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SECTION 25 Phase Equilibria VAPOR-LIQUID EQUILIBRIA Availability of computers, coupled with the more refined Kvalue correlations in modern process simulators, has made the previous GPA convergence pressure charts outdated Complete sets of these charts are available from GPA as a Technical Publication, TP-22 The equilibrium ratio (Ki) of a component i in a multicomponent mixture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 yi ­K i= xi Data for N2-CH4 and N2-C2H6 show that the K-values in these system have strong compositional dependence The component volatility sequence is N2-CH4-C2H6 and the K-values are functions of the amount of methane in the liquid phase For example, at –123°C and 2070 kPa (abs), the K-values depending upon composition vary from: Eq 25-1 For an ideal system (such as ideal gas and ideal solution), this equilibrium ratio becomes the ratio of the vapor pressure of component i to the total pressure of the system P*i ­ i= K P Eq 25-2 This section 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 CO2 separation, alternate methods to obtain K values, and equations of state follow N2 CH4 C2H6 10.2 0.824* 0.0118 3.05 0.635  0.035* ­ here * indicates the limiting infinite dilution K-value See refw erence for the data on this ternary ­ he charts retained in this edition represent roughly 12% of the T charts included in previous editions These charts are a compromise set for gas processing as follows: ­K-DATA CHARTS These charts show the vapor-liquid equilibrium ratio, Ki, for use in example and approximate flash calculations The charts not give accurate answers, particularly in the case of nitrogen They are included only for illustrative purposes and to support example flash calculations and quick estimation of K-values in other hand calculations These charts should not be used in design calculations a hydrocarbons — 3000 psia Pk [20 700 kPa (abs)] b nitrogen — 2000 psia Pk [13 800 kPa (abs)] c hydrogen sulfide — 3000 psia Pk [20 700 kPa (abs)] The pressures in a through c above refer to convergence pressure, Pk, of the charts from the Tenth Edition of this data book They should not be used for design work or related activities Again, they are in this edition for illustration and approximation purposes only; however, they can be very useful in such a role The critical locus chart used in the convergence pressure method has also been retained (Fig 25-8) Previous editions of this data book presented extensive sets of K-data based upon the GPA 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 recognizes 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 The GPA/GPSA sponsors investigations in hydrocarbon 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 FIG 25-1 Nomenclature P* = vapor pressure, kPa (abs) R = universal gas constant, (kPa (abs) • m3) / (kmole • K) T = temperature, K or °C V = ratio of moles of vapor to moles of total mixture xi = mole fraction of component i in the liquid phase yi = mole fraction of component i in the vapor phase Subscripts Ki = equilibrium ratio, yi xi L = ratio of moles of liquid to moles of total mixture N = mole fraction in the total mixture or feed w = acentric factor P = absolute pressure, kPa (abs) Pk = convergence pressure, kPa (abs), psia 25-1 i = component ­Example 25 -1  — Binary System Calculation To illustrate the use of binary system K-value charts, assume a mixture of 60 kmols of methane and 40 kmols of ethane at –87°C and 345 kPa (abs) From the chart on page 25-10, the K-values for methane and ethane are 10 and 0.35 respectively Solution Steps ­From the definition of K-value, Eq 25 -1: xC1 = 0.0674 yC1 = 0.674 ­Also by solving in the same way: xC2 = 0.9326 yC2 = 0.326 To find the amount of vapor in the mixture, let v denote kmols of vapor Summing the kmols of methane in each phase gives: ∑ ­ kmols C1 + C2 = 100 kmols Ni L + VKi = 1.0 Eq 25-5 Other useful versions may be written as Ni L = ∑ + (V/L) Ki Ki Ni ∑ yi = L + VKi = 0.35 ­Solving the above equations simultaneously: Eq 25-4 ­Rewriting for this binary mixture: – xC1 ­L + V = 1.0 ∑ xi = ∑ yC2 ­ C2 = K = 0.35 xC2 – yC1 Eq 25-3 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, yC1 = 10 ­KC1 = xC1 Using the relationships yi ­ i = K xi Eq 25-6 Eq 25-7 ­At the phase boundary conditions of bubble point (L = 1.00) and dew point (V = 1.00), these equations reduce to ∑ Ki Ni = 1.0 (bubble point) Eq 25-8 and ∑ Ni/Ki =1.0 (dew point) Eq 25-9 These are often helpful for preliminary calculations where the phase condition of a system at a given pressure and temperature is in doubt If ∑K­iN­i and ∑N­i/K­i are both greater than 1.0, the system is in the two phase region If ∑K­iN­i is less than 1.0, the system is all liquid If ∑N­i/K­i is less than 1.0, the system is all vapor (yC1 × v) + (xC1 [100 – v]) = 60 kmols Example 25-2 — 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 4140 kPa (abs) to liquefy heavier hydrocarbons prior to cooling to lower temperatures where these components would freeze out as solids (0.674 × v) + (0.0674 [100 – v]) = 60 kmols Solution Steps kmols C1 + kmols C1 = 60 kmols in vapor in liquid v = 87.8 kmols The mixture consists of 87.8 kmols of vapor and 12.2 kmols of liquid ­FLASH CALCULATION PROBLEM The problem below illustrates the calculation of multicomponent vapor-liquid equilibrium using the flash equations and the K-charts in detail The variables are defined in Fig. 25-1 Note that the K-value is implied to be at thermodynamic equilibrium A situation of reproducible steady state conditions in a piece of equipment does not necessarily imply that classical thermodynamic equilibrium exists If the steady composition differs from that for equilibrium, the reason can be the result of timelimited mass transfer and diffusion rates This warning is made because it 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, equilibrium 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 The feed gas composition is shown in Fig. 25-3 The flash equation 25 -5 is solved for three estimated values of L as shown in columns 3, 4, and By plotting estimated L versus calculated ∑x­i, the correct value of L where ∑x­i = 1.00 is L = 0.030, whose solution is shown in columns and The gas composition is then calculated using y­i = K­ix­i in column This “correct” value is used for purposes of illustration It is not a completely converged solution, for x­i = 1.00049 and y­i = 0.99998, columns and of Fig. 25-3 This error may be too large for some applications ­Example 25-3 — 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 (∑N­i/K­i = 1.0) is solved for two estimated dew point temperatures as shown in Fig. 25-4 By interpolation, the temperature at which ∑N­i/K­i = 1.0 is estimated at –41.4°C Note that the heaviest component is quite important in dew point calculations For more complex mixtures, the character25-2 FIG 25-2 Sources of K-Value Charts Charts available from sources as indicated Component Convergence pressures, kPa (abs) [psia] Binary Data Nitrogen * Methane * Ethylene Ethane * Propylene Propane * iso-Butane n-Butane *† iso-Pentane n-Pentane *† Hexane *† Heptane *† Octane Nonane x Decane Hydrogen sulfide Carbon dioxide *  □ × † ** 5500 6900 10 300 13 800 20 700 34 500 69 000 [800] [1000] [1500] [2000] [3000] [5000] [10,000]      × □  □ □ □ □    □  □       ×× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ × × × ** Use KCO2 = √ KC1 • KC2 Binary data from Price & Kobayashi; Wichterle & Kobayashi; Stryjek, Chappelear, & Kobayashi; and Chen & Kobayashi Drawn for 1972 Edition based on available data Note: The charts shown in bold outline are published in this edition of the data book The charts shown in the Reused from 1966 Edition shaded area are published in a separate GPA Technical Reused from 1957 Edition Publication (TP-22) as well as the 10th Edition Prepared for Second Revisions 1972 Edition or revised Limited to CO2 concentration of 10 mole percent of feed or less ­Separation of CO­2 and Methane ization of the heavy fraction as a pseudocomponent such as hexane or octane will have a significant effect on dew point calculations The relative volatility of CO­2 and methane at typical operating pressures is quite high, usually about to From this standpoint, this separation should be quite easy However, at processing conditions, the CO­2 will form a solid phase if the distillation is carried to the point of producing high purity methane ­Carbon Dioxide Early data on CO­2 systems used to prepare K-data (P­k = ) charts for the 1966 Edition were not consistent Later, experience showed that at low concentrations of CO­2, the rule of thumb ­KCO2 = √ KC1 • KC2 Eq 25-10 Fig. 25-5 depicts the phase diagram for the methane-CO­2 binary system.21 The pure component lines for methane and CO­2 vapor-liquid equilibrium form the left and right boundaries of the phase envelope Each curve terminates at its critical point; methane at –83°C, 4604 kPa (abs) and CO­2 at 31°C, 7382 kPa (abs) The unshaded area is the vapor-liquid region The shaded area represents the vapor-CO­2 solid region which extends to a pressure of 4860 kPa (abs) ­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 CO­2 processing See the GPA research reports (listed in Section 1) 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.­23 In general, CO­2 lies between methane and ethane in relative volatility Because the solid region extends to a pressure above the methane critical pressure, it is not possible to fractionate pure methane from a CO­2-methane system without entering the solid formation region It is possible to perform a limited separation 25-3 FIG 25-3 Flash Calculation at 4140 kPa and –30°C Column Component Feed Gas Composition 30°C 4140 kPa L = 0.020 Ki Ni Trial values of L Final L = 0.030 L = 0.060 L = 0.040L Liquid Ni L + V Ki Ni L + V Ki Ni L + V Ki L + Vki xi = Vapor Ni L + V Ki yi C1   0.9010 3.7 0.24712 0.25466 0.25084 3.61900 0.24896   0.0106 1.23 0.00865 0.00872 0.00868 1.22310 0.00867 0.01066 C2   0.0499 0.41 0.11830 0.11203 0.11508 0.42770 0.11667 0.04783 C3   0.0187 0.082 0.18633 0.13642 0.15751 0.10954 0.17071 0.01400 iC4   0.0065 0.034 0.12191 0.07068 0.08948 0.06298 0.10321 0.00351 nC4   0.0045 0.023 0.10578 0.05513 0.07249 0.05231 0.08603 0.00198 iC5   0.0017 0.0085 0.06001 0.02500 0.03530 0.03825 0.04445 0.00038 nC5   0.0019 0.0058 0.07398 0.02903 0.04170 0.03563 0.05333 0.00031 C6   0.0029 0.0014 0.13569 0.04730 0.07014 0.03136 0.09248 0.00013 C7+*   0.0023 0.00028 0.11334 0.03817 0.05712 0.03027 0.07598 0.00002 TOTALS   1.0000 1.17121 0.77714 0.89834 1.00049 0.99998 CO2** C7 0.92117 0.00042 C8 0.00014 * Average of nC7 + nC8 properties ** √ KC1 • KC2 FIG 25-4 Dew Point Calculation at 5500 kPa (abs) Column Feed Estimated T = –45°C Estimated T = –40°C Ni Ki Ni Ki CH4 0.854 2.73 0.313 2.75 0.311 CO2 0.051   0.866 0.059   0.910 0.056 C2H6 0.063   0.275 0.229   0.300 0.210 C3H8 0.032   0.070 0.457   0.080 ∑ 1.000 Component Ki Ni Ki 1.058 0.400 0.977 KCO2 calculated as √ KC1 • KC2 Linear interpolation: Tdew = –40 [–40 – (–45)] Alternative iterate until ∑Ni/Ki = 1.0 1.000 – 0.977 = –41.4°C  1.058 – 0.977  of CO­2 and methane if the desired methane can contain significant quantities of CO­2 At an operating pressure above 4860 kPa (abs), the methane purity is limited by the CO­2-methane critical locus (Fig. 25-6) For example, operating at 4930 kPa (abs), it is theoretically possible to avoid solid CO­2 formation (Fig. 25-7 and 16-36) The limit on methane purity is fixed by the approach to the mixture critical In this case, the critical binary contains 6% CO­2 A practical operating limit might be 10-15% CO­2 One approach to solve the methane-CO­2 distillation problem is to use extractive distillation (See Section 16, Hydrocarbon Recovery) The concept is to add a heavier hydrocarbon stream to the condenser in a fractionation column About 10 GPA research reports present data on various CO­2 systems that are pertinent to the design of such a process ­CO­2-Ethane Separation The separation of CO­2 and ethane by distillation is limited 25-4 by the azeotrope formation between these components An azeotropic composition of approximately 67% CO­2, 33% ethane is formed at virtually any pressure.24 FIG 25-5 Phase Diagram CH4 -Ch2 Binary 21 Fig. 25-7 shows the CO­2-ethane system at two different pressures The binary is a minimum boiling azeotrope at both pressures with a composition of about two-thirds CO­2 and onethird ethane Thus, an attempt to separate CO­2 and ethane to nearly pure components by distillation cannot be achieved by traditional methods, and extractive distillation is required.26 (See Section 16, Hydrocarbon Recovery) ­Separation of CO­2 and H­2S The distillative separation of CO­2 and H­2S can be performed with traditional methods The relative volatility of CO­2 to H­2S is quite small While an azeotrope between H­2S and CO­2 does not exist, vapor-liquid equilibrium behavior for this binary approaches azeotropic character at high CO­2 concentrations25(See Section 16, Hydrocarbon Recovery) FIG 25-6 Isothermal Dew Point and Frost Point Data for Methane-Carbon Dioxide32 25-5 ria for ideal or slightly non-ideal systems; they are not suitable for representation of highly non-ideal systems (e.g., methanol/water systems) FIG 25-7 Vapor-Liquid Equilibria CO2-C2H6 21 They typically are applied only to hydrocarbon mixtures with relatively low concentrations of non-polar or slightly polar fluids Recent advancements have made cubic EOS suitable for handling high concentrations of CO2, H2S, and N2 ● Applicable for prediction of phase equilibria for pure com- ponents (VLE) and mixtures (VLE and VLLE) and for prediction of all thermodynamic properties for vapor and liquid phases Originally developed for handling of pure components, but inclusion and use of various mixing rules, which incorporate binary interaction parameters, have allowed the extension of use to binary and multicomponent mixtures Useful over wide ranges of temperature and pressure, including subcritical, supercritical, and retrograde regions ● Require minimal pure component data Experimental bi- nary data can be used to “tune” binary interaction parameters, usually by regression of experimental data PHASE EQUILIBRIA METHODS ● Major EOS types include cubic, virial, corresponding states, and multi-parameter Descriptions of the more commonly used cubic and virial types are included below: Numerous procedures have been devised to predict phase equilibria (K-values) and the corresponding physical properties of the associated phases These include: Cubic EOS (e.g., van der Waal, Redlich-Kwong, Soave-Redlich-Kwong, Peng Robinson) • Equations of state (EOS) Explicit in pressure (P) with respect to temperature (T) and volume (V) They have separate terms to correct ideal gas predictions for attraction and repulsion forces between the molecules (correcting the real vapor pressure and volume predictions, respectively) When considering the pressure and temperature fixed, the EOS can be algebraically rearranged to give a relationship for V that is a cubic (3rd order) polynomial • Activity coefficient models • Electrolytic models • Combinations of equations of state with liquid theory or with tabular data • Corresponding states correlations A number of methods can be used for the purpose of phase equilibria and thermodynamic property prediction In modern times, calculations are not typically executed by hand, but instead are solved by the use of thermodynamic simulation software (commercial or proprietary) This section describes several of the more popular procedures currently available It does not purport to be all-inclusive or comparative These EOS will include other parameters, specific to each chemical species that are generally determined from the critical properties, Pc and Tc, for the chemical species Additional temperature-dependent functions can be added to more accurately match pure component behavior (i.e., a temperature dependent function correlated to the accentric factor (ω) is normally used to better match a pure component’s vapor pressure versus temperature behavior) Equations of State (EOS) Equations of state have appeal for predicting thermodynamic properties because they provide internally consistent values for all properties in convenient analytical form The section below discusses the basic capabilities of EOS, historical development, and recent advances Multicomponent mixtures are treated with the same EOS parameters that are determined for the pure components present in the mixture The equations used to blend the pure component values are referred to as “mixing rules”, which often include “binary interaction parameters” to account for nonideal interactions between pairs of unlike molecules EOS Capabilities — The following summarizes the basic capabilities and describes the applicability for some of the more commonly used EOS methods ● Although originally developed to describe simple gases, EOS have proven reliable for property prediction of most hydrocarbon-based fluids The EOSs are generally not “tuned” to pure component liquid density data, so they give poor representations of liquid molar volume/liquid den- The simple cubic EOS are generally limited to prediction of thermodynamic properties and phase equilib- 25-6 sity (There are techniques to improve this, such as introducing a volume translation term; see ‘Recent EOS Advancement’ section below) Examples include NBS Steam Tables, Span and Wagner EOS (CO2), Wagner and Pruss EOS (Water), and heat transfer fluid models Not generally accurate for polar compounds or long chain hydrocarbons (There are techniques to improve this, such as introduction of a higher order temperature dependency on the attraction parameter or asymmetric mixing rules; see ‘Recent EOS Advancement’ section below) Historical Development of EOS for Phase Equilibria — 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 fit of vapor-pressure data This original equation is reasonably accurate for light paraffin mixtures at reduced temperatures of 0.6 and above.8 The equation has difficulty with low temperatures, non-hydrocarbons, non-paraffins, and heavy paraffins Virial EOS (e.g., BWR, BWRS, Lee-Kessler-Plocker) Explicit in pressure (P) with respect to temperature (T) and volume (V) Expanded in terms of volume raised to powers much higher than 3rd order Because of the larger number of parameters that can be tuned to pure component data, these EOS can be more accurate than cubic EOS when calculating liquid densities 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 that is capable of predicting light paraffin K-values at cryogenic temperatures For pure components, these EOS can give a more accurate representation than cubic EOS for all thermodynamic data However, the determination of these parameters is more complex, requires more experimental data, and may require a complicated procedure for fitting that experimental data 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 When applied to multicomponent mixtures, the large number of pure component parameters must be blended, perhaps each with their own mixing rules Inaccuracies associated with the application of these mixing rules may make the mixture properties no more accurate than what one would get from a simpler cubic EOS 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 factor11, 18, 19 and by modification of the parameter-combination rules.15, 19 Considering the simplicity of the RedlichKwong equation form, the various modified versions predict Kvalues remarkably well Not usually applicable for polar systems Computerized corresponding states methods may be based on virial EOS for reference fluids The corresponding states methods then provide the framework to blend the properties of the reference fluids to give values for a multicomponent mixture 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 CO2 fractionation processes, and in conventional fractionation plants to predict sulfur compound distribution Other applicable EOS types EOS for Associating Systems The Chao-Seader correlation7 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 to 300°F and pressures to 2000 psia The original (P,T) limitations have been reviewed.12 EOS that include terms for the physical forces (attraction/repulsion) and an associating term that takes into account hydrogen bonding; see Kontogeorgis and Folas34 for reviews of associating EOS The SAFT (Statistical Associate Fluid Theory) family of EOS are based on Wertheims perturbation theory and can be applied to a wide range of fluids including long chain components and hydrogen bonding (e.g., hydrocarbon-alcohol-water systems) Prausnitz and Chueh have developed16 a procedure for highpressure 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 CPA (Cubic-Plus-Association) EOS combines a cubic equation of state (e.g., SRK) for describing physical interactions with an association term similar to SAFT The corresponding states principle10 is used in all the procedures discussed above The principle assumes that the behavior of all substances follows the same equation forms and equation parameters are correlated versus reduced properties and acentric factor An alternate corresponding states approach is to refer the behavior of all substances to the properties of a reference Highly Accurate Pure Component EOS Typically apply only to utility systems within a facility, not to the main processing and separation trains 25-7 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 available experimental data; however, more experimental data is required to allow for a proper fit of the mixing equation Application of more complex mixing rules can make EOS methods adequate for polar/non-ideal systems Specifically, Activity Coefficient methods have been used directly in some mixing rules to more accurately predict binary interactions of mixtures with polar and non-polar components at high pressure, despite the Activity Coefficient method only being fit to available low pressure experimental data (i.e., Wong-Sandler) “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 Sixty-two components have been correlated including olefinic, naphthenic, and aromatic hydrocarbons ● Enhanced binary interaction parameters Group contribution methods have been developed to estimate binary interactions (e.g Predictive SRK) and greatly improve predictions especially for mixtures with polar and non-polar components The Soave Redlich-Kwong (SRK)13 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 a 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 Interaction parameters are typically fitted to experimental data for each specific EOS and mixing rule combination In turn, more quality experimental data in the pressure, temperature, and compositional region of a particular application of interest allows for enhanced binary interaction parameters and improved EOS predictions However, fitting interaction parameters to different sets of data will result in inconsistent predictions from one tool to another Peng and Robinson14 similarly developed a two-constant equation of state in 1976 In this correlation, the attractive pressure term of the semi-empirical van der Waals equation has been modified It predicts the vapor pressures of pure substances and equilibrium ratios of mixtures Binary interaction parameters are often temperature dependent, and may be fit by differing temperature dependency forms, for which proper choice can impact EOS performance The ability for a user to specify binary interaction parameters is included in many of commercially available simulation products A tool that allows for non-constant specification (e.g., includes temperature dependence) will result in improved results In applying any of the above correlations, the original critical/physical properties used in the derivation should be inserted into the appropriate equations It is common for one to obtain slightly different solutions from different computer programs, even for the same correlation This can be attributed to different pure component and binary parameters, iteration techniques, convergence criteria, and initial estimation values, among other items as described in the Recent EOS Advancements sub-section below Determination and selection of a particular equation of state and interaction parameters must be done carefully, considering the system components, the operating conditions, etc ● Additional equation terms Addition of extended or advanced ‘alpha’ functions (intermolecular attraction) to improve fitting of vapor pressure, which can improve the ability of the EOS to handle polar/non-ideal systems Addition of volume translation parameters allow for better prediction of liquid densities for the EOS Recent EOS Advancements — While some of the fundamental, basic equation of state forms are included at the end of this section, there have been many advancements in the prediction of phase equilibria and thermodynamic properties since the last update of this section (pre-1990) As a result, and due to the extensive use of commercial simulation tools, results which differ somewhat will likely be obtained for the same Equation of State, depending on the software chosen, and even options selected within the software In addition to those items listed in the “Historical development of EOS” section above, this is largely due to the advancements made in application of EOS methods The following is a brief summary of some basic reasons for these differences from one software package to another, along with a general description of advanced applications of EOS methods ● Liquid phase property handling Modification of the handling of the term describing real volume of molecules/intermolecular repulsion allow for better prediction of liquid densities for the EOS ● Solids handling (e.g., ice, hydrate, solid CO2, solid hydro- carbon) While EOS are used to represent the fugacity of components in a fluid phase (vapor and liquid), they can be combined with models representing the fugacity in the solid phase to model VSE and LSE See the section titled “Equations of State and the Solid Phase” below for more detailed discussion ● Improved mixing rules A number of different mixing rules can be applied to an EOS, some much more complex than others In general, more complex mixing rules allow for the range of applicability of an EOS extended further beyond the There are a number of multi-parameter equations (i.e., GERG35), that currently exist and are able to model systems to within experimental error However, due to the complexity and computing power required for these, they are not often used in 25-8 facility design simulations One obvious use of an equation of this nature is to generate pseudo-experimental data from which new binary interaction parameters can be regressed for a specific system (P, T, and composition), and in turn used in an EOS to improve its reliability ­BASIC EQUATION OF STATE FORMS Refer to original papers for mixing rules for multicomponent mixtures Z3 – (1 + B) Z2 + AZ – AB = aP A = 2 R T bP B = RT m = 0.37464 + 1.54226 w – 0.26992 w2 R Tc b = 0.0778  Pc  1 R T Co Do Eo P = + Bo R T – Ao – + – V  T T T  V d d + bRT – a – + a a +  T  V  T  V γ c + + V  T   V  ⁄v2 –γ Activity Coefficient Models — Another common method used for the purpose of phase equilibria and thermodynamic property prediction is the use of Activity Coefficient models The following is a brief summary of the basic capabilities and describes the applicability for some of the more commonly used Activity Coefficient methods ● Activity coefficient models are the best method for repre- sentation of highly non-ideal and/or polar systems (i.e., aqueous systems, amines, NH3, caustic, CO2, H2S) and are therefore typically used in the chemicals industry bP B = RT R2 T2c.5 a = 0.42747  Pc  R Tc b = 0.0867  Pc  ● While these models are generally only applicable to pre- diction of phase equilibria for binary and multicomponent mixtures (VLE and LLE), they are not for phase equilibria of pure components However, they require high quality pure component property predictions (e.g., vapor pressure) Depending on the specific Activity Coefficient method, it may not always allow for LLE prediction because tuning the models to VLE specific data or LLE specific data may result in drastically different parameters For this reason, VLLE predictions must also be used with caution ­Soave Redlich-Kwong (SRK)13 Z3 – Z2 + (A – B – B2) Z – AB = aP A = 2 R T bP B = RT a = ac a R2 T2c ac = 0.42747  Pc  a ⁄2 = + m ( – Tr⁄2) m = 0.48 + 1.574 w – 0.176w2 R Tc b = 0.08664  Pc  ● Applicable for prediction of thermodynamic properties for the liquid phase only Vapor properties are unreliable and must be calculated using another method; historically this has been done using ideal gas assumptions for the vapor phase, but commonly includes more advanced EOS methods, as described in the ‘Mixed Models’ section below ● Limited to systems within the pressure and temperature ranges of the experimental data it is correlated against These models are only suitable for low to moderate pressure systems, typical in the chemicals industry, because activity coefficients depend on temperature, but are independent of pressure, while mutual solubilities are in fact dependant on pressure in high pressure LLE systems ­­Peng Robinson31 Z3 – (1 – B ) Z2 + (A – 3B2 – 2B ) Z – (AB – B2 – B3) = aP A = 2 R T a ⁄2 = + m (1 – Tr⁄2) Other Phase Equilibria Methods Z3 – Z2 + ( A – B – B2) Z – AB = aP A = 2.5 R T Note: w, the acentric factor is defined in Section 23 27 R2 T2c a = 64 Pc R Tc b = Pc ­Redlich-Kwong28 R2 T2c a a = 0.45724  Pc  ­Benedict-Webb-Rubin-Starling (BWRS)20, 29 ­van der Waals30 At typical operating pressures, the use of a vapor pressure is not appropriate for light gases above the critical point and instead these light gases are treated as Henry’s components, where Henry’s law coefficients are derived from experimental gas solubility data bP B = RT 25-9 Extrapolation outside the experimental data range is not recommended Hydrogen bonding (i.e., methanol-water-hydrocarbon systems) ● Requires a separate model to determine liquid density ● Natural Gas Dew Point Calculations The Poynting correction is used to account for pressure dependence of a components liquid phase activity ● Other considerations that impact phase equilibria Reactive (i.e., amine gas treating, caustic treating) Some examples of common Activity Coefficient models include: Chien Null, NRTL, Margules, UNIFAC, UNIQUAC, van Laar, and Wilson Electrolytic Models — Electrolytic models are a sub-set of Activity Coefficient models The general purpose of Electrolytic models is to handle systems where dissociation of components is important In general, these components not directly participate in VLE (i.e., ions or solids that not dissolve or vaporize), but they often influence activity coefficients of other species by reaction or interaction and in turn, indirectly participate in impacting the phase equilibria of the system These models generally require high quality experiment data for the specific application they are being applied to (e.g., amine gas treating systems) Mixed Models — Mixed models are commonly used in order to combine the strengths of the various methods described above These models commonly use an EOS method for phase equilibria and prediction of vapor phase thermodynamic properties, but use an Activity Coefficient or Electrolyte method for determination of thermodynamic properties and/or density of the liquid phase(s) However, because of the use of multiple methods, these models not always produce consistent predictions, especially at or near the critical point Practical Application of Phase Equilibria Methods General Considerations — While phase equilibria of typical hydrocarbon systems is modeled very well with currently available commercial computer simulation tools, there are many areas where careful attention to method selection is needed Some systems, due to the thermodynamic complexity, are difficult to model using the basic methods described above, even with more and/or better quality experimental data Some examples of these complex systems in the gas processing industry are shown below The equilibria and thermodynamic property results obtained using a specific method for these systems should be carefully evaluated and compared to commercial experience when used in a process design: Complex Systems to Model — ● Operating conditions that cause divergence from ideal fluid behavior Self associating/dimerizing (i.e., aqueous solutions) Dissociating (i.e., amine gas treating, sour water) Sterically hindered dissociating compounds (i.e., certain amines) Near critical and supercritical fluids (i.e., supercritical natural gas compression, supercritical H2S/CO2 compression, hydrocarbon systems near critical) Combinations of non-polar and polar compounds (i.e., hydrocarbon systems with sour or produced water, dehydration or dew point depression with glycols or methanol) Mass transfer limited (i.e., tertiary and hindered amines) Absorption (i.e., CO2 and H2S adsorption in physical or chemical solvents) Overall, it is important to understand the capabilities and limitation of each method of representing phase equilibria and prediction of thermodynamic properties, and each method’s specific applicability to the gas processing industry for proper choice and use However, it should be noted that tuning of a method to quality experimental data in the region of operation or interest is perhaps more important than the method choice itself, specifically, the choice of mixing rules and quality of binary interaction parameters, which can typically be readily modified in commercial software More specific information relative to phase equilibria methods can be found in Goodwin et al36 and Kontogeorgis and Folas.34 Dew Point Calculation — A thermodynamic dew point is defined as that point where liquid first appears from the gas phase An EOS model actually calculates this point with exactly zero liquid dropout In reality, for natural gas this point cannot directly be measured from experimental methods, but can be estimated from PVT data taken very near the dew point envelope by extrapolating liquid dropout data to 0.0 volume percent liquid In pipeline operations, one can consider a practical dew point that represents a small volume of liquid condensation which does not impact pipeline performance, usually representing a trace of liquid on the pipe wall This practical dew point is what is actually measured by the Bureau of Mines chilled mirror device In recent work for the GPA by Bullin, et al., (RR-213), the practical dew point was defined as 0.00027 m3 liquid per 1000 m3 gas As a natural gas gets leaner, the difference between the EOS predicted dew point of 0.0 volume percent liquid and the practical dew point of 0.00027 m3 liquid per 1000 m3 gas can increase to a much as 5.6°C The practical dew point can be represented by the temperature in a Bureau of Mines chilled mirror device where droplets begin to form Thus, when EOS models are used, both the thermodynamic dew point of 0.0 volume percent liquid and the practical dew point of 0.00027 m3 liquid per 1000 m3 gas should be considered when adjusting the EOS parameters to match the experimental values, and also to better determine the conditions that impact plant/pipeline performance For lean gases where there is more than a 2.2 to 5°C difference in the two values, it may be necessary to only fit to the practical dew point value to evaluate plant/pipeline performance This also points to the fact that when fitting an EOS model to an experimentally obtained dew point value, it should not be assumed that the reported dew point (experimental) represents the thermodynamic dew point of 0.0 volume percent liquid, unless of course it has been confirmed to be extrapolated from multiple experimental points within the two phase envelope 25-10 25-16 25-17 25-18 25-19 25-20 25-21 25-22 25-23 25-24 25-25 25-26 25-27 25-28 25-29 25-30 ... introduction to the general topic of solid phases The Phase Rule”, developed by Gibbs back in the 1870s, still serves as a trusty background to multi -phase equilibria For non-reactive compounds,... fluid phase (vapor and liquid), they can be combined with models representing the fugacity in the solid phase to model VSE and LSE See the section titled “Equations of State and the Solid Phase ... are generally only applicable to pre- diction of phase equilibria for binary and multicomponent mixtures (VLE and LLE), they are not for phase equilibria of pure components However, they require