Constantes de Equilibrio

Tech n ¡cal Pu bl ¡cat ion

TP-22

Convergence Pressure and Vapor-Liquid Equilibrium Ratios

Published by the Gas Processors Association

June, 1999

6526 East 60th Street * Tulsa, Oklahoma 74145 Phone: 918/493-3872

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Fax:

Copyright Gas Processors Association Provided by IHS under license with GPA

Not for ResaleNo reproduction or networking permitted without license from IHS

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GPA Disclaimer

GPA publications necessarily address problems of a general nature and may be used by anyon' desiring to do so. Every effort has been made by GPA to assure accuracy and reliability of the informatioi contained in its publications. With respect to particular circumstances, local, state, and federal laws ani regulations should be reviewed. It is not the intent of GPA to assume the duties of employers manufacturers, or suppliers to warn and properly train employees, or others exposed, concerning healtl and safety risks or precautions.

GPA makes no representation, warranty, or guarantee in connection with this publication and hereb expressly disclaims any liability or responsibility for loss or damage resulting from its use or for thl violation of any federal, state, or municipal regulation with which this publication may conflict, or for an infringement of letters of patent regarding apparatus, equipment, or method so covered.

"Copyright O 1999 by Gas Processors Association. All rights reserved. No part of this report may be

reproduced without the written consent of the Gas Processors Association."



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Convergence Pressure and Vapor-Liquid Equilibrium Ratios

GPA Technical Publication TP-22

Prepared by

Gas Processors Suppliers Association Editorial Review Board 6526 E. 60th Street Tulsa, OK 74145

June, 1999



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Introduction

This Technical Publication, TP-22, serves to archive the GPSA Convergence Pressure Vapor Liquid1 Equilibrium Ratios (K-data) along with other K-data charts. The convergence pressure approach to handling1 composition dependency of K-values was omitted when the eleventh edition of the GPSA Engineering Data, Book was published in 1998. The Editorial Review Board explained this change in the data book with the^ following statement:

Previous editions of this data book presented extensive sets of K-data based on the GPSA Convergence Pressure Pk method. A component's K-data is a strong function of temperature and pressure and a weaker function of composition. The convergence pressure method recognized composition effects in predicting K- data. The convergence pressure technique can be used in hand-calculations, and it is still available as computer correlations for K-data prediction.

There is now general availability of computers. This availability coupled with the more refined K-value1 correlations in modern process simulators, has rendered the previous GPA convergence pressure charts outdated.

The convergence pressure method is presented in dual metric (SI) and English (FPS) units. This TP contains the full explanatory text on convergence pressure from section 25 of the tenth Edition of the GPSA Engineering Data Book (the blue books) supplemented with the SI information from the corresponding material from section 18 in the 1980 SI Engineering Data Book (the green book). All of the standard charts are from the 1980 SI Engineering Data Book (the green book). In addition, other charts from the 1957 edition relating to acid gas components in ethane and propane have been included.

For quick trouble-shooting and evaluation work a set of K-charts can be very helpful. They present an illustration of the effect of temperature and pressure on K-values for light hydrocarbons. A composite set has been retained in the eleventh edition of the data book for both the SI and FPS units.

The charts represent decades of careful comprehensive research directed by GPA and GPSA. Accordingly, the Associations decided that they are to be retained as a resource available to the industry and academia.

Also included in this TP are K-data for High-Boiling Hydrocarbons (Poettman charts), some hydrogen sulfide binanes and hydrogen binaries,

The Equilibrium or K-value curves were first published in chart form by the NGAA and NGSMA (predecessors to the present day GPA and GPSA) in the early 1950's. Much of the basic data was compiled in tabular form under the direction of Dr. G. G. Brown, Dean of Engineering,University of Michigan. The data were taken from numerous published articles and unpublished experimental data. From these tabular data the first charts to be used by the industry were prepared by the Fluor Corporation Engineering Department under the guidance of Blaine Kuist, C. W. Leggett and C. K. Walker. Most of the painstaking draftsmanship was done by Orville McAdams, also of Fluor. Stuart Hadden developed the concept of convergence pressure, thereby making the charts useful for predicting K-values in a mixture containing any number of components. The charts were first published by the NGAA in 1955.



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In anticipation of needing additional information to expand the data and improve the accuracy. NGAA appointed the first Equilibrium Committee chaired by Elliott Organik, United Gas Corporation. Members of the committee were C. L. DePreister, California Research Corporation; Karl Hackmuth, Phillips Petroleum Company; James Kilmer, Stanolind Oil and Gas Company; H. H. Rackford, Humble Oil and Refining Company; Bryon Woertz, The Pure Oil Company; and Charles Webber, Sun Oil Company.

In 1957 the charts were completely redrawn to provide data at lower temperatures and to increase the internai consistency of the K-values. Karl Hackmuth guided the chart revision. By then Byron Woertz had assumed chairmanship of the Equilibrium Committee. E.A. Gromatzky, The Texas Company; R. H. Jacoby, Pan American Petroleum Company; Herbert L. Stone, Humble Oil and Refining Company; and C. J. Walters. California Research Corporation had become members of the committee.

This committee continued to act as a clearing house for new equilibria data and correlation methods. In 1958 these charts were published as Tables of Coefficients for use with early computers. Robert Norman and Brymier Williams presented a paper at the 57th Annual NGAA convention in Dallas April, 1958 that was the basis for the initial work that developed these machine computations.

A set of equations was developed to represent each page in the Data Book. The NGAA prepared a reproduction of the data from these charts in two forms - point cards and coefficient cards. The point cards were IBM cards containing approximately 70,000 data points (pressure, temperature, composition and convergence pressure) punches on 17,000 cards. The coefficient cards were IBM cards that contained the coefficients for the equations used to empirically correlate the point data. This required only 432 cards because each card contained seven coefficients. This drastically reduced the number of cards which in turn reduced the computer storage space required.

Through the years the convergence curves have been modified and improved. They are still published in the GPSA Engineering Data Book and represent decades of careful and comprehensive research directed by GPA and GPSA.



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STDnGPA TP-ZZ-ENGL 1997 382qb97 00201178 1173 =

TP-22

EQUILIBRIUM RATIO (K) DATA

The equilibrium ratio (Ki) of a component i in a multicomponent mixture of liquid and vapor phases is defined as the ratio of the mole fraction of that component in the vapor phase to that in the liquid phase.

K , = - Yi xi

For an ideal system (ideal gas and ideal solution), this equilibrium ratio is reduced to the ratio of the vapor pressure of component i to the total pressure of the system.

pi* Ki = -

P This TP presents an outline procedure to calculate the liquid

and vapor compositions of a two-phase mixture in equilibrium using the concept of a pseudobinary system and the convergence pressure equilibrium charts. Discussion of CO;! separation. alternate methods to obtain K values, and equations of state follow. Finally, three sets of equilibrium rat¡:, charts are presented:

K-values for specific binary systems K-values of various components based on different convergence pressures of hydrocarbon systems K-values of pseudocomponents based on normal boiling point and characterization factor

BINARY SYSTEMS DATA CHARTS

The 1972 edition of the GPSA Engineering Data Book included for the first time several charts representing binary combinations of N2, Cl. C,, C3, and n-C4. These data resulted from GPA-sponsored research at Rice University. Binary data with heavier components and acid gases have been added as revisions as they have become available.

These charts are useful for low temperature (below - 100"F, [-73"C]) calculations. Below the critical temperature of methane (-1 16°F [-82"C]), the convergence pressure of a system no longer has a real significance, since the K-values for heavy components do not reach a value of 1 .O.

Note that the critical data from these binary mixtures were NOT used to adjust the critical loci of Figure 5 . Such an adjustment would necessitate complete cross-plotting and redetermination of all of the convergence pressure charts.

Use of Binary Charts - At temperatures below -100'F [-73T], a light hydrocarbon system is most easily approximated as a pseudobinary. Determine the average molecular weight on a methane-free basis; then interpolate the K-values between the two binaries whose heavy component lies on either side of this pseudocomponent. This calculation becomes poor for systems with N2 content greater than 3 to 5 mole percent.

The ternary methane-ethane-propane system is well defined.5 At temperatures below -75°F [-60°C], (except for the highest pressures 700-800 psia [4800-5500 kPa(abs)] at -75°F [-60"Cl and -100°F [-73"C]. if f=XE(Mp) /xECMi = ratio of ethane mole fraction in the liquid phase of the ternary system to the ethane mole fraction in the liquid phase of the methane- ethane system, then the ternary K-values can be calculated from:

log KM = log KM(P) [KM(P)IKM(E)1 Eq 3

log KP = log KP(M) log [KP(M)/KgME)l Eq 5 Note that xE(MP) I xE(M), and KAo, denotes K-value of A in

the A-B binary system. These are found from pages 102 and 103. The infinite dilution KE(MP) and G,,,, are given on page 104.

These generalizations can be made about the ternary K-

Addition of propane increases the methane K-value and decreases the ethane K-value. Addition of ethane decreases the methane K-value and increases the propane K-value. The propane K-value is highest when at infinite dilution in the ternary (¡.e., a methane-ethane binary). The ethane K-value is highest when no propane is present (¡.e., a methane-ethane binary).

values:

These generalizations can be extended to other ternary (or Figure 1

Nomentclature

K = qulibrium ratio, y/x y = mole fraction in the vapor phase L = ratio of moles of liquid to moles of total mixture N = mole fraction in the total mixture or system ' = pseudobinary

Superscripts

O = acentric factor = log Pv, - 1 .o0

P = absolute pressure, psia [kPa(abs)] where. Pvr = reduced vapor pressure at Tr = 0.7

o0 = at infinite dilution

Subscripts Pk = convergence pressure. psia [kPa(abls)] I = component P* = vapor pressure, psia [kPa(abs)] c = critical R = universal gas constant, (psia CU ft)Ab mole "R) [(kPa m3 ) /(kmole K)] M = methane T = temperature, O R or OF [K or OC] E = ethane V = ratio of moles of vapor to moles of total mixture P = propane x = mole fraction in the liquid phase

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1 Copyright Gas Processors Association Provided by IHS under license with GPA


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STD-GPA TP-ZZ-ENGL 1799 3824b99 0020479 30"

higher) systems: the heaviest component (propane in this case) is analogous to a "lean oil" in an absorber system. The methane- ethane is acting in this case as a "natural gas" with the heaviest component (ethane) being preferentially absorbed into the "lean oil" (propane), with the resulting effects on the K-values. The liquid mole fraction of the middle (volatility) component is a useful parameter. The user should be aware that the composition effect on K-values is very pronounced in the regions covered by the binary and ternary data systems. Accurate values can be obtained only by cross-plots of K-value versus composition.

Limiting conditions should be considered in the analysis and use of K-data. For example, the limits on a binary system are defined by the vapor pressure of the pure components, as shown by the Locus K G and Locus KS of page 102. The limits of a ternary system are the binary systems.

The GPA/GPSA sponsors investigations iL systems of interest to gas processors. Detailed results are given in the annual proceedings and in various research reports and technical publications, which are listed in Section 1 of the Engineering Data Book.

Example 1 - Binary System Calculation

To illustrate the use of binary systerm K-value charts, assume a mixture of 60 moles of methane and 40 moles of ethane at -125OF [-87"C] and 50 psia [345 kPa(abs)]. From the chart on page 102, the K-values for methane and ethane are 10 and 0.35 respectively. This method is valid for either Ibmols or kmols.

Solution Steps

From the definition of K-value, Eq 1 :

Y G2 = z= 0.35 x c2

Rewriting

-- - 0.35 ' yci '- xc1

Solving

xcl = 0.0674

ycl = 0.674

Hence

xc2 = 0.9326

yo = 0.326

To find the amount of vapor in the mixture, let v denote moles of vapor. Summing the moles of methane in each phase gives:

0.674 v + 0.0674 (100 - v ) = 60 v = 87.8

The mixture consists of 87.8 moles of vapor and 13.2 moles of liquid.

K-VALUES CORRELATION BY CONVERGENCE PRESSURE

The convergence pressure chart representation of K-values. used since 1957 by the GPA, provides a useful and rapid graphical approach for engineers. Within limits the values are sufficiently accurate to satisfy many calculations required by the practicing engineer. Moreover. these charts are widely used in industry and are generally preferred over most nomographs.

The charts included in this TP are a mixture of past and present, as shown in Figure 2.

The Pk charts for methane, ethane, and propane are based on extensive cross plots of available experimental data. Properly used, they should represent both absorber oil and condensate systems adequately. Charts for components heavier than propane are selected from previous editions, based on general experience as to whether the 1957 or 1966 edition of the Engineering Data Book gave the best results.

The present convergence pressure charts do not have as many low temperature isotherms as in past editions. This is because no experimental data existed in these regions when the charts were prepared. Calculations of systems at very low temperatures and at high convergence pressures do not have much significance in hydrocarbon systems, and therefore should not be shown.

As an example, refer to Figure 5. Assuming methane as the light componet and methylcyclohexane as the heavy, the critical locus line indicates that the lowest possible temperature for a 3000 Pk is -50°F [-46OC]. This is about the practical limit in real systems; therefore, isotherms significantly below this would be imaginary. This deletion of low-temperature isotherms is intended to warn the user not to apply these charts beyond the range of good practice.

The accuracy of the P, charts depends in part on how carefully the convergence pressure has been determined. At moderate to high values of operating pressurekonvergence pressure (ratio > 0.4), both accurate convergence pressure determination and interpolation between P, charts can be made. K-values from an equation of state are likely to be more reliable. In the case of methane, interpolation between P, charts is a must to obtain accurate K-values. This is true to a lesser degree for heavier components. The revised charts were back-calculated from data and cross-plotted using Figure 5 as the basis of critical locus; therefore, ali determinations of convergence pressure must be made from Figure 5.

If any other source of critical locus is substituted, the revised charts will be less accurate in predicting the system of K-values. Recent binary data for the Cl-nC6, C1-nC,. C,- nC6, Cz-nCIo, C3-nC6, and C3-nC7 systems allowed the addition of their respective critical loci.



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Component Binary Data

Nitrogen * Methane I

I Ethane i * Ethylene

Propylene I

Propane *

*

iso-B utane

--A

Convergence pressure, kPa (abc) [psia] 5500 1 6900 10300 13800 20700 34500 69000

wo1 I [l@JJI [I5001 [2000] j [3000] [SOOO] (iO,OOO] O O O

O O O I

V V V V V X X

O O O V V V V V X X

0 O X X X X X I

V V V V V 1 V

n-Butane *t isoPentane I

O O X X X X X

O O X X X X X

Nonane Decane Hydrogen sulfide ,

I Octane I O i O 1 x l x I x I x X I ~~

i x X X X X

Q O X X X x , x X X

I

I I

Carbon dioxide 1 * v Drawn for Data Book, 1972 Edition based on available data 0 Reused from Data Book, 1966 Edition X Reused from Data Book, 1957 Edition t Prepared for Data Book. Second Revisions 1972 Edition or revised * * Limited to Cop concentration of 1 O mole percent of feed or less

Binary data from Price & Kobayashi; Wichterle & Kobayashi; Stryjek. Chappelear. & Kobayashi; and Chen & Kobayashi

Note: The charts shown in bold outline are published in the 1 1 th edition of the data book. All charts shown in this Figure are published in this Technical Publication (TP-22), as well as the 10th Edition of the data book.

Method of Calculating Convergence Pressure

The vapor-liquid equilibrium K-value of each component in a system is a function of the system temperature, system pressure, and the composition of the vapor and liquid phase. For natural gas processes, the convergence pressure can usually be used as the parameter that represents the compasiton of the vapor and liquid phase in equilibrium. The c.onvergence pressure is, in general, the critical pressure of a system at a given temperature.

At a given temperature, as the system pressure increases, the K-values of ail components of the system converge to unity when the system pressure reaches the convergence pressure. In other words, it is the pressure for a system at a given temperature when vapor-liquid separation is no longer possible. Naturally, it is equally impossible to have a vapor-liquid separation at a given temperture i n which the system pressure is greater than the convergence pressure.

Although the convergence pressure charts include values for methane, ... n-butane ai - 120°F [-84"C] and lower, the use of the

binary K-data is recommended below the critical of methane (-

The convergence pressure of systems as encountered generally in natural gas processes is a function of the temperature and the composition of the liquid phase. This presumes that the liquid composition is known from a flash calculation using a first approximate guess for convergence pressure. Therefore, the method of calculating convergence pressure is an iterative procedure. This calculation procedure is suggested.

Step 1: Assume the liquid phase composition or make an approximation. (If there is no guide, use the total feed composi ton.)

Step 2: Identify the lightest hydrocarbon component which is present at least 0.1 mole 480 in the liquid phase.

Step 3: Calculate the weight average critical temperature and critical pressure for the remaining heavier components to form a pseudobinary system. (A shortcut approach good for most hydrocaròon systems is to calculate the weight average T, only.)

1 16'F [-82OC].



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STDIGPA TP-22-ENGL 1999 3 8 2 q b î 5 002OLi81 Tbö m

, 0.91997 0.01064 0.01183 0.01462 0.00369

~ 0.00203 0.00039 0.00033 O.WO17 0 . m s

0.99972

3.1- I . 13580 0.42770 O. 12409 0.06686 0.05328 0.03854 0.03641 0.03184 0.03064

0.28749 0.00933 0.1 1667 O. I5070 0.09722 0.08446 0.04411 0.05276 0.09107 0.07506

I .O0887

CI CO2 CZ c3 iC1 nC4

nC3 cb

T,+

Taala

ic,

0.9010 0.0106 0.0499 0.0187 0.0065 0.0045 0.00 I7 0.0019 0.0029 0.0023

0.28689 0.00941 O. I 1989 0.14478 0.09183 0.07613 0.04288 0.02173 0.09724 0.08353

1.00431

0.91 804 0.01063 0.04796 0.01520 0.00413 0.00251 0.00056 0.00052 0.00028 0.00005

0.99988

When the convergence pressure so determined is beweein the values for which charts are provided, interpolation between charts may be necessary depending on how close the operating pressure is to the convergence pressure.

If K-values do not change rapidly with P, (Pp>P) then the set of charts nearest to the calculated PI; m a y he used. Otherwise, a cross-plot of K values versus P,. all at constant temperature and pressure, must be constructed for interpolation.

Aromaticity Corrections - These K-chrirrs for which Pk = 800, 1000, and 2000 were based entirely on binary parafin hydrocarbon systems, while those for P, = 4000 and 10.000 were derived from early data on mixtures containing some naphthenes (cycloparaffins) and aromatics. It is believed that small amounts of cyclic components added to mixtures of paraffins do not seriously affect the K-values obtained from these charts. However, as the concentration of cyclics increases the K-values are affected in a manner that is not fully accounted for by the convergence pressure variable.

Step 4: Trace the critical locus of the binary consisting of the light component and pseudoheavy component. When the averaged pseudoheavy component is between two real hydrocarbons. an interpolation of the two critical loci must be made.

Step 5: Read the convergence pressure (ordinate) at the temperature (abscissa) corresponding to that of the desired flash conditions, from Figure 5.

Step 6: Using the convergence pressure determined at Step 5 , together with the system temperature and systrm pressure, obtain K-values for the components from the appropriate convergence-pressure K-charts.

Step 7: Make a flash calculation with the feed composition and the K-values from Step 6, as shown in Figure 3.

Step 8: Repeat Steps 2 through 7 until the assumed and calculated convergence pressures check within an acceptable tolerance, or until the two successive calculations give the same light and pseudoheavy components check within an acceptable tolerance.

Figure 3 - SI Example Flash Calculation at 4000 kPa (abs) and -30°C

Column 6 1 7 1 8 3 1 4 1 5 ' I 2

Feed Gas :omponent I Composition KI tor

Pk = 2000 I

N i 113800 kPaíabs)

Trial values of L Final L = 0.030 Merhane-free basis 1 Liquid 1 Vapor

m s s trac liquid

T,. K ' P,. kPa (absi

L = 0.02

Ni L + V K i 0.28549 0.00932 0.11830 0.16252 0.11356 0.10340 0.05939 0.07286 O. 13265 0.1 I140

1.16889

L = 0.06 Ni -

. + V K , 0.29368 0.00937 O. 1 1203 0.12369 0.06791 0.05451 0.02490 0.02886 0.04694 0.03794

0.79983

L = 0.04

Ni L + V & C.28952 c.00934 O. 1 1508 o. 14047 0.08499 0.01138 0.03509 0.04135 0.06934 0.05661

0.9131 7

kg liquid 7 x 9

MW

0.90 I O 0.0106 0.0499 0.0187 0.0065 0.0055 0.0017 0.0019 0.0029 0.0023

3.2 1.14 0.41 0.097 0.038 0.024 0.0088 0.0062 0.0019 O.ooo66

2 =

44.01 30.07 44.10 58.12 58.12 72.15 72.15 86.18 107.0

0.4106 3.5083 6.6459 5.6504 4.9088 3.1825 3.8066 7.8484 8.0314

43.9929

305.2 30S.J 369.8 408.1 425.2 460.4 469.6 507.4 554.5

7382 4880 424s

3797 3381 3369 3012 261 I

3542

36111

0.0093 0.079X 0.1511 o. 1284 0.1 I I6 0.0723 0.0865 0.1784 O. 1826

1 .oooo 'Avaagc of nC, + nC, properties End of flash calculation Calculation to confirm PI. -! - 448.6li

175.4OC -273.2

Figure 3 - FPS Example Flash Calculation at 600 psia and -20°F

Column 3 1 4

Trial values of L=O.O2 I L=0.06

5 6 7 1 8 ral L = 0.0210 Methane-free basis

:omponeni Composition

IT- __ - . ._ - ._

L = 0.04

Ni L + V K ,

0.28952 0.00942 0.11769 O. I3281 0.078 I3 0.06278 0.03239 0.03831 0.06778 0.05675

0.88559

~~

Tc. "U

557.6 -549.6 665.8 734.2 765.3 828.8 845.5 913.3 998. I

813.11"R

-q-T L + V K i L + V K ,

mass frac.

liquid

0.0093 0.08 I4 0.1441 o. I204 0.0999 0.0698 0.0842 0.1891 0.2017

I .oooo

Pc. psia

~

1071.0 706.5 616.0 527.9 550.6 490.4 4ö8.6 436.9 378.7

SOY.?

Ib liquid l a 9

0.4141 3.6052 6.3846 5.3370 4.4247 3.0935 3.7322 8.3805 8.9377

44.3096

MW

44.01 30.07 44.10 58.12 58.12 72. I5 72.15 86.18 107.00

L + V K ,

3 . 2 m 1.13000 0.4oooO O. 10500 0.01500 0.03300 0.01300 0.01000 0.00290 O.WO55

0.28549 0.29368 o.oO940 0.00945 0.12112 0.11445 0.15216 0.11783 0.10140 0.06354 0.08598 0.04944 0.05192 0.02354 0.06376 0.02738 0.12696 0.04623 0.1 I198 0.03801 i 1.11017 0.78354

3.14060 1.12649 0.4 I620 o. I29 17 0.07079 0.0591 I 0.03965 0.03673 0.02982 0.02754

Avmge of nC, i nCK properties End of flash calculation Calculation IO confirm Pk -459.7 -I 3M.I"F - -



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The GPA has sponsored research to study these effects in absorber systems, and the experimental results of this work are published.' In general. the data show that the presence of aromatics in paraffin mixtures serves to increase the K-values of the paraffins.

HIGH BOILING OIL FRACTIONS

At the end of the convergence pressure series of charts is appended a small set of K-charts developed by Poettman and Mayland2 for high boiling petroleum fractions. They are useful for low pressure, high temperature flash calculations on heavy oils, such as in the reboiler of an absorber oil stripper or an atmospheric crude distillation tower, provided information on ASTM or IBP of the oil is available. At pressures below 100 psia [690 kPa (abs)] K s from these charts are fairly compatible with those from the convergence pressure series, if a plot of K vs. component boiling point is made. The reader is referred to White and Brown3 and Grayson and Streed4 for additional data.

DATA FOR NONHYDROCARBONS

Hydrogen Sulfide

Since the hydrocarbon charts for Pk = 4000 were also eliminated, the Pk = 4000 chart for H2S was reduced to Pk = 3000 by simply adjusting the isotherms at pressures near 3000 psi. The H2S chart for Pk = 1000 was reused from the 1957 edition of the Engineering Data Book.

Nitrogen

The convergence pressure charts for N2 which are based largely on binary and ternary data from the University of Texas were retained from the 1966 editon of the Engineering Data Book, as illustrated in Figure 2.

Data for N2-CH4 and N2-C2H6 are given on pages 109- 1 1 O. These data, and data on the ternary, show that the K-values in this system show strong compositional dependence. In this system, the component order is N2-CH4-C2H6, and the K-values are functions of the amount of methane in the liquid phase. For example, at -190°F [-123"C] and 300 psia [2070 kPa(abs), the K-values vary from:

N2 CH4 C2H6

10.2 0.824* 0.01 18 3.05 0.635 0.035 *

where * indicates the limiting infinite dilution K-value. See TP-45 for the data on this ternary.

FLASH CALCULATION PROBLEM

To illustrate the calculation of multicomponent vapor-liquid equilibrium using the flash equations and the K-charts, a problem is worked out in detail below.

The variables are defined in Figure I . Note that the K-value is implied to be at thermodynamic equilibrium.

A situation of reproducible sieady state condition> in ;1 piecc of equipment does not necessarily imply that c l a s s i ~ i ~ l thermodynamic equilibrium exists. If the steady composition differs from that for equilibrium, the reason can be the result of time-limited mass transfer and diffusion rates. This wming is made because i t is not at all unusual for flow rates through equipment to be so high that equilibrium is not attained or even closely approached. In such cases. equilihrium flash calculations as described here fail to predict conditions in the system accurately, and the K-values are suspected for this failure-when in fact they are not at fault.

Yi X i

Ki = - Eq. 6

L + V = 1.0 Eq. 7

By writing a material balance for each component in the liquid, vapor, and total mixture, one may derive the flash equation in various forms. A common one is.

N. L + VK,

xxi = z I = 1.0

Other useful versions may be written as

KiNi xyyi= ~

L + VK,

Eq. 8

Eq. 9

Eq. 10

At the phase boundary conditions of bubble point (L = 1 .OO)

Eq. 11

and dew point (V = 1.00). these equations reduce to

I;Ki Ni = 1.0 (bubble point)

mi/ Ki = i .O (dew point) Eq. 12

These are often helpful for preliminary calculations where the phase condition of a system at a given pressure and temperature is in doubt. If CK, Ni and LNi/K, are both greater than I .O, the system is in the two phase region. If XK, Ni is less than 1 .O, the system is all liquid. I f If ZKi Ni is less than 1 .O, the system is ail vapor.

and

Example 2-FF'S - A typical high pressure separator gas is used for feed to a natural gas liquefaction plant, and a preliminary step in the process involves cooling to -20°F at 600 psia to liquefy heavier hydrocarbons prior to cooling to lower temperatures where these components would freeze out as solids. Estimate the convergence pressure Pk.

Solution Steps - The feed gas composition, shown in Figure 3-FPS, column 1, is similar to that used in the enthalpy and entropy problems in Section 24 of the Engineering Data Book. Pk is first estimated to be 2000 psia and K's are determined as in column 2. Then the flash equation 8 is solved for three estimated values of L as shown in columns 3, 4, and 5. By plotting estimated L versus calculated Exi, the correct value fo L where Zxi = 1.00 is L=0.027, whose solution is shown in columns 6 and 7. The gas composition is then calculated using



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yi = K,xi in column 8. This "correct" value is used for purposes of illustration. It is not a completely converged solution, for xi = 1.00431 and yi = 0.99988, columns 7 and 8 of Figure 3-FPS. This error will be too large for many applications. To check the convergence pressure assumption. the components heavier than methane in the liquid are converted to a weight fraction basis as shown in columns 10 and 11 using the molecular weights in column 9. Using this composition (column 1 1 ) of the pseudoheavy component, calculate the weight average critical temperature and pressure using the Tci and PCi in columns 12 and 13; the results are shown at the bottom of those columns. By referring to Figure 5 , the calculated T c of 354°F is seen to fall between n-butane and n-pentane, and the calculated P c falls in line with the other paraffin hydrocarbons. By interpolation, as follows, the critical pressure for the pseudobinary, or the convergence pressure, may be calculated at -20°F without actually sketching in a new locus:

Eq. 13

All the Pks are read from the loci at -20°F and the prime quantitites, TCx and P'kx, refer to the pseudobinary. Thus

(813.8 - 765.6)

(845.3 - 765.6) P'k = (2000 - 1700) + 1700

48.2

79.7 PIk= - (300) + 1700 = 1881 psia

This confirms the assumed Pk = 2000 closely enough for this illustration. The more closely the operating pressure approaches the convergence pressure the more accurately the calculated Pk must check the assumed Pk because in this region, the K-values are most sensitive to the Pk value used.

Example 2-SI - A typical high pressure separator gas is used for feed to a natural gas liquefaction plant, and a preliminary step in the process involves cooling to -30°C at 4000 kPa (abs) to knock out heavier hydrocarbons prior to cooling to lower temperatures where these components would freeze out as solids. Estimate the convergence pressure Pk.

Solution Steps - The feed gas composition. shown in Figure 3 SI, column I , is similar to that used in the enthalpy and entropy problems in Section 24 of the Engineering Data Book. Pk is first estimated to be 2000 psia (1 3800 kPa) and K's are read off as in column 2. Then the flash equation 8 is solved for three estimated values of L as shown in columns 3, 4 and 5 . By plotting estimated L versus calculated L xi, the correct value of L where C xi = 1.00 is L = 0.030, whose solution is shown in columns 6 and 7. The gas composition is then calculated using yi = Kixi as in column 8. This "correct" value is used for purposes of illustration. It is not a completely converged solution. for xi = 1.0089 and yi = 0.99972, columns 7 and 8 of Figure 3-SI. This error will bè too large for many applications. To check the convergence pressure assumption. the components heavier than methane in the liquid are converted to a mass fraction basis as shown in columns 12 and i 3 using the molecular masses in column 9. Using this composition (column

13) of the pseudo heavy component. calculate the mash averapt. critical temperature and pressure using the T,, Irnd Pi, I I I

columns 10 and 11; the results are shown at the bottom of those columns. By referring to Figure 3-SI. the calculated T,' ot 175.4"C is seen to fail between n-butane and n-pentane. and the calculated P,' falls in line with the other paraffin hydrocarbons. By interpolation, as follows. the critical pressure for the pseudo binary, or the convergence pressure. may be calculated at -30°C without actually sketching in a new locus:

All the Pks are read from the loci at -30°C and the prime quantitites, Tcx' and Pkx'. refer to the pseudo binary. Thus.

(448'6 - 425.2) (1 3800 - i 1700) + 1700

23.4 (2100) + 11700 = 12800 kPa 44.4

(469.6 - 425.2)

= 1856 psia 12800 6.895

This confirms the assumed Pk = 2000 closely enough for this illustration. The more closely the operating pressure approaches convergence pressure the more accurately the calculated Pk must check the assumed Pk because in this region, the K-values are most sensitive to the Pk value used.

Example 3-FPS - Dew Point Calculation

As gas stream at 100°F and 800 psia is being cooled in a heat exchanger. Find the temperature at which the gas starts io condense.

Solution Steps - The approach to find the dew point of the gas stream is similar to the last example. Assume a convergence pressure of 1000 psia. The equation for dew point condition (CNi/Ki=l.O) is solved for two estimated dew point temperatures as shown in Figure 4-FPS. By interpolation. the temperature at which LNi/Ki = 1.0 is estimated at -44°F. To check the convergence pressure, assume the corresponding saturated liquid consists largely of the heaviest componet, propane. This agrees with the fact that the First condensed liquid droplet from any gas stream is the heaviest component in that stream. From Figure 5 , the convergence pressure at -44°F for the pseudoheavy component, propane, is 1240 psia. This is close to the assumed convergence pressure of 1 O00 psia.

Example 3-SI -Dew Point Calculation

A gas stream at 40°C and 5500 kPa (abs) is being cooled in a heat exchanger. Find the temperature at which the gas starts to condense.

Solution Steps -The approach to find the dew point of the gas stream is similar to the previous example. The equation for dew point condition (ZNi/Ki = 1.0) is solved for two estimated dew point temperatures as shown in Figure 4-SI. By interpolation, the temperature at which cNi/Ki = i .O is estimated at -4 1.4"C.



--`````-`-`,,`,,`,`,,`---

Figure 4 FPS Dew Point Calculation at 800 psia

1

Column 3 - 2 1 -

- Feed P, = loOO. T = -50°F P, = 1000, T = -40°F

Component Ni Ki - Ni Ki - Ni Ki Ki

CH4 0.854 i .420 0.601 1.450 0.589 CO2 0.05 1 0.790 0.065 0.834 0.06 1 C2H6 0.063 0.440 0.143 0.480 0.131 C3H8 0.032 0.145 0.22 1 0.158 0.203

E= 1 .o00 1 .O30 0.984

k o 2 calculated as 4 K,-, G2 Linear interpolation: Tdew = -50 + 1-50 - (-4011

Alternatively iterate until Z N i 6 = 1 .O

Figure 4 SI Dew Point Calculation at 5500 kPa (abs)

Column

3 - 4 5 - 2 - 1 Feed Estimated T = -45°C Estimated T = -40°C

Component Ni Ki - Ni K;

Ki - Ni K:

L

CH4 0.854 2.73 0.3 13 2.75 0.31 1 CO2 0.05 1 0.866 0.059 0.9 1 O 0.056

C3H8 0.032 0.070 0.457 0.080 0.400 C2H6 0.063 0.275 0.229 0.300 0.2 1 o

2 = 1 .o00 1 .O58 0.977

&o2 calculated as 4 QI G2 Linear interpolation: Tdew = -40 - i-40 - (-4511 ( ~ ~ ~ ~ ~ ) =-41.4"C

Alternatively iterate until Z NiKi = 1 .O

Note that the heaviest component is quite important in dew Later, experience showed that at low concentrations of CO2, the point calculations, since it essentially determines the convergence pressure. For more complex mixtures, the characterization of the heavy fraction as a pseudocomponent such as hexane or octane will have a significant effect on dew point calculations.

Carbon Dioxide

Early conflicting data on CO, systems was used to prepare convergence pressure (Pi, = 4000) charts for the 1966 Edition.

rule of thumb

could be used with a plus or minus 10% accuracy. Developments in the use of CO2 for reservoir drive have led to extensive investigations in CO, processing. See the more recent GPA research reports (listed in Section 1 of the Engineering Data Book) and the Proceedings of GPA conventions. The reverse volatility at high concentration of propane and/or butane



--`````-`-`,,`,,`,`,,`---

has been used effectively in extractive distillation to effect CO2 separation from methane and ethane.z2 In general, CO, lies between methane and ethane in relative volatility.

Separation of CO2 and Methane

of the GPSA Engineering Data Book.

C02-Ethane Separation

of the Engineering Data Book.

Separation of CO2 and H2S

of the Engineering Data Book.

See Section 16, Hydrocarbon Recovery, of the 11 th Edition

See Section 16, Hydrocarbon Recovery of the 1 1 th Edition

See Section 16, Hydrocarbon Recovery of the 1 lth Edition

OTHER K-VALUE PROCEDURES

Numerous procedures have been devised to predict K- values. These include state equations, combinations of state equations with liquid theory or wi th tabular data, and corresponding states correlations. This section describes several of the more popular procedures currently available. It does not purport to be all-inclusive or comparative.

State equations have appeal for predicting thermodynamic properties because they provide internally consistent values for all properties in convenient analytical form. Two popular state equations for K-value predictions are the Benedict-Webb-Rubin (BWR) equation and the Redlich-Kwong equation.

The original BWR equation17 uses eight parameters for each component in a mixture plus a tabular temperature dependence for one of the parameters to improve the f i t of vapor-pressure data. This original equation is reasonably accurate for light paraffin mixtures at reduced temperatures of 0.6 and above.* The equation has difficulty with low temperatures, nonhydrocarbons, non-paraffins, and heavy paraffins.

Improvements to the BWR include additional terms for temperature dependence, parameters for additional compounds, and generalized forms of the parameters.

Starling20 has included explicit parameter temperature dependence in a modified BWR equation which is capable of predicting light paraffin K-values at cryogenic temperatures.

The Redlich-Kwong equation has the advantage of a simple analytical form which permits direct solution for density at specified pressure and temperature. The equation uses two parameters for each mixture component, which in principle permits parameter values to be determined from critical properties.

However, as with the BWR equation, the Redlich-Kwong equation has been made useful for K-value predictions by empirical variation of the parameters with temperature and with acentric factor l9 and by modification of the parameter- combination rules. l 5 ~ l 9 Considering the simplicity of the Redlich-Kwong equation form, the various modified versions predict K-values remarkably well.

Interaction parameters for non-hydrocarbons with

hydrocarbon components are necessary in the Redlich-Kwong equation to predict the K-values accurately when high concentrations of non-hydrocarbon components are present. They are especially important in CO, fractionation processes. and i n conventional fractionation plants to predict sulfur compound distribution.

The Chao-Seader correlation' uses the Redlich-Kwong equation for the vapor phase, the regular solution model for liquid-mixture non-ideality, and a pure-liquid property correlation for effects of component identity. pressure. and temperature in the liquid phase. The correlation has been applied to a broad spectrum of compositions at temperatures from -50'F [-46"C] to 300°F [150°C] and pressures to 2000 psia [13.8 MPa(abs)]. The original (P,T) limitations have been reviewed.]?

Prausnitz and Chueh have developed16 a procedure for high- pressure systems employing a modified Redlich-Kwong equation for the vapor phase and for liquid-phase compressibility together with a modified Wohl-equation model for liquid phase activity coefficients. Complete computer program listings are given in their book. Parameters are given for most natural gas components. Adler et al. also use the Redlich-Kwong equation for the vapor and the Wohl equation form for the liquid phase.6

The corresponding states principlelo is used in all the procedures discussed above in that the behavior of all substances is assumed to follow the same equation forms and equation parameters are correlated versus critical properties and acentric factor. An alternate corresponding states approach is to refer the behavior of all substances to the properties of a reference substance, these properties being given by tabular data or a highly accurate state equation developed specifically for the reference substance.

The deviations of other substances from the simple critical- parameter-ratio correspondence to the reference substance are then correlated. Mixture rules and combination rules, as usual, extend the procedure to mixture calculations. Leland and co- workers have developed9 this approach extensively for hydrocarbon mixtures.

"Shape factors" are used to account for departure from simple corresponding states relationships, with the usual reference substance being methane. The shape factors are developed from PVT and fugacity data for pure components. The procedure has been tested over a reduced temperature range of 0.4 to 3.3 and for pressures to 4000 psia r27.6 MPa(abs)]. Sixty-two components have been correlated including olefinic, naphthenic, and aromatic hydrocarbons.

The Soave Redlich-Kwong (SRK)I3 is a modified version of the Redlich-Kwong equation. One of the parameters in the original Redlich-Kwong equation, a, is modified to a more temperature dependent term. It is expressed as function of the acentric factor. The SRK correlation has improved accuracy in predicting the saturation conditions of both pure substances and mixtures. It can also predict phase behavior in the critical region, although at times the calculations become unstable around the critical point. Less accuracy has been obtained when applying the correlation to hydrogen-containing mixtures.



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Peng and Robinson14 similarly developed a two-constant Redlich-Kwong equation of state in 1976. In this correlation, the attractive pressure term of the semi-empirical van der Waals equation has been modified. It accurately predicts the vapor pressures of pure substances and equilibrium ratios of mixtures. In addition to offering the same simplicity as the SRK equation, the Peng-Robinson equation is more accurate in predicting the liquid density.

In applying any of the above correlations, the original critical/physical properties used in the derivation must be inserted into the appropriate equations. One may obtain slightly different solutions from different computer programs, even for the same correlation. This can be attributed to different iteration techniques, convergence criteria, initial estimation values, etc. Determination and selection of interaction parameters and selection of a particular equation of state must be done carefully, considering the system components, the operating conditons, etc.

EQUATIONS OF STATE Refer to original papers for mixing rules for

multicomponent mixtures.

van der Waals"

2 3 - (1 + B) 2 2 + AZ -AB = O aP

R2T2

bP RT

A = -

B = -

27 R*T2, 64 Pc

a =

RTC pc

b = -

Redlich-Kwong"

Z3 - Z2 + (A-B - B2) Z - AB = O aP A = -

~ 2 ~ 2 . 5

bP RT

B = - a = 0.42747 (+) R2T 2.5

Soave Rediich-Kwong (SRK)13

Z3- Z2 + (A - B - B2 ) Z - AB = O

aP A = - R ~ T ~

bP RT

B = -

a = a , a

a, = 0.42747 (q) m = 0.48 + 1.574 o - 0.176 02

b = 0.08664 (7) Peng Robinson27

2 3 - (1 - B) Z2+ (A - 3B* - 2B) Z - (AB - B* - B3) = O

m = 0.37464 + 1.54226 o - 0.26992 a2

b=0.0778 (:> - Benedict-Webb-Rubin-Starling (BWRS)20- 25

C D E

T2 T3 T'l B , R T - A , - o + o - o P = - +

V

+ (bRT-a--$-)$+CX (a+$) 1 V6

- YN2

v3

REFERENCES AND BIBLIOGRAPHY

1. Wilson, G. M., Barton, S. T., NGPA Report RR-2: "K- Values in Highly Aromatic and Highly Naphthenic Real Oil Absorber Systems," (197 I).

Poettman. F. H., and Mayland, B. J.. "Equilibrium Constants for High-Boiling Hydrocarbon Fractions of Varying Characterization Factors," Petroleum Refiner 28.

2.

101-102, July. 1949.

3. White, R. R., and Brown, G. G., "Phase Equilibria of Complex Hydrocarbon Systems at Elevated Temperatures and Pressures," Ind. Eng. Chem. 37, 1162 (1942).

Grayson, H. G., and Streed, C. W., "Vapor-Liquid Equilibria for High Temperature. High Pressure Hydrogen- Hydrocarbon Systems," Proc. 6th World Petroleum Cong., Frankfurt Main, III, Paper 20-DP7, p. 223 (1963).

4.



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5 .

6.

7.

8.

9.

10.

I l .

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

Chappelear, Patsy, GPA Technical Publication TP-4, "Low Temperature Data from Rice University for Vapor-Liquid and P-V-T Behavior," April (1 974).

Adler, S. B., Ozkardesh, H., Schreiner, W. C., Hydrocarbon Proc., 47 (4) 145 (1968).

Chao, K. C., Seader, J. D., AIChEJ, 7,598 (1961).

Bamer, H. E., Schreiner, W. C., Hydrocarbon Proc., 45 (6) 161 (1966).

Leach, J. W., Chappelear, P. S.. and Leland, T. W., "Use of Molecular Shape Factors in Vapor-Liquid Equilibrium Calculations with the Corresponding States Principle," AIChEJ. 14,568-576 (1968).

Leland, T. W., Jr., and Chappelear, P. S., "The Corresponding States Principie-A Review of Current Theory and Practice," Ind. Eng. Chem. 60, 15-43 (July 1968); K. C. Chao (Chairman), "Applied Thermodynamics," ACS Publications, Washington, D.C., 1968, p. 83.

Bamer, H. E., Pigford, R. L., Schreiner, W. C.. Proc. Am. Pet. Inst. (Div. Ref.) 46 244 (1966).

Lenor, J. M., Koppany, C. R.. Hydrocarbon Proc. 46,249 (1967).

Soave, Giorgio, "Equilibrium Constants from a Modified Redlich-Kwong Equation of State," Chem. Eng. Sci. 27,

Peng, D. Y., Robinson, D. B., Ind. Eng. Chem. Fundamentals 15 ( 1976).

Spear, R. R., Robinson, R. L.. Chao, K. C., IEC Fund., 8 ( 1 ) 2 (1969).

Prausnitz, J. M., Cheuh, P. L., Computer Calciilations for High-pressure Vapor-Liquid Equilibrium, Preiitice-Hall (1968).

Benedict, Webb, and Rubin, Chem. Eng. Prog. 47,419 ( 1 95 1 ).

Wilson, G. M., Adv. Cryro. Eng., Vol. II, 392 (1966).

Zudkevitch. D., Joffe. J.. AIChEJ., 16 ( I ) 112 (1970).

Starling, K. E., Powers, J. E., IEC Fund., 9 (4) 531 (1970).

Hwang, S . C., Lin. H. M.. Chappelear. P. S . , and Kobayashi, R., "Dew Point Values for the Methane Carbon Dioxide System," GPA Research Report RR-21 (1976).

Price, B. C., "Looking at COz Recovery", Oil & Gas J., p. 48-53 (Dec. 24, 1984).

Nagahana, K., Kobishi, H., Hoshino, D., and Hirata, M., "Binary Vapor-Liquid Equilibria of Carbon Dioxide-Light Hydrocarbons at Low Temperature," J. Chem. Eng. Japan 7, No. 5, p. 323 (1974).

Redlich, O.. Kwong, J. N. S. , Chem. Rev. 44,233 (1 949).

Benedict. M.. Webb, G. B., Rubin, L. C., "An Empirical Equation for Thermodynamic Properties of Light

1 197-1 203 (1972).

Hydrocarbons and Their Mixtures,'' Chem. Eng. Prog. 47. 419-422 (1951); J. Chem. Phys. 8.334 (1940).

26. van der Waals, J., "Die Continuitat des Gasformigen und nussigen Zustandes," Barth, Leipzig ( 1899).

27. Peng, D. Y., Robinson, D. B.. "A New Two-Constant Equation of State," Ind. Eng. Chem. Fundamentals 15. 59- 64 (1976).

ADDITIONAL REFERENCES

See listing in Section I of the Engineering Data Book for GPA Technical Publications (TP) and Research Reports (RR). Note that RR-64, RR-77, and RR-84 provide extensive evaluated references for binary, ternary. and multicomponent systems. Also as a part of GPNGPSA Project 806. a computer data bank is available through the GPA Tulsa office.

Another extensive tabulation of references only is available from Elsevier Publishers of Amsterdam for the work of E. Hala and I. Wichterle of the Institute of Chemical Process Fundamentais, Czechoslovak Academy of Sciences, Prague- Suchdol. Czechoslovakia.

Also, Hiza, M. J., Kidnay, A. J., and Miller. R . C., Equilibrium Properties of Fluid Mixtures Volumes I and II. IFyPienum, New York, 1975. See Fluid Phase Equilibria for various symposia.

CHARTS FOLLOW AS LISTED BELOW

Convergence Pressure Charts: 80W to 10000# pages I l through 101

Binaries: Pages 102 - 1 1 1 Methane-Ethane Methane-Propane Methane-Ethane & Methane-Propane Infinite Dilution Methane-n-Butane Methane-n-Pentane Methane-n-Hexane Methane-n-Heptane Nitrogen-Methane Nitrogen-Ethane Methane-Carbon Dioxide

Poettmann Charts: Pages 112 through 117

1957 Engineering Data Book Hydrogen Sulfide-Ethane 118 Hydrogen Sulfide-Propane 119 Hy drogen-Ethane 120 Hydrogen-Propane 121 Hydrogen-Isobutane 122 Hydrogen-n-Bu tane 123 Hydrogen-Octane through Dodecane 124 Hydrogen-Benzene 125 Hydrogen-Toluene 126



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