artigo - calculating and applying k-values

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. Several case studies were presented to demonstrate their application. Based on the VLE calculations results, for light hydrocarbon systems, an equation of state should be used and for polar systems, an activity coefficient model is recommended.

Introduction

Modeling and design of many types of equipment for separating gas and liquids such as flash separators 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 offugacity 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 it in this form for practical applications. For calculation purposes, Equation (1) is transformed to a more commonly used expression:

Page 1 of 15Calculating and Applying K-Values

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(2)

In Equation (2), Ki is 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 the K-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 determine K-values. Obviously, experimental measurement is the most desirable; however, it is expensive and time consuming. Alternatively, there are several graphical or numerical tools that can be used for determination of K-values. This paper presents a history of the development of many of those graphical methods 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 plus temperature, pressure, composition, and degree of polarity affect the accuracy and applicability, and hence 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 involve using 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 for light hydrocarbons is presented in reference [1]. In these charts, K-values for individual components are plotted on the ordinate as a function of temperature on the abscissa with pressure as a parameter. In each chart the pressure range is from 10 psia to 1000 psia and the temperature range is from 40 F to 500 F.

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 all components converged toward a common value of unity (1.0) at some high pressure. This pressure was termed the Convergence Pressure of the system and has been used to correlate the effect of composition on K-values. Plotting this way permits generalized K-values to be presented in a moderate number of charts.

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

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

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



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

(3)

Raoults Law

Raoults Law is based on the assumptions that the vapor phase behaves as an ideal gas and the liquid 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 of temperature. 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 be avoided because they should not be used to extrapolate to temperatures beyond the critical temperature of each component. Also, Raoults law is applicable to only to low-pressure systems (up to about 50 psia) and to systems in which the components are very similar such as benzene and toluene. This method is simple but it suffers when the temperature of the system is above the critical temperature of one 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 computer calculations. Therefore, scientists and engineers have developed numerous curve fitted expressions forcalculation of K-values. However, these correlations have limited application because they are specific to a certain system or applicable over a limited range of conditions. Some of these are polynomial or exponential correlations in which K-values are expressed in terms of pressure and temperature. One of these correlations presented by Wilson [9], is:



(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 each component is determined by using an EOS. The same EOS describes both phases. The - approach is a powerful tool and it is relatively accurate if used appropriately. This approach is widely used in industry 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 using an 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 are known. 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.



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 the NRTL (Non-Random Two Liquid) model [12]. The required parameters for a few binary systems are given in reference [13]. In order to calculate the K-values by Equation (13), the mole fractions in both phases, in addition to pressure and temperature, are needed. Normally not all of these variables are known. As is the case for the - approach, calculations are trial and error. The - approach is applicable 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 liquid phase 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 for engineering 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 an EOS is trial and error and, consequently tedious and time consuming for hand calculation, the computeris ideally suited for this task.

The first computer generated K-values were based on the Chao-Seader EOS [14]. An equation similar 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 of light hydrocarbons as well as some selected non-hydrocarbon compounds such as nitrogen, carbon dioxide and hydrogen sulfides. This program which was based on the SRK EOS was well received by industry due to its accuracy, reliability and flexibility.

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



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 streams is calculated in only a few minutes. Accurate generation of K-values by an EOS enables engineers to have 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 this temperature 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, the pressure is known and temperature is to be calculated. In the other case the temperature is known and the 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 pressures from 14.7 to 1700 psia using.

Table 1 presents the summary calculated bubble point results for pressures 14.7 to 1700 psia. Table 1 indicates 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 pure component 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 2.0

n-Pentane 2.0

n-Hexane 2.0

Total 100.0



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 both vapor 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 gas mixture 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 SRK14.7 -252.49 -252.60 -254.15 -253.84100 -195.66 -196.38 -196.71 -197.23300 -147.91 -147.96 -148.40 -147.10600 -107.23 -107.12 -107.12 -103.76800 -87.16 -86.89 -86.03 -81.65

1000 -69.97 -69.54 -68.49 -61.961200 -54.74 -54.13 -53.91 -43.081400 -40.92 -40.13 -41.76 -23.241600 -28.17 -27.18 -31.54 0.251700 -22.13 -21.10 -26.5 16.27



Table 2 presents the summary calculated dew point results for pressures 14.7 to 1870 psia. The analysis of 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 by writing 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

Raoult WKVC NGPA SRK14.7 5.90 -3.42 5.77 5.27100 77.65 74.12 67.68 70.83300 134.65 133.68 109.63 113.32600 180.27 180.05 134.38 136.15800 202.23 202.06 142.09 141.82

1000 220.77 220.44 146.19 142.901200 237.06 236.44 147.81 140.051400 251.69 250.75 147.52 133.181600 265.10 263.77 145.72 120.961700 271.44 269.87 144.32 111.511800 277.54 275.77 142.63 97.121870 281.70 279.77 141.30 59.81



performed and the results are presented in Table 4. This table indicates that the Raoults law and Wilson correlation 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 5.7629Ethane 0.6566 0.6377 0.7014 0.7168Propane 0.1065 0.1060 0.1469 0.1571n-Butane 0.0189 0.0201 0.0331 0.0347n-Pentane 0.0037 0.0044 0.0078 0.0080n-Hexane 0.0007 0.0011 0.0022 0.0012

P, psia

T, F

Liquid fraction, L/F

Raoult WKVC NGPA SRK14.7 -50 0.0417 0.0368 0.0433 0.0415100 -50 0.0967 0.0969 0.0961 0.0955300 -50 0.1833 0.1873 0.1743 0.1768600 -50 0.3096 0.3131 0.2954 0.3007800 -50 0.4131 0.4179 0.4111 0.4204

1000 -50 0.5663 0.5745 0.5862 0.63911200 50 0.2016 0.2044 0.1502 0.15281400 50 0.2381 0.2147 0.1684 0.17101600 50 0.2778 0.2827 0.1858 0.17951700 50 0.2993 0.3048 0.1943 0.17051800 50 0.3220 0.3283 0.2026 0.1213



Case No. 5

Yarborough and Vogel [18] reported experimental VLE measurements for a gas mixture of the following composition at several pressures and temperatures. Composition of mixture B is the experimental 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 flash calculation 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 accurate liquid fraction.

P, psia 110 255 535 552 1032 1033 1547 2044 2543 3035Mixture 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 0.18 Ethane 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



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 and PR 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



Consistency Test of K-Values

There are several ways of testing the accuracy and validity of K-Values calculated from measured 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 in Figures 1. For each isotherm, the K-values for each component all fall on a smooth curve if they are consistent.

Another method is to plot at the specified pressure and temperature the log of the K-value for each 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 the Yarborough and Vogel data at 3035 psia and 200 F. This figure indicates that the Yarborough and Vogel 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



Conclusions:

Several methods for determination of K-values were reviewed. Several case studies were presented 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 predicted temperature is large and as the pressure increases the difference becomes larger. Case No. 2 indicates that the difference in the predicted dew point temperature is even larger. Case No. 4 shows that as the pressure 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 the systems tested, all three EOS generally give relatively accurate results; however, with the exception ofvery 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 hydrocarbonsystems, 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



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: Advanced Techniques 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, 4th 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 datacalculations, 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 in Hydrocarbon 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



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)

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

xi 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.

