calculating and applying

8/2/2019 Calculating and Applying

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Calculating and Applying K-Values

By

Abdulreda Al-Saygh and Mahmood Moshfeghian

Department of Chemical Engineering

University of Qatar, Doha, Qatar

And

Robert N. Maddox

School of Chemical Engineering

Oklahoma State University

Stillwater, Oklahoma, U.S.A.

ABSTRACT

Several methods for determination of K-values were reviewed and sample results are presented. Severalcase studies were presented to demonstrate their application. Based on the VLE calculations results, forlight hydrocarbon systems, an equation of state should be used and for polar systems, an activitycoefficient model is recommended.

Introduction

Modeling and design of many types of equipment for separating gas and liquids such as flashseparators at the well head, distillation columns and even a pipeline are based on the phases presentbeing in vapor-liquid equilibrium. The criteria for thermodynamic equilibrium between vapor and liquidphases are equality of temperature in both phases, equality of pressure in both phases, and equality ofugacity of each component in both phases. The mathematical expression for the last equality in terms

of the fugacity of component i, is written as:(1)

Equation (1) is the foundation of vapor-liquid equilibrium calculations, however, we rarely use itin this form for practical applications. For calculation purposes, Equation (1) is transformed to a morecommonly used expression:

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(2)

In Equation (2), Kiis called the vaporliquid equilibrium constant. Equ. (2) is also called "Henrys law"

and K is frequently referred to as Henrys constant. For the more volatile components in a mixture theK-values are greater than 1.0, whereas for the less volatile components they are less than 1.0.

Depending on the system under study, any one of several approaches may be taken to determineK-values. Obviously, experimental measurement is the most desirable; however, it is expensive andtime consuming. Alternatively, there are several graphical or numerical tools that can be used fordetermination of K-values. This paper presents a history of the development of many of those graphicalmethods and numerical techniques.

In general K-values for all components in a mixture are function of the pressure, temperature,and composition of the vapor and liquid phases present. The components making up the system plustemperature, pressure, composition, and degree of polarity affect the accuracy and applicability, andhence the selection, of an approach to estimating the K values. The widely used approaches are K-value

charts, Raoults law, the - approach and the - approach [1-5]. The last two approaches involveusing an equation of state.

Methods for Determining K-Values

K-Value Charts

There are several forms of K-value charts. One of the earliest forms of the K-value charts forlight hydrocarbons is presented in reference [1]. In these charts, K-values for individual components areplotted on the ordinate as a function of temperature on the abscissa with pressure as a parameter. Ineach chart the pressure range is from 10 psia to 1000 psia and the temperature range is from 40 F to 500F.

Early high pressure experimental work revealed that, if a hydrocarbon system of fixed overallcomposition was held at constant temperature and the pressure was increased, the K-values of allcomponents converged toward a common value of unity (1.0) at some high pressure. This pressure wastermed the Convergence Pressure of the system and has been used to correlate the effect ocomposition on K-values. Plotting this way permits generalized K-values to be presented in a moderatenumber of charts.

In more recent publications [2], the K-values are plotted as a function of pressure on the abscissawith Convergence Pressure and temperature as parameters.

In order to use these charts, one should determine the Convergence Pressure first. Thedetermination of convergence Pressure is a trial-and-error procedure. Illustrative example calculationscan be found elsewhere [6].

In 1958, for computer use, these K-Value charts were curve fitted to the following equations by




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academic and industrial experts collaborating through the aegis of the Natural Gas Association oAmerica [7].

(3)

Raoults Law

Raoults Law is based on the assumptions that the vapor phase behaves as an ideal gas and theliquid phase is an ideal solution. Under these conditions the fugacities are expressed as

(4)

(5)

Substituting from Equations (4) and (5) into Equation (1) gives

(6)

The vapor pressure may be read from a Cox chart or calculated from a suitable equation in terms oftemperature. A typical Cox chart can be found in reference [8]. The Antoine [5] equation is

recommended for calculating vapor pressure.

Complex vapor pressure equations presented by Wagner [5], even though more accurate, should beavoided because they should not be used to extrapolate to temperatures beyond the critical temperatureof each component. Also, Raoults law is applicable to only to low-pressure systems (up to about 50psia) and to systems in which the components are very similar such as benzene and toluene. Thismethod is simple but it suffers when the temperature of the system is above the critical temperature ofone or more of the components in the mixture. At temperatures above the critical point of a component,one must extrapolate the vapor pressure, which frequently results in erroneous K-values. In addition,this method ignores the fact that the K-values are composition dependent.

Correlation Method

As mentioned earlier, determination of K-values from charts is not suitable for computercalculations. Therefore, scientists and engineers have developed numerous curve fitted expressions forcalculation of K-values. However, these correlations have limited application because they are specificto a certain system or applicable over a limited range of conditions. Some of these are polynomial orexponential correlations in which K-values are expressed in terms of pressure and temperature. One othese correlations presented by Wilson [9], is:




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(7)

This correlation is applicable to low and moderate pressure, up to about 3.5 MPa (500 psia), and the K-values are assumed to be independent of composition.

- Approach

The - approach is based on using an Equation Of State (EOS). The fugacity of eachcomponent is determined by using an EOS. The same EOS describes both phases. The - approach isa powerful tool and it is relatively accurate if used appropriately. This approach is widely used inindustry for light hydrocarbon and non-polar systems. In this approach the fugacities are expressed by

(8)

(9)

Substitution of fugacities from Equations (8) and (9) in Equation (1) gives

(10)

In equation (10), fugacity coefficients must be determined from a generalized chart or calculated usingan EOS. In order to calculate the fugacity coefficients for a mixture by an EOS, the mole fractions in

both phases are needed in addition to pressure and temperature. Normally not all of these variables areknown. Therefore, calculation of K-values using an EOS is a trial and error procedure. The -approach is applicable to non-polar systems and yields good results up to about 15,000 psia.

- Approach

The so-called -approach is also based on using an EOS, but requires that the vapor phase non-ideality be described through the fugacity coefficient, with an activity coefficient model being used toaccount for the non-ideality of the liquid phase. This approach is widely used in industry even for polar

systems exhibiting highly non-ideal behavior. Using this model the fugacities are expressed by

(11)

(12)

The saturation fugacity coefficient for a component in the system, , is calculated for purecomponent i at the temperature of the system but at the saturation pressure of that component.




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Normally, an EOS is used to calculate both and . Substitution of fugacities from Equations (11)and (12) in Equation (1) gives

(13)

Activity coefficients are calculated by an activity coefficient model such as that of Wilson [11] or theNRTL (Non-Random Two Liquid) model [12]. The required parameters for a few binary systems aregiven in reference [13]. In order to calculate the K-values by Equation (13), the mole fractions in bothphases, in addition to pressure and temperature, are needed. Normally not all of these variables areknown. As is the case for the -approach, calculations are trial and error. The -approach isapplicable to polar systems such as water ethanol mixtures from low to high pressures.

Approach

Normally, for low pressures, we can assume that the vapor phase behaves like an ideal gas;

therefore both and are set equal to 1.0. Under such circumstances, Equation (13) reduces to

(14)

Equation (14) is applicable for low-pressure non-ideal solutions and polar systems. Assuming the liquidphase is an ideal solution, i becomes unity and Equation (14) is reduced further to a simple Raoults

law.

The Impact of Computers on calculating K-values

The accuracy of any process simulation by computer software depends directly on the accuracyof the K-values used. The K-values are the essential ingredient for design and simulation of a separationsystem involving distillation columns, flash separators, etc. As computers were developed forengineering calculations, scientists and engineers strived for the development of generalized andaccurate EOSs. Since the K-value charts are limited and simplified and calculation of K-values based anEOS is trial and error and, consequently tedious and time consuming for hand calculation, the computer

is ideally suited for this task.

The first computer generated K-values were based on the Chao-Seader EOS [14]. An equationsimilar to Equation (13) was used for this purpose. Later Erbar and Maddox [15] developed the K&HMod II software marketed by the Gas Processors Association for generating K-values and enthalpies olight hydrocarbons as well as some selected non-hydrocarbon compounds such as nitrogen, carbondioxide and hydrogen sulfides. This program which was based on the SRK EOS was well received byindustry due to its accuracy, reliability and flexibility.

Due to the development of computers and software, a complete phase envelope for a reservoir




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fluid based on EOS generated K-values can be plotted in a few seconds; something that would havetaken months before the 1970s. Similarly, today, a complete rigorous computer simulation of adistillation column for multicomponent systems with multi-stage, multi feed and multi-product streamsis calculated in only a few minutes. Accurate generation of K-values by an EOS enables engineers tohave a more accurate, economically sound, flexible, and reliable design.

Application of K-Values

Bubble Point

For a mixture, the bubble point is defined as the temperature at which the first bubble of vaporforms at a given pressure. At the bubble point, the liquid phase composition is known. At thistemperature vapor and liquid phases are at equilibrium; therefore Equations (1) and (2) hold.

(15)

Depending on the situation, two kinds of bubble point calculations may be required. In one case, thepressure is known and temperature is to be calculated. In the other case the temperature is known andthe bubble point pressure is to be calculated. In either case the calculation is trial and error and may becarried out by hand or computer.

Case No. 1

In order to show typical prediction accuracy for each method, the bubble point temperature and

vapor phase mole fraction composition of the following natural gas mixture were calculated at pressuresfrom 14.7 to 1700 psia using.

Table 1 presents the summary calculated bubble point results for pressures 14.7 to 1700 psia. Table 1indicates that at low pressures up to 800 psia, all methods give close answers for bubble pointtemperature; however, as the pressure increases, the difference becomes so large.

(a) Raoults Law K-values (based on purecomponent vapor pressures).

(b) Wilson K-Value Correlation (WKVC)

(c) NGPA K-Value Chart based on a

convergence pressure of 5000 psia.

(d) SRK EOS using EzThermo [16]simulation software.

Component Mole %

Methane 76.0

Ethane 13.0

Propane 5.0

n-Butane 2.0n-Pentane 2.0

n-Hexane 2.0

Total 100.0




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Table 1- Predicted bubble point temperature

Dew Point

For a mixture, the dew point is defined as the temperature at which the first drop of liquid formsat a given pressure. At the dew point, the vapor phase composition is known. At this temperature bothvapor and liquid phases are at equilibrium; therefore Equations (1) and (2) hold.

(16)

Depending on the situation, two cases of dew point calculation may occur. In one case, the pressure is

known and temperature is to be calculated or in the other case the temperature is known and the dewpoint pressure is to be calculated. In either case the calculation is trial and error and may be carried outby hand or by computer.

Case No. 2

Calculate the dew point temperatures and compositions of the first drop of liquid of natural gasmixture of Case No. 1 for pressures from 14.7 to 1870 psia using:

(a) Raoults Law K-value

(b) Wilson K-Value Correlation (WKVC)

(c) NGPA K-Value Chart based on convergence pressure of 5000psia

(d) SRK EOS using EzThermo simulation software

P, psia

Bubble point temperature, F

Raoult WKVC NGPA SRK

14.7 -252.49 -252.60 -254.15 -253.84

100 -195.66 -196.38 -196.71 -197.23

300 -147.91 -147.96 -148.40 -147.10

600 -107.23 -107.12 -107.12 -103.76

800 -87.16 -86.89 -86.03 -81.65

1000 -69.97 -69.54 -68.49 -61.96

1200 -54.74 -54.13 -53.91 -43.08

1400 -40.92 -40.13 -41.76 -23.24

1600 -28.17 -27.18 -31.54 0.25

1700 -22.13 -21.10 -26.5 16.27




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Table 2 presents the summary calculated dew point results for pressures 14.7 to 1870 psia. The analysisof Table 2 indicates that at Raoults law and Wilson correlation give very close results; however, theyare much different from NGPA and SRK results even at low pressures.

Table 2- Predicted dew point temperature

Flash Calculations

Separating a two phase stream into a vapor stream and a liquid stream is frequently required.Normally, the feed composition, total feed rate, temperature, and pressure are known and flow rates andcomposition of vapor and liquid must be calculated. The governing equations can be obtained bywriting the overall and component material balances and the equilibrium relationship.

Case No. 3

Flash calculation results for the natural gas mixture of Case No. 1 at 300 psia (2058.4 kPa) and -9.7 F (250 K) for for several methods were performed and the results are presented in Table 3.

Case No. 4

A series of flash calculations for the natural gas mixture of Case No. 1 for several methods were

P, psia

Dew point temperature, F


14.7 5.90 -3.42 5.77 5.27

100 77.65 74.12 67.68 70.83

300 134.65 133.68 109.63 113.32

600 180.27 180.05 134.38 136.15

800 202.23 202.06 142.09 141.82

1000 220.77 220.44 146.19 142.901200 237.06 236.44 147.81 140.05

1400 251.69 250.75 147.52 133.18

1600 265.10 263.77 145.72 120.96

1700 271.44 269.87 144.32 111.51

1800 277.54 275.77 142.63 97.12

1870 281.70 279.77 141.30 59.81




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performed and the results are presented in Table 4. This table indicates that the Raoults law and Wilsoncorrelation give similar liquid fraction but they are different from those predicted by NGPA and SRK.Up to pressure of 800 psia, both NGPA and SRK give close answers but as pressure increases, theirresults deviate from each other drastically.

Table 3- Predicted K-values at 300 psia (2058.4kPa) and -9.7 F (250 K)

Table 4- Predicted liquid fraction formation

Component

K-Values at 300 psia and -9.7 F

Raoult WKVC K-Chart SRK

Methane 8.2872 8.1934 6.1792 5.7629

Ethane 0.6566 0.6377 0.7014 0.7168

Propane 0.1065 0.1060 0.1469 0.1571

n-Butane 0.0189 0.0201 0.0331 0.0347

n-Pentane 0.0037 0.0044 0.0078 0.0080

n-Hexane 0.0007 0.0011 0.0022 0.0012

P, psia T, F

Liquid fraction, L/F


14.7 -50 0.0417 0.0368 0.0433 0.0415

100 -50 0.0967 0.0969 0.0961 0.0955

300 -50 0.1833 0.1873 0.1743 0.1768

600 -50 0.3096 0.3131 0.2954 0.3007

800 -50 0.4131 0.4179 0.4111 0.4204

1000 -50 0.5663 0.5745 0.5862 0.6391

1200 50 0.2016 0.2044 0.1502 0.1528

1400 50 0.2381 0.2147 0.1684 0.1710

1600 50 0.2778 0.2827 0.1858 0.1795

1700 50 0.2993 0.3048 0.1943 0.1705

1800 50 0.3220 0.3283 0.2026 0.1213




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Case No. 5

Yarborough and Vogel [18] reported experimental VLE measurements for a gas mixture of thefollowing composition at several pressures and temperatures. Composition of mixture B is theexperimental repeat for composition of mixture A. Flash calculations for this natural gas mixture at the

following reported pressures and 200 F were carried out using three popular equations of state:

A summary of the error analyses for the liquid fraction expressed as moles of liquid to moles of feed(L/ F), is presented in Table 6. The experimental liquid fraction was not reported; a simple flashcalculation using experimentally determined K-values was performed to calculate the experimental L/F.Table 6 indicates that with the exception of the last two pressures, SRK EOS gives the most accurateliquid fraction.

P, psia 110 255 535 552 1032 1033 1547 2044 2543 3035

Mixture B B B A A B B A B A

(a) Soave-Redlich-Kwong (SRK) EOS [10]

(b) Peng-Robinson(PR) EOS [19]

(c) Nasrifar-Moshfeghian (NM) EOS[20]

Component

Mole %

Mixture A%

Mixture B%

Nitrogen 0.480 0.470

Methane 80.64 80.31

Carbon dioxide 0.15 0.18Ethane 5.93 5.93

Propane 2.98 3.00

n-Pentane 4.30 4.45

n-Heptane 3.08 3.14

n-Decane 2.44 2.52

Total 100.0 100.0




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Table 6- A summary of error analyses for the liquid fraction and the moles of liquid to moles of feed (L/F)

A summary of the error analysis for all K values is shown in Table 7. This table indicates that SRK andPR give comparable results. A graphical comparison between predicted K-values and experimental K-values for the NM EOS are shown in Figures 1.

Table 7- Summary of the error analysis for all K-values

P

Psia Mix.

Liquid Fraction, L/F at 200 F Percent Error in (L/F)

Exp SRK PR NM SRK PR NM

110 B 0.0271 0.0250 0.0242 0.0243 7.75 10.70 10.33

255 B 0.0451 0.0438 0.0430 0.0421 2.88 4.66 6.65

535 B 0.0665 0.0647 0.0633 0.0607 2.71 4.81 8.72

552 A 0.0631 0.0629 0.0615 0.0589 0.32 2.54 6.66

1032 A 0.0854 0.0853 0.0823 0.0760 0.12 3.63 11.01

1033 B 0.0899 0.0891 0.0860 0.0795 0.89 4.34 11.57

1547 B 0.1072 0.1086 0.1027 0.0908 -1.31 4.20 15.30

2044 A 0.1110 0.1174 0.1071 0.0897 -5.77 3.51 19.19

2543 B 0.1006 0.1298 0.1102 0.0867 -29.03 -9.54 13.82

3035 B 0.0214 0.0905 0.0390 0.0393 -322.90 -82.24 -83.64

AAPE 37.37 13.02 18.69

P

Psia Mix. Average Absolute Percent Error in All K-Values

SRK PR NM

110 B 20.77 21.23 22.77

255 B 16.72 17.74 20.16

535 B 9.46 8.48 11.76

552 A 7.65 9.06 11.39

1032 A 5.55 8.59 12.36

1033 B 7.79 7.49 10.70

1547 B 6.23 11.35 10.39

2044 A 3.82 5.61 6.59




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Consistency Test of K-Values

There are several ways of testing the accuracy and validity of K-Values calculated frommeasured experimental liquid and vapor phase mole fractions or generated by any EOS, charts orcorrelation. In one method, K-values are plotted as a function of pressure on a log-log scale as shown inFigures 1. For each isotherm, the K-values for each component all fall on a smooth curve if they areconsistent.

Another method is to plot at the specified pressure and temperature the log of the K-value foreach hydrocarbon component as a function of its critical point absolute temperature squared. If the K-values are consistent and accurate, they will fall on a straight line. Figure 2 presents such a test for theYarborough and Vogel data at 3035 psia and 200 F. This figure indicates that the Yarborough andVogel data are of good quality.

2543 B 7.42 5.59 9.39

3035 B 5.37 2.48 11.39

AAPE 9.07 9.76 12.76




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Conclusions:

Several methods for determination of K-values were reviewed. Several case studies werepresented to demonstrate their application. The data used clearly show that the accuracies of these

methods are not the same and care must be taken to choose the right and proper method for a givenapplication. In Case No. 1, for bubble point pressures above 800 psia, the difference in predictedtemperature is large and as the pressure increases the difference becomes larger. Case No. 2 indicatesthat the difference in the predicted dew point temperature is even larger. Case No. 4 shows that as thepressure increases, the amount of liquid predicted by different methods differs considerably.

In Case No. 5,two widely used EOS and a new EOS were tested. The results show that for thesystems tested, all three EOS generally give relatively accurate results; however, with the exception overy high pressure, the SRK gives the most accurate VLE calculation results.

Based on the cases studied, our conclusion is that for VLE calculations of light hydrocarbon

systems, an equation of state should be used and for polar systems, an activity coefficient model isrecommended.

Always, the K-Values should be checked for accuracy and consistency before accepting thecalculation results.

References

1. Proceedings Natural Gasoline Supply Mens Association, 20th Annual Convention, April 23-25,1941.

2. Engineering Data Book, 10th and 11th Editions, Gas Processors and Suppliers Association DataBook, Tulsa, Oklahoma, (1998).

3. Prausnitz, J. M.; R. N. Lichtenthaler, E. G. de Azevedo, Molecular Thermodynamics of Fluid




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Phase Equilibria, 3rd Ed., Prentice Hall PTR, New Jersey, NY, 1999.

4. Maddox, R. N. and L. L. Lilly, Gas Conditioning and Processing, Volume 3: AdvancedTechniques and Applications, John M. Campbell and Company, Norman, Oklahoma, USA,1994.

5. Reid, R. C.; J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids, 4

th

Ed.,McGraw Hill, New York, 1987.

6. Engineering Data Book, 7th Edition, Natural Gas Processors Suppliers Association, Tulsa,Oklahoma, 1957.

7. Equilibrium Ratio Data for Computers, Natural Gasoline Association of America, Tulsa,Oklahoma, (1958).

8. Natural Gasoline and the Volatile Hydrocarbons, Natural Gasoline Association of America,Tulsa, Oklahoma, (1948).

9. Wilson, G., A modified Redlich-Kwong equation of state applicable to general physical data

calculations, Paper No15C, 65th AIChE National meeting, May, (1968).

10. G. Soave, Chem. Eng. Sci. 27, 1197-1203, 1972.

11. Wilson, G. M., J. Am. Chem. Soc. Vol. 86, pp.127-120, 1964

12. Renon, H. and J. M. Prausnitz, AIChEJ., Vol. 14, pp.135-144, 1968.

13. Gmehling et al., Vapor-Liquid Equilibrium Data Collection, Chemistry Data Series, Vol.1,

part 1a, 1b, 2c, and 2e, DECHEMA, Frankfurt / Main 1981-1988

14. Chao, K. C. and J. D. Seader, A General Correlation of Vapor-Liquid Equilibria inHydrocarbon Systems, AIChE, 7, 598, 1961

15. Erbar, J.H. and R.N. Maddox, GPA K & H Mod II Program, Natural Gas Processors andSuppliers Association, Tulsa, Oklahoma, 1962

16. Moshfeghian, M. and R.N. Maddox, EzThermo Software, 2002

17. Freshwater, D. C. and K. A. Pike, J. Chem. Eng. Data, Vol. 12, pp. 179-183, 1967

18. Yarborough, L. and John L. Vogel, "A New System for Obtaining Vapor and Liquid SampleAnalyses to Facilate the Study of Multicomponent Mixtures at Elevated Pressures," Chem. Engr.Symposium Series, No. 81, Vol.63,

19. D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. Vol. 15, pp. 59-64, 1976

20.Nasrifar, Kh. and M. Moshfeghian, A New Cubic Equation of State for Simple Fluids: Pure andMixtures, J. of Fluid Phase Equilibria, 190, pp. 73-88, 2001




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Symbols

the fugacity of component i in the vapor phase.

the fugacity of component i in the liquid phase.

Ki

the vaporliquid equilibrium constant (K-value)

i. mole fraction of one component i in the vapor phase.

i mole fraction of component i in the liquid phase.

T absolute temperature.

P absolute pressure.

bij the fitted values coefficients in Equation (3)

the saturation pressure of a component i.

Tci

the critical temperature.

Pci the critical pressure.

i the acentric factor.

the fugacity coefficients for component i in the vapor phase.

the fugacity coefficients for component i in the liquid phase.

the fugacity coefficients for pure component i at the saturation pressure.

i

the activity coefficient for component i.


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