Application of Group ContributionâNRTL Model with Closure to Predict

Article pubs.acs.org/jced

Application of Group Contribution−NRTL Model with Closure to Predict LLE Behavior of an Oil/Brine/Surfactant System Bahar Dadfar† and Davoud Biria*,† †

Department of Biotechnology, Faculty of Advanced Sciences and Technologies, University of Isfahan, P.O. Box 8174673441 Hezar Jarib Avenue, Isfahan, Iran ABSTRACT: Distribution of surfactants in an aqueous/nonaqueous two phase system is of the great importance especially in enhanced oil recovery (EOR) processes. In this paper, the application of genetic algorithm to estimate the binary interaction parameters of the nonrandom two liquid (NRTL) activity coefficient model for a brine/oil/ionic surfactant system has been investigated. The presence of ionic surfactant in the system has been taken into account by employing the modified Debye−Huckel model. Moreover, the binary interaction parameters used in NRTL activity coefficient model were found to be interdependent and related to each other by a set of linear equations known as closure equations. These equations were considered as constraints to the optimization calculations. On the other hand, when experimental data were not available, the LLE data were estimated primarily through the Scatchard−Hildebrand activity coefficient model. These data were used to evaluate the GC−NRTL model parameters and calculate equilibrium mole fractions. The estimated binary interaction parameters using GA (genetic algorithm) showed a proper fitness with experimental values and the application of closure equations exhibited lower root-mean-square deviations. In addition, employing the Scatchard−Hildebrand model predictions for the modified GC−NRTL model calculations resulted in an acceptable accuracy. Accordingly, the presented model in this work can be utilized as a powerful method to study liquid−liquid equilibrium systems including surfactants.

1. INTRODUCTION In the past three decades, application of enhanced oil recovery (EOR) processes to petroleum reservoirs aiming at the production of incremental amounts of crude oil in response to the increasing global demands have gained a special attention.1 Consequently, various methods such as carbon dioxiade injection, surfactant flooding, polymer flooding, alkaline-surfactant−polymer flooding, steam injection and microbial enhanced oil recovery (MEOR) have been proposed and implemented practically.1 In fact, the past field experiences revealed that the application of these techniques, typically leads to 5% to 15% increase in oil recovery,2 which is high enough to motivate the investigation of the involved phenomena in the recovery processes in detail. Among the EOR processes, surfactant flooding has been more focused on because it can create advantageous conditions in order to mobilize the trapped oil in reservoirs. Surfactants can alter the wettability of reservoir rocks and at the same time they can reduce the interfacial tension between the oil/brine phases significantly. These mechanisms are considered as the main causes of the residual oil mobilization and increasing the recovery after the surfactant application.3 Distribution of surfactants in the two phases and their influence on formation of emulsions, make it necessary to study their phase behavior in a liquid−liquid equilibrium (LLE) system (oil/brine/surfactant). Therefore, a reputable model should be utilized to estimate the components activity coefficients. Nonrandom two liquid (NRTL) model is one of the successful molecular thermodynamic models, which has been © XXXX American Chemical Society

widely used to describe the phase behavior of highly nonideal systems.4 The model which was originally proposed by Renon and Prausnitz is an activity coefficient model based on the local composition hypothesis. This model and its modifications have been widely used to describe phase behaviors of highly nonideal systems. However, there are two important problems in the estimation of adjustable parameters of the activity coefficient models such as NRTL. First, the selected parameters should be precise enough so that the results have a minimum deviation from the experimental data. This can be achieved by defining an appropriate objective function for the model parameters and optimizing them through suitable optimization algorithms. Concerning phase equilibrium calculations, several optimization methods including evolutionary algorithms have been suggested in the literature.5 For instance, Sahoo et al.6,7 calculated the interaction parameters for NRTL and UNIQUAC models in ternary, quaternary, and quinary LLE systems based on genetic algorithm (GA) and showed that the obtained results were better than the other previously reported techniques. Alvarez et al.8 mentioned that stochastic optimization techniques had often been found to be as powerful and effective as deterministic methods in many engineering applications and used GA for parameter estimation in Wilson model for a VLE system. Received: January 28, 2015 Accepted: July 31, 2015

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DOI: 10.1021/acs.jced.5b00087 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Second, reputable experimental data have not been reported for a large number of (bio) chemical compounds including (bio) surfactants. In fact, experimental data are required to calculate and optimize the activity model parameters. Consequently, a method with the capability of the calculation of reasonable values for the activity coefficients from some available properties of the system is extremely beneficial. Fortunately, hydrophilic lipophilic balance (HLB), which is a measure of solubility of surfactants in water and oil phases, can be used to calculate solubility parameters and activity coefficients9 in a water/oil/surfactant system. The HLB value, which is readily available for many surfactants, can be calculated for a wide range of nonionic and anionic surfactants.10 It should be noted that HLB has been defined as a molecular property and its correlation with temperature, which has been studied by Kunieda and Shinoda11 can be used to consider the temperature effect on the phase equilibrium. In this paper, the accuracy of the GC−NRTL model with GA optimized parameters for a system of water/surfactant/oil was examined. Moreover, Scatchard−Hildebrand solubility parameter of the surfactant was estimated using the HLB value to fill in the gaps in equilibrium data. These data were used to evaluate the modified GC−NRTL model parameters and calculate the equilibrium mole fractions. Validity of the applied procedure was investigated by comparing the obtained results with the experimental data.

τmj =

(gmj − gjj) RT

ln γi =

∑ (ln

− ln

n

∑ j = 1 (τjmGjmxj)

⎛ 2AM d ⎞⎡ 1 ln γnD − H = ⎜ 3 n s ⎟⎢1 + bI1/2 − ⎝ b d n ⎠⎣ 1 + bI1/2 ⎤ − 2ln(1 + bI1/2)⎥ ⎦

n

∑k = 1 (Gkmxk) ⎛ ⎜⎜τmj − ⎝

⎡

∑⎢ j=1

A = 1.327757 × 105

ds1/2 (εT )3/2

(7)

and ds1/2 (εT )1/2

(8)

where ε is the dielectric constant of the mixture and can be calculated from a local composition model offered by Liu et al.15 as below: 14

ε=

1 (1 + 9p + 3 9p2 + 2p + 1 ) 4

(9)

where p is the molar polarization per unit volume: p=

(ε − 1)(2ε + 1) 9ε

(10)

16

Wang and Anderko suggested a quadratic mixing rule for molar polarization per unit volume

(xjGmj)

⎢⎣ ∑nk = 1 (Gkjxk)

n ∑k = 1 (xkτkjGkj) ⎞⎤ ⎟⎥ n ∑k = 1 (Gkjxk) ⎟⎠⎥⎦

(6)

where Mn is the molecular weight of solvent (kg/mol), ds is the density of ion-free solution in (kg/m3), dn is the density of pure solvent (kg/m3), A and B will be calculated by the equations below:

(1)

n

+

(5)

where ln γn is the overall activity coefficient of the solvent in electrolyte solution, ln γGC−NRTL is the activity coefficient of n solvent in absence of ions which in this paper calculated from the GC−NRTL model and ln γD−H is the Debye−Huckel n activity coefficient modification due to the presence of ions and can be calculated from the following relation:

where, the γim is the activity coefficient of group m in molecule i, γi,r m is the activity coefficient of group m in molecule i in a reference solution containing only molecule i, and the summation is over all groups present in the molecule i. Considering the γi,r m in the summation is necessary to satisfy the normalization criterion that the activity coefficient of molecule i becomes equal to unity as its mole fraction goes to one. The activity coefficients in the above equation are calculated as ln γm =

(4)

ln γn = ln γnGC − NRTL + ln γnD − H

b = 6.359696

m=1

(3)

T

In general, Ajj = Amm = 0, Amj ≠ Ajm and αij = αji. Values of αij usually vary5 from 0.15 to 0.5. However, γi calculated from eq 1 needs to be modified to consider the effect of ions in the system. The ionization of inorganic salts in the aqueous phase to cations and anions will affect the phase equilibrium and the activity coefficient model presented by eq 1 cannot describe the phase behavior of such systems. Debye and Huckel proposed an activity coefficient model to consider the influence of electrolytes on the activity coefficient. Accordingly, the overall activity coefficient for the solvent (e.g., water in an aqueous solution) can be calculated by the following equation:

l

γmi,r)

A mj

Gjm = exp( −αjmτjm)

2. THERMODYNAMIC MODELS 2.1. GC−NRTL Model. One of the modifications made on the original NRTL model to produce more proper outputs is the employing of group contribution (GC) concept. In this approach, the smaller number of binary interaction parameters between functional groups have been used instead of the binary interaction parameters between chemical compounds. Clearly, the number of functional groups is much less than the chemical compounds and a large number of mixtures can be formed from just a few functional groups.12 The GC−NRTL model was used to correlate the experimental LLE data in this work. The activity coefficient γi, is given by γmi

=

n

p=

n

∑ ∑ xixj(νp)ij with (νp)ij i=1 i=1

1 (νip + νjpj )(1 + kij) 2 i (11)

(2)

where νi is the molar volume of component i, and kij is the binary parameter available in Table 1. The mixing rule reduces to linear form when all of the (kij = 0).15

For this model, the binary interactions, Amj, and nonrandomness parameters, αmj, have been defined as13 B



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Table 1. Binary Parameter, kij binary

compound

kij

1−2 1−3 2−3

oil−water oil−surfactant water−surfactant

0.242−0.2042 −1.8553 −1.9095

The binary interaction parameters of the system were estimated and optimized by genetic algorithm. Genetic algorithm is one of the most prominent and widely used computational models of evolution in artif icial life systems. GA, differing from conventional search techniques, start with an initial set of random solutions called population satisfying boundary or system constraints to the problem. Each individual in the population is called a chromosome (or individual), representing a solution to the problem at hand. Chromosome is a string of symbols usually, but not necessarily, a binary bit string. The chromosomes evolve through successive iterations called generations. During each generation, the chromosomes are evaluated, using some measure of f itness. To create the next generation, new chromosomes, called offspring, are formed by either merging two chromosomes from the current generation using a crossover operator or modifying a chromosome using a mutation operator. A new generation is formed by selecting, according to the fitness values, some of the parents and offspring and rejecting others so as to keep the population size constant. Fitter chromosomes have higher probabilities of being selected. After several generations, the algorithms converge to the best chromosome, which hopefully represents the optimal or suboptimal solution to the problem.19,20 In general, parameter estimation is carried out by minimization of a proper objective function. The objective function is defined to minimize the deviation between the experimental and calculated mole fraction of components in the phase equilibrium and can be expressed by the following equation:

In the systems in which the binary parameters are unknown, the mixture dielectric constant can be obtained using Oster mixing rule,17 which for a binary mixture, we have ⎤ ⎡ (ε − 1)(2ε2 + 1) x ′V εm ≈ ε1 + ⎢ 2 − (ε1 − 1)⎥ + 2 2 2ε2 V ⎦ ⎣ (12)

where V = x1′V1 + x 2′V2

(13)

where V1 and V2 are the molar volumes of pure components 1 and 2, respectively. x′1 is the ion-free mole fraction of solvent 1 in a binary mixture. The ionic strength (I) is defined as 1 I= ∑ Zi2mi 2 i = ion (14) where Zi is the valences of ions and mi is the molality of ion i (mol/kg of water). In this work, we assumed full dissociation for surfactant and salt, so the ionic strength can be obtained as follows: I = 3 × msalt + msurf (15)

n

OF =

The binary interaction parameters for NRTL model have been found to be interdependent and their relationship has been called a “closure equation”. Syed Akhlaq Ahmad et al.18 derived the closure equations for ternary, quaternary, and quinary systems. Application of the closure equations will allow us to eliminate a few binary interaction parameters, Aij, equal to the number of closure equations.6 Kumar Sahoo et al.6 used closure equations for NRTL parameters in several systems and achieve lower rmsd (root mean square deviation) values in their results. For a ternary system, six binary interaction parameters exist. The closure equation, which describes the relationship between these six binary interaction parameters, is A12 − A 21 + A 23 − A32 + A31 − A13 = 0 (16)

(18)

A12 − A 21 + A 25 − A52 + A51 − A15 = 0

(19)

A 23 − A32 + A34 − A43 + A42 − A 24 = 0

(20)

A 23 − A32 + A35 − A53 + A52 − A 25 = 0

(21)

A 24 − A42 + A45 − A54 + A52 − A 25 = 0

(22)

nc

j=1 l=1 i=1

(23)

where n is the number of tie lines, p is the number of phases, and nc is the total number of components in the system. The goodness of fit is usually measured by the root mean square deviation (rmsd) and the lower the rmsd, the better the fitness. Rmsd has been defined as ⎛ ⎞1/2 OF rmsd = ⎜ ⎟ ⎝ p × nc × n ⎠

(24)

For optimizing the binary interaction parameters by genetic algorithm, GA toolbox of Matlab software package (version R2009a) was utilized. The experimental data and objective function are properly defined. With the first guess of binary interaction parameters by GA (the closure equations and the bounds of interaction parameters should be defined for GA), the first equilibrium data are calculated and the objective function are evaluated. If the objective function value were not to satisfy the determined condition (the objective function value should be lower than 10−6), the binary interaction parameters would be changed and calculations would repeat until the specified condition is achieved. At last, the optimized binary interaction parameters and the calculated equilibrium data would be presented. We put the number of initial population and the generation size equal to 150 and 100, respectively. Other parameters were equal to the preferences of the GA toolbox. 2.2. Scatchard−Hildebrand Model. In order to predict the phase behavior of a surfactant/brine/oil ternary system at a condition that the experimental data for this system are not available, Scatchard−Hildebrand activity coefficient model with

This means that only five out of six binary interaction parameters are independent. For a quaternary system, we have 12 binary interaction parameters and three closure equations and for a quinary system, there are 20 binary interaction parameters and six closure equations A12 − A 21 + A 23 − A32 + A31 − A13 = 0 (17) A12 − A 21 + A 24 − A42 + A41 − A14 = 0

p

l l (j , i) − xcalc (j , i))2 ∑ ∑ ∑ (xexp

In this work, we had a quinary system because of using functional groups instead of chemical compounds in the group contribution approach, with six closure equations. C



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a few modifications can be utilized to obtain a set of LLE data as a first guess for the GC−NRTL model. Optimization of binary interaction parameters of GC−NRTL model with the obtained data can produce a more accurate estimation for phase equilibrium compositions. Scatchard−Hildebrand activity coefficient has been suggested for regular solutions and expressed as RT ln γi = υi(δi − δ ̅ )2

Table 2. Physical Properties of Compounds in the System comp. 1 2 3 4

i

(26)

Φ is the volume fraction of component j in the system which is defined as xjυj Φj ≡ m ∑i xiυi (27) The solubility parameters of chemical compounds present in surafctant/brine/oil system can be obtained in two ways: (1) Hansen−Hildebrand solubility parameters and (2) using a correlation between the solubility parameter and HLB of a surfactant suggested by Little.9 In the solubility parameter approach proposed by Hansen− Hildebrand, the solubility behavior of a solute is described by using three parameters. The summation of energies to overcome the dispersion forces δd, dipole forces δp and hydrogen bond breaking δh is considered equal to the solubility parameter as explained in eq 28 δ

2 2 2 2 =δ +δ +δ t d p h

(28)

The applied method to estimate δd, δp, and δh is based on the structural functional group contributions. Equation 29 shows the correlation between HLB of a surfactant and its solubility parameter as suggested by Little δ=

118.8 + 6.0 54 − HLB

d20

V20

g/cm3

cm3/mol

134 18 272.38 110.9

0.78 1.0 1.0

171.8 18.0 272.38

ε 2.074 78.54 10.0

Table 3. Initial Mixture Molar Compositions for All Three Sets of Experiments, zja

m

∑ Φiδi

oil water surfactant salt

M g/mol

(25)

where δi and υi are the solubility parameter and molar volume of component i in the solution respectively and δ̅ is the volumefraction average of the solubility parameters of all the components in the solution which can be calculated from the following equation: δ̅ =

compound

J

oil

water

surfactant

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

0.0055 0.0116 0.0257 0.0433 0.0658 0.0957 0.1372 0.1988 0.2997 0.4955 0.0055 0.0116 0.0258 0.0436 0.0663 0.0964 0.1384 0.2010 0.3041 0.5058 0.0056 0.0118 0.0261 0.0441 0.0672 0.0980 0.1411 0.2057 0.3132 0.5279

0.9943 0.9883 0.9741 0.9565 0.9340 0.9041 0.8626 0.8009 0.6999 0.5039 0.9942 0.9881 0.9738 0.9561 0.9333 0.9031 0.8610 0.7984 0.6951 0.4930 0.9938 0.9876 0.9732 0.9552 0.9320 0.9011 0.8579 0.7931 0.6851 0.4696

0.0001 0.0001 0.0002 0.0002 0.0002 0.0002 0.0003 0.0003 0.0004 0.0006 0.0003 0.0003 0.0003 0.0004 0.0004 0.0004 0.0005 0.0006 0.0008 0.0012 0.0006 0.0006 0.0006 0.0007 0.0008 0.0009 0.0011 0.0013 0.0017 0.0024

a

For experiments 1−10, surfactant wt % = 0.5. For experiments 11−20, surfactant wt % = 1. For experiments 21−30, surfactant wt % = 2.

(29)

3. EXPERIMENTAL DATA The phase equilibrium data of a system containing oil (n-decane), brine, and ionic surfactant (sodium dodecyl sulfonate) were taken from Riyazi and Moshfeghiyan experiments.21 The system consists of mainly two phases, the microemulsion phase which involved oil, brine, and surfactant and the oil phase. The physical properties of these components are available in Table 2. The water contained CaCl2 salt at concentration of 500 mg/L, which is equivalent to 0.138 wt %. All of the experiments were conducted at atmospheric pressure and 20 °C. Three sets of experiments were conducted by three surfactant concentrations (0.5 wt %, 1 wt %, and 2 wt %). In each set, 10 different oil volumes were used from 5 % to 90% of the total volume of system. The initial mole fractions of the components have been shown in Table 3.

4. LLE CALCULATION In the LLE calculations, the oil-rich phase was specified by “o”, and the water-rich phase by “w”. Therefore, the equilibrium relation for all components in the system can be written as xioγio = xiwγi w xwi

(30)

γwi

Where and are the mole fraction and activity coefficient of component i in the water-rich phase, respectively. We assume that φ reperesent the mole ratio of oil phase to the total number of moles in the system, through material balance between two phases, we have zi xiw = 1 + ϕ(K i − 1) (31) D



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where zi is the mole fraction of component i in the whole mixture and Ki is an equilibrium ratio defined as γi w xio = xiw γio

Ki =

Considering all three components of the system, there are five functional groups (CH3 (1), CH2 (2), Na+ (3), SO−3 (4), H2O (5)). The relevant values of of γi were calculated from eqs 1 and 2, and the corresponding binary interaction parameters were obtained through eqs 17−22.

(32)

When two the phases (oil and microemulsion) are in equilibrium, φ will vary between 0 and 1, and

∑ xiw = 1

5. RESULTS AND DISCUSSION In this work, GC−NRLT model was used to obtain liquid− liquid equilibrium mole fractions in a system of oil/brine/ surfactant. The activity coefficients were modified by Debye− Huckel model for electrolytes to emend the GC−NRTL predictions. The binary interaction parameters of the GC− NRTL model were calculated by minimizing a proper objective function eq 23 using genetic algorithm. The obtained binary interaction parameters from GA have been shown in Table 4. A wide variation boundary (−5000 to 5000) was considered for binary interaction parameters (τij) and the nonrandomness parameters (αij) was assumed to be from 0.15 to 0.5. The lower and upper bound of τij was changed to get better responses and the mentioned boundary returned the lowest rmsd. The experimental oil phase data were required to evaluate the objective function and because the mole ratio of oil phase to the total number of moles (ϕ) was reported as experimental data, we had to calculate the oil phase composition and we assumed that if the calculated composition was less than zero, it should be replaced with 10−5 to have a non-negative

(33)

i

To calculate the LLE, the function below must be zero Ω=

∑ xiw − 1

(34)

i

In our calculations, we assumed that salt is only dissolved in water (brine solution) Table 4. Binary Interaction Parameter Obtained Using GA with Closure Equation, Aij, for Surfactant/Oil/Brine System CH3 CH2 Na+ SO−3 H2O

CH3

CH2

Na+

SO−3

H2O

0.00 1300.41 19.35 419.20 −760.43

−292.56 0.00 −733.75 −89.12 1900.20

−38.99 800.59 0.00 84.43 −763.07

283.77 1368.42 7.33 0.00 −849.55

809.39 5062.99 864.79 855.41 0.00

Table 5. Experimental and Calculated Molar Composition, in Percent, of Microemulsion Phase from Modified GC−NRTL Model experimental data

predicted data from GC−NRTL

exp. no.

oil

water

surfactant

100 × ϕ

oil

water

surfactant

100 × φ

oil (vol %)

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3

0.18 0.43 0.15 0.09 0.36 0.15 0.33 0.77 1.05 1.37 0.22 0.33 0.48 0.44 0.51 0.48 0.64 1.13 1.51 2.57 0.18 0.36 0.54 0.70 0.88 1.07 1.38 1.71 3.15 7.19

99.81 99.56 99.83 99.90 99.62 99.82 99.64 99.19 98.89 98.51 99.75 99.64 99.49 99.52 99.45 99.47 99.30 98.80 98.37 97.19 99.76 99.58 99.40 99.22 99.03 98.84 98.50 98.13 96.61 92.33

0.014 0.015 0.016 0.018 0.021 0.024 0.030 0.039 0.057 0.112 0.029 0.030 0.030 0.037 0.042 0.049 0.060 0.078 0.115 0.230 0.058 0.060 0.066 0.074 0.084 0.099 0.121 0.159 0.236 0.480

0.37 0.73 2.42 4.25 6.24 9.43 13.43 19.26 29.23 48.85 0.33 0.83 2.12 3.93 6.15 9.21 13.29 19.19 29.34 49.28 0.38 1.00 2.44 4.24 6.57 9.66 13.98 20.47 31.27 52.83

0.20 0.49 0.30 0.25 0.46 0.51 0.66 0.89 1.06 1.28 0.30 0.44 0.53 0.58 0.59 0.74 0.71 1.21 1.56 2.70 0.30 0.32 0.40 0.45 0.48 0.52 0.60 0.73 2.36 7.58

99.77 99.47 99.69 99.73 99.52 99.33 99.31 99.07 98.86 98.64 99.65 99.53 99.34 99.36 99.36 99.19 99.24 98.71 98.32 97.06 99.64 99.61 99.53 99.48 99.43 99.38 99.29 99.11 97.39 92.66

0.020 0.010 0.021 0.021 0.021 0.022 0.023 0.037 0.057 0.077 0.030 0.030 0.041 0.031 0.043 0.055 0.069 0.074 0.113 0.238 0.060 0.061 0.072 0.073 0.086 0.100 0.116 0.151 0.247 0.527

0.25 0.50 2.22 4.09 6.08 9.37 13.41 19.29 29.33 49.09 0.20 0.60 1.85 3.71 5.97 9.10 13.24 19.21 29.42 49.40 0.26 1.04 2.50 4.33 6.65 9.68 13.95 20.31 30.79 52.74

5 10 20 30 40 50 60 70 80 90 5 10 20 30 40 50 60 70 80 90 5 10 20 30 40 50 60 70 80 90

E



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Figure 1. Oil composition in the microemulsion phase at 20 °C and atmospheric pressure.

Figure 2. Water composition in the microemulsion phase at 20 °C and atmospheric pressure.

experimental results. In fact, optimization of the GC−NRTL model binary parameters through the application of a powerful optimization tool (genetic algorithm) in combination with Debye−Huckel modifications have been produce a proper framework to describe the phase behavior of the system. Moreover, the effect of considering closure equations on the model results was examined by introducing these equations as constraints to the utilized genetic algorithm optimization. It was observed that using such equations can be helpful to obtain lower rmsd following better fitness, which was in agreement with the results of Kumar et al.6 The obtained rmsd with closure equations was 0.0013, which is lower than 0.0017 that calculated without closure equations. Riazi and Moshfeghian21 have been examined a modified NRTL model without the closure equations with the same experimental data set as this

composition and then the compositions were normalized again. Riazi and Moshfeghian21 guessed that the oil phase to be nearly pure and they kept xooil ≈ 1. However, their results showed that this assumption was invalid. Therefore, we skipped this assumption and calculate the oil phase mole fractions too. The mixture initial molar compositions have been given in Table 3 and the predicted molar compositions of the microemulsion phase as well as the calculated ϕ value in percent have been shown in Table 5. In addition, the dielectric constant of mixture was calculated through the local composition model suggested by Liu et al.15 The model results for equilibrium mole fractions (in percent) of oil, brine, and surfactant as well as the experimental data have been shown in Figures 1, 2, and 3, respectively. As can be seen, there is a good proximity between the calculated and F



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Figure 3. Surfactant composition in the microemulsion phase at 20 °C and atmospheric pressure.

Figure 4. Comparison between equilibrium data predicted by Scatchard−Hildebrand, GC−NRTL, and experimental data for oil in 0.5% sodium dodecyl sulfonate/oil/brine system (at 20 °C and atmospheric pressure).

containing an electrolyte solution.13 Then, the obtained data were used to optimize the binary interaction parameters of the GC−NRTL model using genetic algorithm. The solubility parameter for sodium dodecyl sulfonate can be obtained from its HLB value using Little equation.9 The HLB of sodium dodecyl sulfonate was reported to be 13, so the solubility parameter for this surfactant is 8.897. The solubility parameter for water and n-decane is 23.5 and 7.614, respectively.22 The calculated mole fractions of the system components from Scatchard−Hildebrand and GC−NRTL models as well as the experimental data have been illustrated in Figures 4, 5, and 6. As expected before, Scatchard−Hildebrand model

work. We compared the obtained results with their calculation results and it was observed that the closure equations have a positive impact on the accuracy of the model results. Accordingly, the application of closure equations can be suggested to obtain binary interaction parameters through optimization by GA for activity coefficient models such as NRTL. To obtain a suitable model for a surfactant/brine/oil system when experimental data are not available, the Scatchard− Hildebrand model was utilized to produce a primary estimate for the LLE data. Because this model has been developed for nonpolar compounds with different molecular sizes, it seems to be inappropriate for phase behavior prediction of a system G



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Figure 5. Comparison between equilibrium data predicted by Scatchard−Hildebrand, GC−NRTL model, and experimental data for water in 0.5% sodium dodecyl sulfonate/oil/brine system (at 20 °C and atmospheric pressure).

Figure 6. Comparison between equilibrium data predicted by Scatchard−Hildebrand, GC−NRTL model, and experimental data for surfactant in 0.5% sodium dodecyl sulfonate/oil/brine system (at 20 °C and atmospheric pressure).

predictions have not been fitted to experimental data and have the minimum accuracy. It is obvious that this model does not consider the surfactant effects on increasing the oil solubility in aqueous phase. Although the GC−NRTL model parameters have been calculated on the basis of the Scatchard−Hildebrand model results, it has produced better predictions because of the Debye−Huckel modification on the system components activity coefficients. Results of the modified GC−NRTL model for the oil mole fractions (Figure 4) have been fitted completely to those of Scatchard−Hildebrand model. In fact,

this was expectable because oil has not been ionized in the system. The overall trend of the predicted results for water mole fractions by the modified GC−NRTL model is in agreement with the experimental results, though their values are not exactly the same (Figure 5). The observed lack of fit between the model and experimental results can be explained by the weakness of the Scatchard−Hildebrand driven data for optimization of the binary parameters of GC−NRTL model. However, distance of the predicted mole fractions by the model from the experimental results are less than 0.015 for all the H



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δ = solubility parameter ε = dielectric constant ν = molar volume τ = adjustable binary parameter φ = mole ratio of oil phase to the total moles in the system Φ = volume fraction

results. Results of the modified GC−NRTL model for the surfactant have been obtained close to the experimental data. The calculated determination factor (R2) for the surfactant mole fraction predicted by modified GC−NRTL model was 0.956, which shows that the model results are fitted to the experimental results properly (Figure 6).

Superscript and Subscript

6. CONCLUSION In this paper LLE data of a brine/oil/surafctant system was calculated using a modified GC−NRTL model. The binary interaction parameters of the model were obtained through minimizing an objective function by genetic algorithm with the closure equations as the constraints. Results revealed that the applied procedure can estimate the phase behavior of the system properly. Considering the fact that the closure equations resulted in a lower rmsd and a better fitness. To predict the LLE phase behavior of a similar system without experimental data, solubility parameter of the surfactant was calculated from its HLB value through the Little equation and the LLE data was approximated by the Scatchard−Hildebrand model. The modified GC−NRTL model was run with these data. Results indicated that the model can predict the equilibrium mole fractions of the system components with an acceptable accuracy, especially for the surfactant. Accordingly, the applied procedure in this work can be used to predict surfactant distributions in the two phase system of EOR processes even without experimental data.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +98 (31) 37934373. Notes

The authors declare no competing financial interest.

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calc = calculated value D−H = Debye−Huckel model exp = expermental value i,j = component NRTL = NRTL model n = solvent s = solution

NOMENCLATURE

List of Symbols

Aij = binary interaction parameter A = constant b = constant d = density [kg/m3] G = energy parameter g = NRTL energy interaction term [J/mol] I = ionic strength Ki = equilibrium ratio kij = binary parameter M = molecular weight [kg/mol] mi = molality of ion i [mol/kg of water] n = number of tie lines nc = number of total components OF = objective function p = molar polarization per unit volume P = number of phases R = universal gas constant [J/mol·K] rmsd = root mean square deviation T = absolute temperature [K] x = molar composition Zi = valences of ion i zi = initial mixture composition Greek Letters

α = nonrandomness parameter γ = activity coefficient I



Article

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