Fuel 115 (2014) 1–16

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Fuel

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Development of a thermodynamically consistent, robust and efficientphase equilibrium solver and its validations

0016-2361/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.fuel.2013.06.039

⇑ Corresponding author. Tel.: +1 608 262 0145; fax: +1 608 262 6707.E-mail addresses: lqiu4@wisc.edu (L. Qiu), wang46@wisc.edu (Y. Wang),

qjiao2@wisc.edu (Q. Jiao), hwang259@wisc.edu (H. Wang), reitz@engr.wisc.edu(R.D. Reitz).

1 Tel.: +1 662 617 9132.

Lu Qiu 1, Yue Wang, Qi Jiao, Hu Wang, Rolf D. Reitz ⇑Engine Research Center, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI 53705, USA

h i g h l i g h t s

� A robust and efficient equilibrium solver is developed.� The solver is based on rigorous thermodynamics.� Complex phase behaviors of multi-component mixtures are predicted.� Liquid–liquid and vapor–liquid–liquid equilibria are predicted.� Feasible region method is applied to accelerate three-phase flash.

a r t i c l e i n f o

Article history:Received 3 May 2013Received in revised form 25 June 2013Accepted 25 June 2013Available online 12 July 2013

Keywords:Phase equilibriumPhase stabilityPhase splittingFlash calculation

a b s t r a c t

An applied phase-related equilibrium (APPLE) solver using only the Peng–Robinson equation of state isdeveloped based on rigorous classical thermodynamics. The solver is theoretically and thermodynami-cally consistent with the stringent equilibrium criterion. It is mainly composed of phase stability andphase splitting calculations, which will be called routinely in the course of searching for the globally sta-ble equilibrium state. It also makes use of various robust and efficient numerical methods. To demon-strate its performance, the solver is tested against various mixtures, such as oil and gas mixtures,hydrocarbon mixtures and hydrocarbon–nitrogen mixtures. Phase diagrams of these mixtures are con-structed and verified with available experimental data or other researchers’ calculations. Results showthat the APPLE solver is reliable and fast to solve phase equilibrium problems, including three-phaseequilibrium. Finally, its potential applications to droplet evaporation and computational fluid dynamics(CFD) calculations are discussed.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The phase transition/change process is an important physicalprocess that occurs in many engineering applications, such asCO2 injection and CO2 storage in enhanced oil recovery, and liquidphase combustion upon evaporation. Considerations on thedynamics of phase transition, which requires analyses of non-equi-librium thermodynamics such as the nucleation process, areincomplete if the equilibrium state (‘‘destination state’’) is notknown. Phase equilibrium, hence, is a starting point for furthernon-equilibrium analyses and it is of primary importance fromthe point view of classical thermodynamics. In reality, phase equi-librium thermodynamics-based calculations have been used in awide range of industrial fields, such as for distillation columns,

and in reservoir simulation. An equation of state, as an importantpart of thermodynamics needed to quantify the state properties,is generally used to study non-ideal gas behaviors of mixtures.

From the fundamental thermodynamic postulate, the entropy isa continuous and monotonically increasing function of energy [1].In terms of the Gibbs free energy G for a homogeneous and opensystem, the second law of thermodynamics limits the possible pro-cesses through:

dG ¼ VdP � SdT þX

li � dni 6 0 ð1Þ

Here, T is temperature; P is pressure; V is volume; S is entropy, l ischemical potential; n is mole number and i is the species index. Atthe equilibrium state, the equality sign in the above equation holds.It is well known that the Gibbs free energy at the equilibrium stateis the minimum in the phase space composed of T, P and ~n.Mathematically, the problem now becomes to search for a globalminimum point in multi-dimensional space. This problem is usuallytackled by solving for the equality of the chemical potential, whichis the partial molar Gibbs free energy. In actual calculations,

2 L. Qiu et al. / Fuel 115 (2014) 1–16

fugacity equality is commonly used for identifying the equilibriumstate by requiring, for a two-phase equilibrium,

f 1i ¼ f 2

i ; i ¼ 1; . . . ; Nc ð2Þ

Here, j is the index of phase; f is the fugacity and Nc is the num-ber of components. The advantage of this method is that fugacitycoefficient can be relative easily calculated from various equationsof state, such as the Peng–Robinson equation of state [2].

However, Eq. (2) is not a sufficient condition for phase equilib-rium. Mathematically, the zero derivative location only indicates alocal extreme but does not guarantee that it is also the global ex-treme. Hence, the equality of fugacity in Eq. (2), standing for a localstationary point in the phase space, is not necessarily the globalminimum at the same time. In other words, solution of Eq. (2)could represent a ‘‘false’’ equilibrium state, which only correspondsto some local minima. As a result, fugacity equality is a necessarybut not sufficient condition for phase equilibrium. While for simplemixtures, the local extreme could happen to be a global extreme,the situation quickly becomes complex when a multi-componentmixture is considered since the phase dimension and the numberof local extrema will increase. This is especially the case that occursquite often at liquid–liquid and vapor–liquid–liquid equilibria [3].All these noteworthy points are quantitatively illustrated in thework of Baker et al. [4], benefiting from the original work of Gibbs[5]. They suggested using Gibbs free energy directly for phase equi-librium to guarantee the ‘‘true’’ equilibrium state, rather than thetraditional way with fugacity. They further proved that the neces-sary and sufficient condition for a system to be stable, for specifiedtemperature, pressure and species, is that the tangent plane at thespecies feed composition to the Gibbs free energy surface should atno other point intersect (lie above) the Gibbs free energy, but onlythe tangent point.

This paper is organized as follows. The equation of state modelis introduced first, since it is used for both phases throughout allthe calculations. After this, the methodologies adopted for the cur-rent development of a robust and efficient applied phase equilib-rium solver, named APPLE here, are presented in detail. We thenevaluate the performance of the solver for various cases from theliterature. Some potential applications of the solver to computa-tional fluid dynamics (CFD) calculations are remarked upon. Sum-mary and conclusions are then made.

2. Equation of state

The equation of State is a relationship between thermodynamicproperties at a specified state. To consider the non-ideality of ther-modynamic properties at high pressures, a proper equation of stateother than the ideal gas law is needed. While there are no generalguidelines for selecting a specific equation of state, the Peng–Rob-inson equation of state (PR EOS) [2] was chosen for all the calcula-tions here for several reasons. First, it is a simple form of the cubicequation of state so it is easy to implement for engineeringcalculations. Second, it has better performance for the predictionof vapor–liquid phase equilibrium properties over other cubicequation of states. More importantly, it also has been widely usedin the oil and gas industry with great successes.

Theoretically, using only one equation of state for both theliquid and gas phases would bring some numerical difficultiessince more solutions of cubic equations are possible. However, ithas the benefit of using one unified treatment for both phasesand hence it is easier to handle and elegant to implement. In addi-tion, the non-random mixing model, such as UNIFAC, is not usedhere because it is specially designed for handling the non-idealityof compressed liquids or solutions; and it is mainly used for sub-critical conditions where the molecular interactions are strong

and surface tension effects are important. An equation of statemodel, on the other hand, can be used for a wide range of condi-tions (from subcritical to supercritical). Besides, there is no needto specify the reference state for equation of state models sinceideal gas limit is used exclusively. Finally, enthalpy and some otherthermodynamic properties can be determined directly from anequation of state model in a consistent manner.

The PR EOS [2] is of the form:

P ¼ RTv � b

� avðv þ bÞ þ bðv � bÞ ð3Þ

Here, v is the molar volume. a and b are the two parameters deter-mined from:

a ¼ 0:457235 R2T2c

Pc� a

b ¼ 0:077796 RTcPc

a ¼ ½1þ jð1�ffiffiffiffiffiTrpÞ�2

8>><>>: ð4Þ

With

j ¼ 0:37464þ 1:54226x� 0:26992x2 ð5aÞ

Here, Tc, Pc and x are the critical temperature, pressure andacentric factor, respectively. We also adopted a formula that is la-ter expanded [6], which uses, when x is greater than 0.5.

j ¼ 0:3796þ 1:485x� 0:1644x2 þ 0:01667x3 ð5bÞ

In the case of a mixture, the classical Van der Waals mixing ruleis used:

a ¼X

i

Xj

xixjaij

b ¼X

i

xibi

aij ¼ ð1� di;jÞffiffiffiffiaip ffiffiffiffi

ajp

8>>>>><>>>>>:

ð6Þ

Here, xi is the mole fraction of species i in the mixture. di,j is thebinary interaction parameter between components i and j, and it isgenerally assumed to be independent of pressure or temperaturefor a mixture. Temperature-dependent interaction parameterscan be used to improved vapor–liquid equilibrium predictions withthe group contribution method discussed in Ref. [7,8]. This groupcontribution method is used in the petroleum industry to findinteraction parameters for hydrocarbons (Avaullee et al. [9]), butit is not pursued here. More advanced mixing rules are not pursuedin the current work, either. Cubic equations generally have threeroots at sub-critical conditions. In all the calculations here, whena compressibility root is to be solved, only the one with minimumGibbs free energy is chosen.

3. Methodology

The equilibrium condition shown in Eq. (1) is written in termsof Gibbs free energy, which has been well applied to isothermal–isobaric flash (TPn) calculations. This kind of flash calculation hasbeen investigated by many researchers and there are many well-developed theories and numerical methods. There is a growing rec-ognition that robust and efficient phase equilibrium calculationsshould be composed of two essential parts: phase stability andphase splitting calculations [3,10,11].

3.1. Phase stability

The traditional two-phase flash calculation assumes that a two-phase solution exists. If no solution or trivial solution is found, asingle phase is then believed to exist. This becomes more and moretroublesome when the mixture approaches its thermodynamic

L. Qiu et al. / Fuel 115 (2014) 1–16 3

critical point, where most flash techniques are likely to fail due tothe similarity between the vapor and liquid phases. Another issueis that sometimes a blind flash search may even yield the wrongsolution at a local extreme of Gibbs free energy. This point was alsomade by Baker et al. [4], as mentioned above. However, their crite-rion is more descriptive than useful because they did not discloseany numerical methods that can actually be implemented. Sincethe actual number of phases coexisting at a specified thermody-namic state is not known a priori, Michelsen [12] suggests that aphase stability test be conducted before the flash calculation.

Similar to the Gibbs free energy analyses of Baker et al. [4],Michelsen [12] used the Gibbs free energy of mixing to develop amethod called the Tangent Plane Distance (TPD). The TPD functionis defined as:

TPDð~xÞ ¼XNc

i¼1

xi � ðlið~xÞ � lið~zÞÞ ð7Þ

Here,~x is the mole fraction array and~z is the feeding composi-tion array. Geometrically, the TPD represents the vertical distancefrom the tangent hyper-plane (to the molar Gibbs free energy sur-face of ~z) to the molar Gibbs free energy surface of ~x. It can beproved that the stability test requires the TPD to be non-negative:

TPDð~xÞP 0 ð8Þ

for any accessible ð~xÞ subject to the species conservationconstraints:

xi P 0 andX

xi ¼ 1 ð9Þ

If TPDð~xÞ is always non-negative for all feasible ð~xÞ, then the ori-ginal mixture is thermodynamically stable and exists as a singlephase. On the other hand, if TPDð~xÞ has some negative values, theoriginal mixture is not stable and the phases will separate. Sinceall the minima are in the region of the physical constraint, TPD willbe non-negative if it is non-negative at all the stationary points,where the derivative with respect to all independent variables iszero. Therefore, this method reduces the search space from thewhole domain to the local extremes (i.e., local maxima, minimaor saddle points) and hence an exhaustive search of all possiblef~xg is not needed. It is remarked that the TPD is strictly zero at sat-uration point line; Implementation of such criterion to find the sat-uration temperature or pressure directly without building thecomplete phase envelop can be found in Ref. [13].

Even with this treatment, the actual search is still challenging,especially if the dimension of the phase space is large. For a mul-ti-component mixture, geometrically the Gibbs free energy of mix-ing is a hyper-surface and the tangent plane is a hyper-plane [4].When an equation of state is used, Eq. (7) can be expressed interms of fugacity coefficient

TPDð~xÞ ¼XNc

i¼1

xi � ½ln xi þ ln uið~xÞ � hi�P 0 ð10Þ

where

hi ¼ ln zi þ ln uið~zÞ ð11Þ

Michelsen [12] formulated a unconstrained minimization prob-lem in terms of a variable X:

TPD�ð~xÞ ¼ 1þXNc

i¼1

Xi � ½ln uið~XÞ þ lnðXiÞ � hi � 1� ð12Þ

where

Xi ¼ xi expð�KÞ ð13Þ

Xi here can be considered as mole number and K is a constant.Geometrically, K stands for the vertical distance between the

tangent hyper-plane of the energy surface at the stationary pointand the tangent hyper-plane at the feeding composition. It has beenproved that TPDð~xÞ has the same stationary points as TPDð~xÞ andthey have the same sign at the stationary points [12]. Nonetheless,TPDð~xÞ is more favorable because it now converts the constrainedminimization problem to an unconstrained one (except thatX > 0). This treatment bypasses the numerical difficulties of the con-strained minimization problem such as the boundary constraints.Hence, in the current calculation, Eq. (10) is selected as the objec-tive function for the minimization. To avoid negative mole fractionsduring the iterations, Michelsen [12] suggested to change the itera-tion variable as:

ai ¼ 2ffiffiffiffiffiXi

pð14Þ

The above conversion bypasses the negative X issue. It shouldbe mentioned that, during the iteration, X is still used when thefugacity coefficient is needed. The first order derivative of TPDð~xÞis needed and it is calculated according to

@

@aiTPD� ¼

ffiffiffiffiffiXi

p� ½lnuið~XiÞ þ lnðXiÞ � hi� ð15Þ

Even with all these numerical treatments, convergence is some-times still quite difficult, and multi-sided initial guesses are furthersuggested [3,12]. Hoteit and Firoozabadi [14] applied the Broyden–Fletcher–Golfarb–Shanno (BFGS) algorithm to find local extremes.This Quasi-Newton type method approximates the Hessian matrixrather than calculating it as in a standard Newton’s method, so it iscomputationally inexpensive. With each initial guess, a local sta-tionary point, if any, is determined. The global minimum is thenthe minimum of all these candidates. The updated scheme for thismethod is:

akþ1 ¼ ak þ ~Fðgk; sk; FkÞ ð16Þ

With

gk ¼ Fðakþ1Þ � FðakÞsk ¼ akþ1 � ak

(ð17Þ

Here F stands for the gradient of TPDð~xÞ mentioned above.Therefore, a local extreme indicates that F is zero in each direction.~F is the update matrix simplified from the BFGS algorithm. Detailsabout the numerical implementation can be found in [14], as wellas its application in a reduction method. This method, as will beshown later, is found to perform quite well in the non-linear min-imization and it is fast to converge. There is another category ofmethod using global stochastic search methods, such as theHomotopy method [15]. Nichita et al. [16] used a global optimiza-tion method called Tunneling to escape local extremes and saddlepoints. A recent review on Generic and Monte Carlo algorithms canbe found in Zhang and Bonilla-Petriciolet [17]. Most of the globaloptimization methods, however, do not guarantee convergence,either. Even if a global minimum can be found, it can only beachieved after exhaustive and cumbersome searches, so there is al-ways a heavy load on the computation time and resources. Sincewe are pursuing a fast and robust phase equilibrium solver thatwill be coupled with computational fluid dynamics (CFD) calcula-tions, these stochastic and global searching methods are not pur-sued. Especially when considering reacting flows, a large amountof computer time is spent on chemistry mechanism, which is al-ready complicated by the stiffness of the reaction system.

3.2. Phase splitting

Phase splitting occurs whenever a phase stability test yields anegative TPD value, indicating that the original mixture is non-sta-ble (either stable or metastable). It is traditionally called flash

4 L. Qiu et al. / Fuel 115 (2014) 1–16

calculation, which is mainly based on the Rachford–Rice algorithm[18]. For a two-phase equilibrium, the Rachford–Rice equation isXNc

i¼1

ziðki � 1Þ1þ kðki � 1Þ ¼ 0 ð18Þ

Here, ki is the phase equilibrium ratio of species i and k is themole fraction of one phase. The objective function is a monotoni-cally decreasing, but not continuous, function. Together there areNc poles for k. Hence, caution is needed to window the searchingregion between two asymptotes and the physical solution isbounded between 0 and 1. At the same time, depending on the ini-tial guess, the trivial solution, showing identical composition inboth phases, can also be generated. More seriously, an incorrectsolution can be generated by a blind search, especially at liquid–li-quid and three-phase equilibriums.

While implementation of this algorithm is straightforward fortraditional two-phase vapor–liquid equilibrium, the multi-phaseflash is not a trivial task. The multi-phase equilibrium, however,occurs quite often in real situations, such as in carbon dioxide(CO2) utilization for enhanced oil recovery, as well as when oil,gas and water coexist at low temperatures. These situations willbe shown in the following calculations. Analogous to Eq. (18), thecorresponding three-phase Rachford–Rice function is [19,20]:XNc

i¼1

ziðkij � 1Þhi

� �¼ 0; j ¼ 1; . . . ; ðNp � 1Þ ð19Þ

where

hi ¼ 1þXNp�1

j¼1

½ðkij � 1Þbj�; i ¼ 1; . . . ; Nc ð20Þ

kij ¼xij

xiNp

; i ¼ 1; . . . ; Nc; j ¼ 1; . . . ðNp � 1Þ ð21Þ

Here, Np is the number of phases; kij is the phase equilibrium ra-tio of component i in phase j relative to phase Np; bj is the mole frac-tion of phase j. Notice that the notation above implicitly assumesthat phase Np is the reference phase. Methods for three-phase equi-librium calculations before the TPD method seem not to be widelyused and are limited by numerical methods, which lack a system-atic thermodynamic treatment. For example, Mehra et al. [21] stud-ied multiphase equilibrium up to four phases using first-order andsecond-order methods. Depending on the numerical conditions,switches between these methods are needed. However, the numer-ical solutions are not guaranteed to be thermodynamically stablestates. When two solutions are found, a further comparison onthe mixture Gibbs free energy is needed. With the phase stabilitytest introduced, a systematic way of calculating phase equilibriumis possible. Michelsen [12] suggested to check phase stability beforethe flash calculations and to use the outcomes of stability as an ini-tial guess in the phase splitting calculations. Many calculations[10,11,19] also prove that the phase stability test can supply a verygood initial guess for convergence in flash calculations. Nghiem andLi [22] applied the TPD stability test and applied it to a quasi-Newton Successive substitution (QNSS) scheme to study multi-phase equilibria of reservoir oil and CO2 mixtures. Haugen et al.[23] showed that a combination of bisection and Newton methodsis another feasible approach that works quite well for a number ofthree-phase calculations on reservoir oils.

Okuno et al. [20] recently published a robust and efficient mul-ti-phase flash method named the feasible region algorithm. Thealgorithm is based on minimization of a non-monotonic convexfunction with Nc linear constraints. The feature of this method isthat it limits the solution in a small feasible region constrained by:

aTi b 6 bi ð22Þ

where bi = {1 � kij}, b = {bi}, and bi ¼minf1� zi;minjf1� kijzigg fori ¼ 1; . . . ; ðNp � 1Þ. This feasible region does not contain any poleswhere the Hessian matrix is ill conditioned, so a minimizationscheme with the aid of line search using the Hessian matrix is appli-cable. In addition, the constraint SC ¼ fbjaT

i b 6 bi; i ¼ 1; . . . ; Ncgcreates a much smaller feasible region compared to the traditionalconstraint LC ¼ fbjhi P 0; i ¼ 1; . . . ; Ncg developed by Leiboviciand Neoschil [24]. For positive flash, another constraintPC ¼ fbjbj P 0; j ¼ 1; . . . Npg is imposed. Independent of the num-ber of phases for both positive and negative flash, this method guar-antees convergence [20]. Generally, it requires only 6–7 iterationsfor convergence once the feasible region is determined; makingthe method is very robust and practical. For three-phase flash calcu-lations, only when close to the bi-critical point will the iterationsslow down, which is mainly because the lines from linear con-straints are close to parallel with each other. The above-mentionedthree widely used methods for three-phase flash, bisection method,direct Newton’s method and Okuno’s feasible region method, havebeen studied very recently by Li and Firoozabadi [11]. Their calcu-lations show that the direct Newton’s method is the most efficientone.

3.3. Summary

For the development of the APPLE solver, we adopted the fol-lowing methodologies. For a TPn flash at specified temperature,pressure and species mole numbers, we apply an equilibrium cal-culation in a stage-wise manner. Starting from one phase, the sta-bility of the mixture is tested. If it is proven non-stable from thetest, another phase is added with phase splitting and a flash calcu-lation is conducted. The above procedures continue until the mix-ture is stable. Formulated in this way, the ‘‘true’’ equilibriumsolution is guaranteed since a new phase is added if and only ifit decreases the mixture Gibbs free energy. This methodology isalso mentioned in Li and Firoozabadi [11].

For the phase stability part, the above-mentioned BFGS local-ized minimization scheme [14] is implemented to benefit fromits fastness and effectiveness. For the initialization, instead ofsearching for a global minimum using some stochastic methods,the multi-sided test suggested by Li and Firoozabadi [10] isadopted. For a mixture composed of Nc components, togetherNc + 4 tests are needed for a full phase stability test. The totalCPU time, however, is actually very cheap even with multi-sidedguesses because the Hessian matrix calculation is not needed,which generally requires large amounts of computational effort.While the initialization scheme can locate a maximum of threenon-trivial solutions [10], it is sufficient to locate the global mini-mum for stability testing based on our experience.

For the phase splitting part, we solve the Rachford–Rice equa-tions after the stability test. In a two-phase flash, with the initialguess from the one phase stability, either the bisection or Newton’smethod is used. The two-phase stability test can be done on eitherphase, and the phase with the higher molecular weight is selected.For the Rachford–Rice three-phase flash, both successive substitu-tion and Okuno’s method are implemented. The latter is used as abackup if the former fails, though this rarely happens in actual cal-culations. Details on the initialization strategies for the numericalmethods can be found in [10,11] and their suggestions were foundto be reliable.

4. Results and discussion

The tangent plane distance theory for stability testing and thenumerical methods for phase splitting summarized in Section 3.3were implemented in a Fortran program to formulate the applied

L. Qiu et al. / Fuel 115 (2014) 1–16 5

phase-related equilibrium (APPLE) solver. In the following, wepresent tests on a variety of mixtures. Most of them are for reser-voir mixtures, as a large amount of available data come from thisfield. The phase diagram, which is considered a ‘‘stability map,’’is critical to characterize the phase transition processes, and willbe the focus in the following sections to evaluate the performanceof APPLE solver. Dependence of phase equilibrium state on pres-sure will also be discussed. Since the Peng–Robinson equation ofstate is used, it is important to document clearly the propertiesand parameters used [25]. The thermodynamic properties usedas inputs to the solver, such as critical properties and binary inter-action parameters, are all documented in tables for each test. In allthe calculations of phase diagrams, unless mentioned separately,the phase boundary is recorded when a new phase appears or anold phase disappears if pressure is changed by 0.001 bar.

4.1. Test problem 1: CH4–H2S

Calculation of the CH4–H2S binary mixture is first performed toillustrate the usefulness of the stability test for the flash calcula-tions to get the correct solution. The mixture properties are sum-marized in Table 1. This classical problem has been also tested inRefs. [12,15,16,26]. The tangent plane distance is shown inFig. 1(a) for the 50:50 (by mole) mixture, it is noticed that it hasfive local extremes. The negative sign of the distance indicates thatthe binary mixture cannot stabilize itself in one phase, neither asingle liquid nor a vapor phase. However, if the numerical methodbecomes stuck at a local extreme at zero TPD, a false one-phasesolution will be found. Even if a non-stable phase is detected, ifthe phase splitting calculation is not robust or the initial guess isnot good, the calculation may still converge to another false va-por–liquid two-phase solution. This point is best illustrated by con-sidering the Gibbs free energy of mixing, shown in Fig. 1(b). It isnoticed that mixture could exhibit both liquid–liquid and vapor–li-quid behaviors, and only the former corresponds to the minimumof Gibbs free energy. At this situation, phase stability is critical insupplying a good initial guess for the subsequent flash calculation.The solution from the current equilibrium solver can read directlyfrom Fig. 1(b), which agrees well with the reference values in Refs.[12,15,16,26].

4.2. Test problem 2: C1–CO2–H2S–H2O

The mixture composed of C1–CO2–H2S–H2O is studied. Thethermodynamic data and calculation results are tabulated in Ta-bles 2 and 3, respectively. Convergence was always achieved. Thenumber of calculation iterations increase a little from a two-phaseflash to a three-phase flash. In Table 3, T and F stand for stable andnon-stable mixture, respectively. The numbers in the table are themole fractions of each phase. The phase stability results of thethree systems in Table 3 are consistent with those in Ref. [10].

4.3. Test problem 3: acid gas

The APPLE solver was applied to study the behaviors of acid gasand its mixing effects with CO2. The inputs needed for the solverare documented in Table 4. This mixture turns out to exhibit verycomplex behaviors, as shown in Fig. 2(a). Note that since the acid

Table 1Inputs for binary C1–H2S mixture at 190 K and 40.53 bar.

Species Composition Tc (K) Pc (bar)

C1 0.5 190.6 46.0H2S 0.5 373.2 89.4

gas already has CO2 in it, the data shown in the figure is the totalamount of CO2. Depending on the conditions, the mixture can beat one phase, two-phase and three-phase states. For the two-phaseequilibrium, it can be at a vapor–liquid or a liquid–liquid equilib-rium. The three-phase region spans a wide range of conditions.At low pressure, the mixture is in the one-phase region. As pres-sure increases, phase separation occurs and the mixture can enterboth two-phase and three-phase regions. To better characterize thecomplex phase behaviors, especially in the three-phase region, weadopt the ‘‘saturation line’’ concept, which is an original conceptfor two-phase mixtures. For an existing mixture, whether onephase or two-phase, when a fresh liquid phase forms across a line,we name this line the dew point line. Similarly, when a fresh vaporphase forms across a line, we name it the bubble point line. There-fore, the meeting point of the bubble point and dew point lines,such as the point at (83%, 22.94 bar, shown by the circle inFig. 2(a)), does not indicate that it is necessarily the critical point,which indeed will be for mixtures that only show one-phase ortwo-phase regions. This situation also occurs for the other mix-tures. These saturation lines are also explicitly labeled in Fig. 2and all the subsequent phase diagrams. It is noticed that the singlevapor phase only exists at very low pressures and the liquid phasestarts to form with pressure increase. Therefore, the phase bound-ary is mainly characterized by the six branches of the dew pointlines. The upper two dew point lines, however, have a large slopeso that the mixture is very sensitive to pressure effects. The bubblepoint line only shows up at higher pressures when the vapor phasedisappears. Note the deep penetration of the dew point line intothe low-pressure region. Fig. 2(b) shows the disappearance of thevapor phase from the three-phase to the liquid–liquid region with60% CO2 feeding. The fraction and percentage in all the calculationsin this paper indicate the mole fraction. It is noticed that the spe-cial location (95%, 0.7 bar) (shown in the blow up plot) is the pointwhere the single vapor and vapor–liquid–liquid regions have thesmallest distance, and these two regions do not intersect.

It is of great interest to study retrograde behaviors. This specialbehavior occurs when two saturation point lines are not separatedby the critical point, the location where bubble and dew point linesmeet. For this mixture, it is noticed that retrograde condensationoccurs at the three-phase region along the phase border of thetwo vapor–liquid regions. Due to the curvature effects of the twodew point lines, retrograde condensation occurs at both the left(when CO2 feeding is between 0.261 and 0.469. The starting andending points were highlighted using yellow circles) and rightbranches (when CO2 feeding is between 0.832 and 0.857. Thestarting and ending points were highlighted using green circles).Usually, the maximum amount of liquid phase is located some-where between the dew points for a given feeding composition.This retrograde condensation can be shown to lie along anisothermal process. As an example, Fig. 3(a) shows retrogradecondensation of the CO2-rich liquid phase (L2) at the left branchwhen the CO2 feeding is 40%. Before reaching the 26.034% maxi-mum liquid amount at 6.0 bar, increase of pressure leads toincrease of phase L2 due to compression and condensation. Furtherincrease in pressure does not lead to liquid phase formation. On thecontrary, phase L2 starts to decrease and the CO2 lean liquid phase(L1) builds up. The CO2 mole fractions for each phase during thisprocess are plotted in Fig. 3(b). A similar plot is shown inFig. 3(c) for retrograde condensation of phase L1 at 84% CO2

x (–) MW (g/mole) ki;C1ki;H2S

0.011 16 0.0830.097 34 0.083

Fig. 1. Gibbs energy analyses of the binary mixture CH4–H2S at 190 K and 40.53 bar. (a) The tangent plane distance (TPD) function. (b) Gibbs free energy of mixing.

Table 2Inputs for the mixtures of C1–CO2–H2S–H2O.

Species Tc (K) Pc (bar) x (–) MW [g/mole) ki;C1ki;CO2

ki;H2 S

C1 190.6 46.0 0.008 16CO2 304.2 73.8 0.225 44 0.095a, 0.1005b, 0.13c

H2S 373.2 89.4 0.1 34 0.0755b, 0.095c 0.0999b, 0.097c

H2O 647.3 220.5 0.344 18 0.4928 0.04

a C1–CO2 mixture.b All the cases of C1–CO2–H2S–H2O mixture.c C1–CO2–H2S mixture.

Table 3Equilibrium solution of the C1–CO2–H2S–H2O mixtures.

Mixture T (K) P (bar) Composition 1P-stab 2P-splitting 2P-stab 3P-splitting

C1–CO2 220.0 60.8 (0.9,0.1) T(0.8,0.2) F (0.9701, 0.0299) T(0.7,0.3) F (0.5695, 0.4305) T(0.57,0.43) F (0.0488, 0.9512) T(0.4,0.6) T

C1–CO2–H2S 190.16 26.82 (0.4989, 0.0988, 0.4023) F (0.4323, 0.5677) F (0.4092, 0.0285, 0.5623)

C1–CO2–H2S–H2O 310.95 76.0 (0.1488, 0.2991, 0.0494, 0.5027) F (0.50404, 0.49596) T380.35 129.3 (0.1496, 0.3009, 0.0498, 0.4997) F (0.4936, 0.5064) T310.95 62.6 (0.0504, 0.0503, 0.3986, 0.5008) F (0.4956, 0.5044) F (0.4938, 0.0797, 0.4265)

Table 4Component properties and non-zero interaction parameters for the acid gas.

Species Composition Tc (K) Pc (bar) x (-) MW (g/mole) ki;CO2ki;N2

ki;H2 S

CO2 0.70592 304.211 73.819 0.225 44.0 �0.02 0.12N2 0.07026 126.2 33.9 0.039 28.0 �0.02 0.2H2S 0.01966 373.2 89.4 0.081 34.1 0.12 0.2C1 0.06860 190.564 45.992 0.01141 16.0 0.125 0.031 0.1C2 0.10559 305.322 48.718 0.10574 30.1 0.135 0.042 0.08C3 0.02967 369.825 42.462 0.15813 44.1 0.150 0.091 0.08

6 L. Qiu et al. / Fuel 115 (2014) 1–16

feeding. Notice that the amount of phase L1 is very low, so it isincreased by 50 times for a clear illustration. In comparison, it is re-marked that the left retrograde region occupies a larger area, indi-cating that the condensation rate is less at the left branch than atthe right one. In other words, careful pressure control is neededto avoid another liquid phase forming at high CO2 concentrations.The corresponding change of CO2 in each phase is shown inFig. 3(d). In both cases, the vapor phase amount keeps decreasing,as well as the CO2 mole fraction in this phase. In addition, a furthercomparison between Fig. 2(b) with Fig. 3(a) and (c) shows that inthe retrograde region, non-linear behavior due to curvature effectsis more obvious.

4.4. Test problem 4: Maljamar reservoir oil and Maljamar separator oil

The Maljamar reservoir oil and separator oil, both exhibitinginteresting three-phase behaviors, have been well studied in[10,23,27]. The component properties and parameters for thesetwo oils are given in Tables 5 and 6, respectively. The phasediagram of the Maljamar reservoir oil is shown in Fig. 4(a). Twobi-critical points with coordinates (64.6%, 94.02 bar) and (99.6%,75.00 bar) are shown at the lower right and upper left portionsof the phase boundary. The bi-critical point here is defined as thepoint at which intensive properties of two phases, such as density,compressibility, composition and hence molar weight, in the

Fig. 2. (a) Phase diagram of the acid gas (6 components). Symbols V, L1 and L2 denote the vapor, CO2-lean and CO2-rich liquid phases. Also shown are the bubble and dewpoint lines. Meeting points between the bubble and dew point lines are not critical points. (b) Evolutions of the three phases during a pressurizing isothermal process at 60%CO2 feeding.

Fig. 3. Two different retrograde behaviors of the acid gas. (a) Retrograde condensation of CO2-rich liquid phase L2 at 40% CO2 feeding. (b) CO2 mole fraction in the three phasesduring the retrograde condensation process in (a). Note that CO2 fraction in L2 is reduced by 50%. (c) Retrograde condensation of CO2-lean liquid phase L1 at 84% CO2 feeding.Note that phase fraction in L1 is increased by 50 times. (d) CO2 mole fraction in the three phases during the retrograde condensation process in (c). Note that CO2 fraction in L2

is reduced by 50%.

Table 5Component properties and non-zero interaction parameters for Maljamar reservoir oil.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole) ki;CO2ki;C1

CO2 0.0 304.211 73.819 0.225 44.0 0.115C1 0.2939 190.6 45.4 0.008 16.0 0.115C2 0.1019 305.4 48.2 0.098 30.1 0.115C3 0.0835 369.8 41.9 0.152 44.1 0.115nC4 0.0331 425.2 37.5 0.193 58.1 0.115C5–7 0.1204 516.667 28.82 0.2651 89.9 0.115 0.045C8–10 0.1581 590.0 23.743 0.3644 125.7 0.115 0.055C11–14 0.0823 668.611 18.589 0.4987 174.4 0.115 0.055C15–20 0.0528 745.778 14.8 0.6606 240.3 0.115 0.06C21–28 0.0276 812.667 11.954 0.8771 336.1 0.115 0.08C29+ 0.0464 914.889 8.523 1.2789 536.7 0.115 0.28

L. Qiu et al. / Fuel 115 (2014) 1–16 7

Table 6Component properties and non-zero interaction parameters for Maljamar separatoroil.

Species Composition Tc (K) Pc

(bar)x (–) MW (g/

mole)ki;CO2

CO2 0.0 304.211 73.819 0.225 44.0C5–7 0.2354 516.667 28.82 0.2651 89.9 0.115C8–10 0.3295 590.0 23.743 0.3644 125.7 0.115C11–14 0.1713 668.611 18.589 0.4987 174.4 0.115C15–20 0.1099 745.778 14.8 0.6606 240.3 0.115C21–28 0.0574 812.667 11.954 0.8771 336.1 0.115C29+ 0.0965 914.889 8.523 1.2789 536.7 0.115

8 L. Qiu et al. / Fuel 115 (2014) 1–16

three-phase mixture become identical. This also means the equilib-rium ratios of species in these two phases are the same. It shouldbe pointed out that these phases, determined from the three-phasesplitting calculations, exist in finite amounts that are not negligi-ble. Some other criteria can also be used, such as the tangent planedistance as used by Li and Firoozabadi [10]. Therefore, it is also ex-pected that the bi-critical point is located at the location where thebubble and dew point lines meet, analogously to the critical pointfor a two-phase mixture. It is noticed that, for a wide range of CO2

feedings up to 65%, the mixture is in vapor–liquid state. AnotherCO2 rich liquid phase comes out at higher concentrations. Thethree-phase peninsula penetrates all the way up to almost pureCO2. Above the bubble point line, due to the disappearance of thevapor phase, only liquid–liquid equilibrium exists. To illustratethe evolutions of the three phases, Fig. 4(b) shows the dynamicchange of these phases as a function of pressure with 80% CO2

feeding. Away from the dew point line, the fraction of the vaporphase gradually decreases and reaches zero at the bubble point.The CO2 lean liquid phase amount also decreases but not too muchduring the pressuring process. Therefore, the overall effect is to in-crease the CO2 rich liquid phase continuously.

The pressure-composition phase diagram of the Maljamar sep-arator oil is presented in Fig. 5(a). There are two distinctive behav-iors when compared with the reservoir oil. First, a single liquidphase is found when CO2 feeding is less than 67%. Along the sin-gle-phase boundary, a bubble point line exists at low pressureswhile a dew point line is found at high pressures. Secondly, notic-ing the different scales in the vertical coordinates, the three-phaseregion now is stretched to higher CO2 feedings and its areadecreases. Thirdly, only one bi-critical point exists at the closeboundary between the liquid–liquid and vapor–liquid regions.The three-phase region ends at this bi-critical point. The phaseevolution at 90% CO2 feeding is shown in Fig. 5(b). A similar trendis observed as for the reservoir oil: the vapor phase decreases, L1

phase decreases and L2 phase increases. A blow up plot (indicated

Fig. 4. (a) Phase diagram of Maljamar reservoir oil (11 components). Symbols V, L1 and Land dew point lines. (b) Evolutions of three phases across the three-phase region at 80%

by green circle) is used to illustrate the details in the contiguous re-gion between phases L1 and V-L1-L2. It is seen from the plot that thesingle phase region does not intersect with the three-phase region.

4.5. Test problem 5: Oil B

The phase diagram of Oil B was also constructed. The thermody-namic properties used for this oil are documented in Table 7. Thismixture was used by Shelton and Yarborough [28] to investigatethe mixing effects of CO2 with reservoir oil used in enhanced oilrecovery. The general shape of the phase diagram is similar to thatof Maljamar separator oil, as presented in Fig. 6(a). There are one-phase, two-phase and three-phase regions, too. Interestingly, thereis a point where the four regions almost touch at the point (58%,67.9 bar) and this kind of behavior also appears in some other oils[29]. The three-phase region ends at a bi-critical point at the two-phase boundary very close to pure CO2. Below the bubble pointline, mixture is in the vapor–liquid region for a wide range ofCO2 feedings. At higher pressures, the singe phase dominates untilreaching the dew point line across which a CO2 rich liquid phaseforms when the CO2 feeding is around 73%. Between the liquid–li-quid and vapor–liquid regions, there is a three-phase region actinglike a ‘‘transition region’’. This kind of phase behavior seems to begeneral since no sharp phase transition is found, as also can be ob-served from Figs. 2(a), 4(a) and 5(a). There are no common meetingpoints between the single phase and the three-phase regions. Anenlargement plot (indicated by green circle) around point (70.8%,86.77 bar) shows the nuances of phase behaviors close to the phaseborders.

Specifically for the three-phase flash, two examples show thefeasible region method with this oil. Shown in Fig. 7(a) is the per-formance of this method at 75% CO2 feeding and 82.4 bar. The in-puts, such as the equilibrium ratios are from the two-phasestability test, are tabulated in Table 8, so this is the first iterationin the three-phase flash. The temperature and pressure corre-sponds to a point close to the dew point line between the three-phase and vapor–liquid regions. It is seen that the feasible regionmethod greatly reduces the search space to a limited region andthe optimization algorithm converges quickly around six itera-tions. It is worth mentioning that some constraint lines of becomeparallel. This is because the mixture is close to the phase boundary.Fig. 7(b) shows results at 95% CO2 feeding at 76.4 bar. The equilib-rium ratios are also from the two-phase stability test. This temper-ature and pressure is located at the upper bubble point line of thethree-phase region where the liquid–liquid region just terminates.It is easy to notice that the lines of constraints now are more par-allel with each other, compared to the case in Fig. 7(a). This againdemonstrates the challenges for the liquid–liquid equilibrium

2 denote the vapor, CO2-lean and CO2-rich liquid phases. Also shown are the bubbleCO2 feeding.

Fig. 5. (a) Phase diagram of Maljamar separator oil (7 components). Symbols V, L1 and L2 denote the vapor, CO2-lean and CO2-rich liquid phases. Also shown are the bubbleand dew point lines. Meeting points, excepted for the specially noted, between the bubble and dew point lines are not critical points. (b) Evolutions of the three phases acrossthe three-phase region at 90% CO2 feeding.

Table 7Component properties and non-zero interaction parameters for Oil B.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole) ki;CO2ki;N2

ki;C1

CO2 0.0011 304.211 73.819 0.225 44.01 �0.02 0.075N2 0.0048 126.2 33.5 0.04 28.01 �0.02 0.08C1 0.1630 190.6 45.4 0.008 16.04 0.075 0.08C2 0.0403 305.4 48.2 0.098 30.07 0.08 0.07 0.003C3 0.0297 369.8 41.9 0.152 44.1 0.08 0.07 0.01iC4 0.0036 408.1 36.0 0.176 58.12 0.085 0.06 0.018nC4 0.0329 425.2 37.5 0.193 58.12 0.085 0.06 0.018iC5 0.0158 460.4 33.4 0.227 72.15 0.085 0.06 0.025nC5 0.0215 469.6 33.3 0.251 72.15 0.085 0.06 0.026C6 0.0332 506.35 33.9 0.299 84.0 0.095 0.05 0.036PC1 0.181326 566.55 25.3 0.3884 112.8 0.095 0.1 0.049PC2 0.161389 647.06 19.1 0.5289 161.2 0.095 0.12 0.073PC3 0.125314 719.44 14.2 0.6911 223.2 0.095 0.12 0.098PC4 0.095409 784.93 10.5 0.8782 304.4 0.095 0.12 0.124PC5 0.057910 846.33 7.5 1.1009 417.5 0.095 0.12 0.149PC6 0.022752 919.39 4.76 1.4478 636.8 0.095 0.12 0.181

Fig. 6. (a) Phase diagram of Oil B (16 components) at 307.6 K. Symbols V, L1 and L2 denote the vapor, CO2-lean and CO2-rich liquid phases. Also shown are the bubble and dewpoint lines. Meeting points, excepted for the specially noted, between the bubble and dew point lines are not critical points. (b) Evolutions of the three phases across thethree-phase region at 80% CO2 feeding.

L. Qiu et al. / Fuel 115 (2014) 1–16 9

computation compared to the common vapor–liquid equilibrium.Faced with this difficulty, the feasible region is more stretchedand narrow, and common flash calculation is likely to fail becauseof a poor initial guess and uncontrolled step size. However, theminimization method combined with a line search in the feasibleregion quickly found the solution in about five iterations.

4.6. Test problem 6: Bob Slaughter Block oil and North Ward Estes oil

The Bob Slaughter Block oil and North Ward Estes oil were stud-ied by Khan et al. [29] as verification of their fluid characterization

scheme. The generated parameters for these two West Texas oilmixtures, including the interaction parameter and the tuned criti-cal constants of methane, are tabulated in Tables 9 and 10, respec-tively. Since these parameters were already tuned withexperimental data, we studied the mixing of impure CO2 (95%CO2 and 5% C1) with the Bob Slaughter Block oil at 313.71 K (105F). The pressure-composition phase diagram is shown in Fig. 8(a).There is a small three-phase region. A single liquid phase that islean in CO2 is found to span a wide range of pressures and feedingcompositions until reaching the saturation line. At low pressures,the vapor–liquid phase dominates. Interestingly, the two liquid

Fig. 7. Performance of the feasibility region method in the three-phase flash on Oil B. (a) 75% CO2 feeding and 82.4 bar. (b) 95% CO2 feeding and 76.4 bar. Notice the change ofscales from (a) to (b) and the close-to-parallel lines in (b). ‘‘S limit’’ and ‘‘P limit’’ stand for the SC and PC constraints in the last paragraph of Section 3.2, respectively.

Table 8Inputs for the three-phase flash using the feasible region method for the two example cases for Oil B.

Species Example case 1 Example case 2

Equilibrium ratio 1 Equilibrium ratio 2 Equilibrium ratio 1 Equilibrium ratio 2

CO2 0.124763 � 10+01 0.123846 � 10+01 0.132727 � 10+01 0.134109 � 10+01

N2 0.311587 � 10+01 0.163727 � 10+01 0.185050 � 10+01 0.337815 � 10+01

C1 0.208726 � 10+01 0.138068 � 10+01 0.151460 � 10+01 0.226808 � 10+01

C2 0.854931 � 10+00 0.871704 � 10+00 0.829260 � 10+00 0.846343 � 10+00

C3 0.527766 � 10+00 0.718108 � 10+00 0.646657 � 10+00 0.507804 � 10+00

iC4 0.380655 � 10+00 0.621078 � 10+00 0.534334 � 10+00 0.356389 � 10+00

nC4 0.327115 � 10+00 0.584798 � 10+00 0.496377 � 10+00 0.304867 � 10+00

iC5 0.232398 � 10+00 0.507828 � 10+00 0.413095 � 10+00 0.211426 � 10+00

nC5 0.207699 � 10+00 0.485316 � 10+00 0.390260 � 10+00 0.187989 � 10+00

C6 0.135667 � 10+00 0.394919 � 10+00 0.300911 � 10+00 0.119582 � 10+00

PC1 0.601892 � 10�01 0.277618 � 10+00 0.188930 � 10+00 0.495008 � 10�01

PC2 0.159753 � 10�01 0.156128 � 10+00 0.890264 � 10�01 0.118259 � 10�01

PC3 0.354171 � 10�02 0.799940 � 10�01 0.367214 � 10�01 0.228396 � 10�02

PC4 0.676409 � 10�03 0.374895 � 10�01 0.132442 � 10�01 0.364778 � 10�03

PC5 0.129447 � 10�03 0.166739 � 10�01 0.425159 � 10�02 0.542617 � 10�04

PC6 0.274828 � 10�04 0.657881 � 10�02 0.100869 � 10�02 0.721841 � 10�05

Table 9Component properties and non-zero interaction parameters for Bob Slaughter Blockoil.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole) ki;CO2

CO2 0.0337 304.2 73.77 0.225 44.01C1 0.0861 160.0 46.0 0.008 16.04 0.055PC1 0.6478 529.03 27.32 0.481 98.45 0.081PC2 0.2324 795.33 17.31 1.042 354.2 0.105

Table 10Component properties and non-zero interaction parameters for North Ward Estes oil.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole) ki;co2

CO2 0.0077 304.2 73.77 0.225 44.01C1 0.2025 190.6 46.0 0.008 16.04 0.12PC1 0.118 343.64 45.05 0.13 38.4 0.12PC2 0.1484 466.41 33.51 0.244 72.82 0.12PC3 0.2863 603.07 24.24 0.6 135.82 0.09PC4 0.149 733.79 18.03 0.903 257.75 0.09PC5 0.0881 923.2 17.26 1.229 479.95 0.09

10 L. Qiu et al. / Fuel 115 (2014) 1–16

phases merge to one single liquid phase at very high CO2 feedingsand its borderline is very sharp. This extra liquid phase does notappear in the original pure CO2 mixing with the oil. Only one bi-critical point exists at the three-phase boundary at the liquid–li-quid region side.

The impure CO2 mixing with the North Ward Estes oil is alsopresented. Its phase diagram is shown in Fig. 8(b). The overall plotlooks similar to the Bob Slaughter Block oil plot. However, anotherbi-critical point appears on the saturation line with the departureof the single-phase mixture. Also, no single liquid phase L2 is ob-served at the proximity of pure CO2. TPn flash calculations alongthe left phase boundary do not find a point where the two phaseshave identical intensive properties in the pressure range shown;therefore, a two-phase critical point does not exist.

4.7. Test problem 7: reservoir oil of Stenby et al. [30] and North Sea gascondensate

The phase behaviors of another two mixtures that was studiedby Nichita et al. [31] on developing a delumping procedure werealso considered. The first mixture is the reservoir oil used by Sten-by et al. [30]. Its phase diagram is shown in Fig. 9(a) and its prop-erties are tabulated in Table 11. The calculated phase envelope issimilar to that of Stenby et al. [30]. It is noteworthy that thereexists a small and slim three-phase region at low temperatureand pressure conditions for this mixture, which is not reportedby Stenby et al. [30]. The North Sea gas condensate mixture iscomposed of 161 components from experimental analysis. How-ever, Leibovici et al. [32] reduced the original mixture to 27 com-ponents, including two pseudo-components for characterizingphase behaviors with available data. Their characterization hasbeen adopted by Nichita et al. [31] and is also used here. Details

Fig. 8. (a) Phase diagram of the Bob Slaughter Block Oil (4 components). (b) Phase diagram of the North Ward Estes Oil (7 components). Symbols V, L1 and L2 denote the vapor,CO2-lean and CO2-rich liquid phases. Also shown are the bubble and dew point lines. Meeting points, excepted for the specially noted, between the bubble and dew point linesare not critical points.

Fig. 9. (a) Phase diagram of the 20 components reservoir oil. Note that there is a slim three-phase region at very low temperatures. (b) Results of TPn flash at 344.26 K and177.65 bar. (c) Phase diagram of the 27 components North Sea gas condensate. (d) Results of TPn flash at 398.15 K and 220.0 bar.

L. Qiu et al. / Fuel 115 (2014) 1–16 11

on the component properties can be found in Table 12. The phasediagram of this mixture is plotted in Fig. 9(c). The experimentalpoints are sampled from Nichita et al. [31]. There is only one satu-ration line and no critical point appears. More data is needed toclarify the low-pressure phase behavior, though the experimentalhigh-pressure, low-temperature behavior is well captured. TwoTPn flash calculations are made: one on the reservoir oil at344.26 K and 177.65 bar and the other is for the gas condensateat 398.15 K and 220 bar, and the results are shown in Fig. 9(b)and (d). The calculation results are identical to those from Nichitaet al. [31]. (Note that they swapped the mole fraction of the firstspecies (N2) and the second one (CO2) for the gas condensate mix-ture by mistake in their plot.).

4.8. Test problem 8: reservoir oil of Hoffman et al. [33]

To construct the global phase diagram, the above calculationscall the equilibrium solver for a range of conditions. If only the sat-uration point at a specified temperature or pressure is of interest, itcan be calculated directly by using the tangent plane distancemethod. That is, (i) the tangent plane distance at the feeding com-position must be parallel to that at another composition, and (ii)the distance between these tangent planes must be strictly zero.This method has been applied by Nghiem and Li [13] on the reser-voir oil from Hoffman et al. [33]. A comparison with their calcu-lated phase diagram is shown in Fig. 10. The properties data aretabulated in Table 13. The interaction parameter is calculated using

Table 11Component properties and non-zero interaction parameters of Stenby reservoir oil.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole) ki;N2ki;co2

ki;c1

N2 0.00403 126.20 33.60 0.04 28.01CO2 0.01000 304.20 72.90 0.23 44.01C1 0.45396 190.60 45.40 0.01 16.04 0.02 0.12C2 0.04202 305.40 48.20 0.10 30.07 0.06 0.15C3 0.00887 369.80 41.90 0.15 44.09 0.08 0.15iC4 0.00561 408.10 36.00 0.18 58.12 0.08 0.15nC4 0.00518 425.20 37.50 0.19 58.12 0.08 0.15iC5 0.00647 460.40 33.40 0.23 72.15 0.08 0.15nC5 0.00294 469.60 33.30 0.25 72.15 0.08 0.15C6 0.01011 507.40 29.30 0.30 86.18 0.08 0.15 0.0298C7 0.13117 567.16 29.01 0.52 111.89 0.08 0.15 0.0350C11 0.07127 633.70 21.51 0.66 163.22 0.08 0.15 0.0442C14 0.03847 671.11 18.92 0.75 198.71 0.08 0.15 0.0488C16 0.06005 710.30 17.18 0.85 239.54 0.08 0.15 0.0512C20 0.03352 752.38 15.90 0.96 289.22 0.08 0.15 0.0544C23 0.03340 790.47 15.14 1.06 337.54 0.08 0.15 0.0565C27 0.02870 835.97 14.54 1.17 399.66 0.08 0.15 0.0586C32 0.02179 887.86 14.12 1.27 475.59 0.08 0.15 0.0609C38 0.01892 956.19 13.85 1.35 581.43 0.08 0.15 0.0627C48+ 0.01351 1090.01 13.80 1.24 797.11 0.08 0.15 0.0800

Table 12Component properties and non-zero interaction parameters of North Sea gas condensate.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole) ki;N2ki;co2

ki;c1ki;c2

N2 3.2430 � 10�03 126.20 33.94 0.0400 28.01 0.15CO2 1.9549 � 10�02 304.20 73.77 0.2250 44.01 0.15C1 7.6153 � 10�01 190.60 46.00 0.0115 16.04 0.12 0.12C2 7.7367 � 10�02 305.40 48.84 0.0908 30.07 0.12 0.15C3 3.6177 � 10�02 369.80 42.45 0.1454 44.10 0.12 0.15iC4 5.6670 � 10�03 408.10 36.48 0.1760 58.12 0.12 0.15nC4 1.3258 � 10�02 425.20 38.00 0.1928 58.12 0.12 0.15iC5 4.4800 � 10�03 460.26 33.83 0.2271 72.15 0.12 0.15nC5 6.1000 � 10�03 469.60 33.74 0.2273 72.15 0.12 0.15CC5 5.8000 � 10�04 511.60 45.09 0.1923 70.14 0.12 0.15PC6 6.0010 � 10�03 503.79 30.07 0.2860 86.18 0.12 0.15CC6 4.1480 � 10�03 547.41 39.90 0.2215 84.16 0.12 0.15AC6 1.7350 � 10�03 562.10 48.94 0.2100 78.11 0.12 0.15PC7 4.2390 � 10�03 536.44 27.60 0.3364 100.21 0.12 0.15CC7 5.0560 � 10�03 566.27 34.69 0.2451 98.19 0.12 0.15AC7 3.0630 � 10�03 591.70 41.14 0.2566 92.14 0.12 0.15PC8 3.2910 � 10�03 565.05 25.02 0.3816 114.23 0.12 0.15CC8 2.5400 � 10�03 594.05 29.74 0.2391 112.21 0.12 0.15AC8 2.5640 � 10�03 619.46 35.84 0.3228 106.16 0.12 0.15PC9 2.6300 � 10�03 590.64 23.29 0.4230 128.25 0.12 0.15CC9 1.6400 � 10�03 621.21 28.39 0.2998 125.97 0.12 0.15AC9 1.2170 � 10�03 644.06 32.08 0.3725 120.16 0.12 0.15PC10 2.4110 � 10�03 613.72 21.46 0.4646 142.28 0.12 0.15CC10 4.2700 � 10�04 621.58 26.25 0.4058 140.20 0.12 0.15AC10 1.0490 � 10�03 670.83 29.72 0.3642 133.80 0.12 0.15CN-1 2.6703 � 10�02 711.04 18.75 0.8000 240.00 0.12 0.15 0.03CN-2 3.3380 � 10�03 848.08 16.33 1.3000 450.26 0.12 0.15 0.05 0.03

12 L. Qiu et al. / Fuel 115 (2014) 1–16

the relationship between the critical specific molar volume data ofthe pure component according to

dij ¼ 1� 2ðvcivcjÞ1=6

v1=3ci þ v1=3

cj

" #h

ð23Þ

The parameter h = 1.2126. The match with their calculations isquite good except that there is some over prediction around(200 K, 120 bar), which could be due to some minor differencesin the model parameters, such as the coefficients used to deter-mine the energy and volume parameters for the PR EOS. Again, itis remarked that the APPLE solver detects a three-phase region pe-netrating into the two-phase region in a similar slim fashion at lowtemperature and pressure conditions, which is not shown by

Nghiem and Li [13]. This type of behavior seems to exist for mix-tures with very high and opening pressure limits at low tempera-tures. This appearance of three-phase can only be verified withexperimental data and needs more investigations. Also shown inthis figure is the stability limit line. The stability limit can be foundwhenever a saturation point is calculated, but the compositionsolution is trivial. Only some calculation points are shown sincesaturation point calculation is not the main topic here. The criticalpoint on stability limit curve is also the point where the bubble anddew point lines meet. Since the critical point is between the cricon-dentherm and the cricondenbar, retrograde behavior exists for thismixture. The isothermal process at 680 K on the dew point line andthe isobaric process at 200 bar are shown in Fig. 10(b) to illustratethis type of behavior. For the isothermal process, a maximum

Fig. 10. Comparison on the phase diagram of Hoffman reservoir oil (12 components) calculated with Nghiem and Li. Also shown are the stability limit and critical point. Notethat there is a branch of the three-phase region penetrates into the two-phase region at very low temperatures.

Table 13Component properties of Hoffman reservoir oil.

Species Composition Tc (K) Pc (bar) x (–) MW (g/mole)

C1 0.52 190.6 46.00155 0.0080 16.043C2 0.0381 305.4 48.83865 0.0098 30.070C3 0.0237 369.8 42.45517 0.1520 44.097iC4 0.0076 408.1 36.477 0.1760 58.124nC4 0.0096 425.2 37.99688 0.1930 58.124iC5 0.0069 460.4 33.84255 0.2270 72.151nC5 0.0051 469.6 33.74123 0.2510 72.151C6 0.0206 507.5 32.8901 0.2637 86.000PC1 0.11972 597.0 28.50272 0.3661 120.910PC2 0.12007 696.1 21.49103 0.5237 182.310PC3 0.08687 757.2 15.97895 0.6658 250.580PC4 0.04174 846.3 11.86516 0.9301 361.040

L. Qiu et al. / Fuel 115 (2014) 1–16 13

amount of liquid phase is determined at 125 bar corresponding to26.78% liquid phase and it quickly drops at higher pressures. Forthe isobaric process, the maximum amount of the vapor phase,24% occurs at around 475 K.

4.9. Test problem 9: hydrocarbon mixtures

Calculations on vapor–liquid equilibrium of three hydrocarbonmixtures were also performed. Here, we focused on binarymixtures with methane. The component properties can be foundin Table 14. Fig. 11(a) and (b) shows the pressure–composition dia-gram of binary CH4 and C2H4 and CH4 and C3H8 mixtures, respec-tively. Experimental data are from Refs. [34,35] and are shownwith symbols. The open symbols are for the vapor phase whilethe closed ones are for the liquid phase. In all the computations,the equilibrium solver quickly finds the equilibrium compositionof two phases. This shows that the APPLE solver is useful inpredicting the vapor–liquid equilibrium. Equilibrium calculationresults for another binary mixture composed of CH4 and nC10H22

are shown in Fig. 11(c) and (d) at higher and lower temperatures,respectively. Experimental data are taken from Ref. [36]. It is

Table 14Component properties and non-zero interaction parameters for binary CH4–C2H4 andCH4–nC10H22 mixture.

Species Tc (K) Pc (bar) x (–) MW (g/mole) ki;c1

C1 190.564 45.99 0.0115 16C2 282.34 50.41 0.0862 28 0.023C3 369.83 42.48 0.1523 44 0.015nC10 617.7 21.1 0.4923 142 0.043

noticed that CH4 mainly exists in the vapor phase, while the liquidphase is occupied mostly by heavy molecules of decane.

4.10. Test problem 10: binary hydrocarbon and nitrogen mixtures

Finally, vapor–liquid equilibrium calculations were performedfor binary hydrocarbon species with nitrogen. This type of mixturehas been widely used in calculations of droplet evaporation [37],non-reacting mixing in high-pressure n-heptane injections, andturbulent mixing modeling [38]. Two sets of mixtures are consid-ered: n-pentane-N2 and n-heptane-N2. Fig. 12(a) and (c) showscomparisons between calculations and the experimental data thatare adopted from Refs. [39] and [40]. Their suggested interactionparameters are used. All the other thermodynamic data are sum-marized in Table 15.

The vapor–liquid equilibrium behavior of this kind of mixture israther difficult to model with a cubic equation of state due to thevery different molecular structures. It is noticed from the plots thatthe match with experimental data is good at low pressures but failsat high pressures, especially close to the critical point. Since theonly tunable parameter for the equation of state model is the bin-ary interaction parameter, it is expected that using temperature-dependent interaction parameters could improve the accuracy[39,40]. A recent discussion on addressing the difficulty of model-ing mixtures with nitrogen using the group contribution methodcan be found in Ref. [41]. It is noteworthy to mention that the crit-ical pressure increases while the critical temperature decreases. Inaddition, a linear relation between the critical point of a mixtureand that of each pure species does not exist.

The non-ideal behavior is further shown in Fig. 12(b) and (d) forthese two mixtures. The pure species data for the latent heat is ta-ken from the DIPPR database [42]. The latent heat, originally usedfor pure component is shown here to illustrate the non-idealbehaviors at high pressures. It is defined here as the difference be-tween the partial molar enthalpy of the hydrocarbon in the vaporand liquid phases. In the pure species limit, only the hydrocarbonexists, so the latent heat of pure species is recovered. In both plots,an increase of pressure leads to a substantial decrease of latentheat, which is because the composition in the liquid and vaporphases becomes similar. For a specified pressure, latent heat de-creases with temperature increase. This is because the mixture isapproaching the critical state. At the thermodynamic critical point,the latent heat is strictly zero.

5. Final remarks

In internal combustion engine related research, there have beenattempts to apply an equation of state to account for real gas

Fig. 11. Vapor–liquid equilibrium of hydrocarbon mixtures. (a) Vapor–liquid equilibrium of CH4–C2H4 at three temperatures. (b) Vapor–liquid equilibrium of CH4–C3H8 atseven temperatures. Note that temperature is in Fahrenheit. (c) Vapor–liquid equilibrium of CH4–nC10H22 at three higher temperatures. (d)Vapor–liquid equilibrium of CH4–nC10H22 at two lower temperatures.

Fig. 12. Phase equilibrium of hydrocarbons with nitrogen. (a) Vapor–liquid equilibrium of nC5H12 and nitrogen. (b) Pressure effects on latent heat for nC5H12 and nitrogen. (c)Vapor–liquid equilibrium of nC7H16 and nitrogen. (b) Pressure effects on latent heat for nC7H16 and nitrogen.

14 L. Qiu et al. / Fuel 115 (2014) 1–16

Table 15Component properties and non-zero interaction parameters for binary nC5H12–N2 andnC7H16–N2 mixture.

Species Tc (K) Pc (bar) x (–) MW (g/mole) ki;N2

nC5 469.7 33.7 0.2515 72.0 0.0657nC7 540.3 27.36 0.3495 100.0 0.0971N2 126.2 33.9 0.0377 28.0

L. Qiu et al. / Fuel 115 (2014) 1–16 15

behavior at high-pressure conditions in the combustion chamber.Researchers were mainly focused on the droplet evaporation pro-cess between the liquid phase and the ambient gas phase. This isbecause state-of-the-art computational fluid dynamics (CFD) sim-ulations in direct-injection (DI) diesel engines use an Eulerian gasand Lagrangian (liquid) droplet treatment. Liquid fuel is injectednot as a continuous liquid phase, but as discrete droplet of similarsize to the nozzle diameter. To match the boundary conditions indroplet evaporation, phase equilibrium is generally applied to pro-vide mass transfer between the two phases [43,44].

However, additional simplifications are added beyond the equi-librium assumption. For instance, in developing discrete evapora-tion model for multi-component evaporation, the phaseequilibrium criterion reduces to the relationship between satura-tion pressure and ambient gas pressure for each component [45].The interactions between components are fully neglected, so a Le-wis mixture is actually assumed. In some cases, the iso-fugacity ofeach component assumption at the droplet surface is applied[46,47]. However, these calculations are limited to binary mixturesof n-heptane, n-pentane or n-nonane with nitrogen. It is doubtful ifthe flash calculation procedure is robust enough to handle mix-tures of more species. Since liquid–liquid equilibrium is not ex-pected for such simple mixtures, the vapor–liquid equilibriumalways yields the final state. Therefore, the equilibrium calculationdoes not need to consider potential phase separations. Nonethe-less, as pointed out above, thermodynamically consistent phaseequilibria should have much more stringent meaning and hencerequires more careful treatments. Another benefit of the presentapproach is that the phase stability test provides a good initialguess to accelerate the flash calculation, which is complicated bythe interactions of many different components. The advantage ismore obvious, especially when it is used in computational fluid dy-namic (CFD) simulations such as in simulations of supercritical gasjets [48].

6. Summary and conclusion

In summary, an applied phase-related equilibrium solver,named APPLE, that is able to handle complex multicomponentand multiphase equilibrium has been developed and validated.To formulate a thermodynamically consistent equilibrium solver,rigorous analyses of phase equilibrium based on the equilibriumand stability criterion of fundamental thermodynamics wereapplied. Because the solver is based on minimization of the Gibbsfree energy in the TPn flash calculations, it is theoretically correctand thermodynamically consistent. APPLE solver is composed ofphase stability test and phase splitting when the stability testshows a negative tangent plane distance. Treated in this way, anew phase is only added when it reduces the Gibbs free energy.Phase stability test results then serve as the starting guess forthe splitting calculations. Several efficient numerical methods,such as the BFGS algorithm for solving the tangent plane distanceobjective function for phase stability, Newton’ method and feasibleregion method for efficient multi-phase flash calculations areadopted. Verifications are performed on various mixtures includ-ing reservoir oils and hydrocarbon mixtures that show intricate

phase behaviors, including liquid–liquid and two-phase to three-phase transitions. Retrograde condensation, which is critical inthe CO2 injection process used in enhanced oil recovery, is also wellpredicted by the solver. Finally, potential applications of the APPLEsolver to spray jet simulation, such as droplet evaporation, werediscussed. An recent computational fluid dynamics simulation thatcouples the equilibrium solver on jet condensation also proves therobustness of the APPLE solver and its usability in more intensivecalculations.

Acknowledgements

Financial support from the DOE Sandia National Laboratories isacknowledged. One of the authors, Qiu, L., would like to express hisgratitude to Dr. Li (now with ExxonMobil) and Prof. Firoozabadi atthe Reservoir Engineering Research Institute (RERI) for helpful dis-cussions of phase equilibrium calculations.

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