

(Xi )1(Y1)

0.8 1 .o

Fl&u•• U , Prt41dlon ,1 lb, ,4.t1111r1um pr111111M1111pttltlon dt,.,am for (lllltllint • n-4tul'III 1111111 accordfns t• "1• pmu1 •14 Ilk �11tlo!lf o1 ,1,1, :r).lllff.UX, 1ht �llll\� ebb npl)rttd 0� 1h111'1Ph Id 111111 frOII lttf.

350

-•

.., •..

'IIII• .. IL

e" �'" •

E to

lW

140

N&W T • 344.28 K

120 A( 0 0 0 Exp,

100 0

0 80

0 80 0

40 0

0

0,2 0,4 0.6 0.8 1 .o

(X1 ),(Y1)

flgvrt U, P!1111ctlon of 1h1 ,qulllbrlum preuurc-coa1po1lllon dlllfllll to, (urbDd dioxide• n'lltcwl ty1lt1111ccor,llng to llit pm,nl and RK •,u,Uona of 1111, !kr,at MUI K. Th, up1rlm1ntal d1t1 rtpor11d on 1h11 gr1ph m t1ktn from 11,1. 110.

.. O.I 02 .,

(X1 ),(VI)

0

l'lt1H NIW RK

EMP 0

u 0.1

JflWf l& • P11411tlvn �, tht t\tlllbrl1111 p1ttm1-cot1pt1l1ten dhgrun forW1rttt11 + ,ui,on dloatikl 1,alrai 1m1rdlos lo Ibo ptMUI 1nJ � ,..,,don, of 11111 lkif1L ti T • m K. lbt rrperimental d1ta ,.,1111,d np 1� IIIPh IN lllctn hOm 1,i112�


SPE Paper # 26667, Proceed. 68th SPE Ann. Tech. Conf. & Exhib., Houston, TX, USA, 3-8 Oct. 1993____________________________________________

659

A Simple Cubic Equation of State for Hydrocarbons and ...hydrocarbons, hydrocarbon mixtures, and...

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