An equation of state for electrolyte solutions. 3. Aqueous solutions

Jan 1, 1991 - Statistical Associating Fluid Theory Coupled with Restricted Primitive Model to Represent Aqueous Strong Electrolytes: Multiple-Salt Sol...
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Ind. Eng. Chem. Res. 1991,30, 240-248

Wardrop, A. B. Anatomical Aspects of Lignin Formation in Plants. The Eckman Days 1981. Int. Symp. Wood Pulp. Chem., Stockholm, Sweden, 1981; Vol. 1, pp 44-51. Wend, H. F. J. Anatomy and Physical Properties of Wood. The Chemical Technology of Wood;Academic Press, Inc.: New York, 1970; Chapter 2, pp 32-88. Winston, M. H. Characterizationof the Lignin Residue from Hy-

drolysis of Sweetgum Wood with Superconcentrated Hydrochloric Acid. Ph.D. Thesis, North Carolina State University, Raleigh, 1987, 113 pages.

Received for review January 5, 1990 Revised manuscript received June 30, 1990 Accepted July 17, 1990

An Equation of State for Electrolyte Solutions. 3. Aqueous Solutions Containing Multiple Salts Gang Jint and Marc D. Donohue* Department of Chemical Engineering, The Johns Hopkins Unioersity, Baltimore, Maryland 21218

An equation of state for mixtures containing electrolytes was derived by taking moleculemolecule interactions, charge-charge interactions, and charge-molecule interactions into account by using perturbation expansions based on their potentials. Previous calculations for a large number of aqueous solutions containing single strong or volatile weak electrolytes demonstrated the potential of this equation of state. In this paper, we give a new physical description of this model and extend the work to aqueous systems containing multiple salts. The only adjustable parameter in the model is related to the ionic size. In our previous papers, we had one parameter for each electrolyte (i.e., each ion pair). In this work, we have used a single adjustable parameter for each ion. These parameters were determined from mean ionic activity coefficient data for a large number of single-salt aqueous systems. The values then are used in the calculations for multisalt systems without any additional adjustable binary or ternary parameters. Preliminary calculations are presented for several double-salt aqueous solutions and one triple-salt system, CaS04-MgC12-NaC1-H20. Our calculations are compared with experimental data and with calculations made with other models, which contain a number of adjustable parameters. This equation of state shows good agreement with experimental data over wide ranges of temperature and composition.

I. Introduction There is widespread interest in the solubility of calcium sulfate and its hydrates (CaS04,CaS04-H20,CaS04.2H20, and CaS04.4H20)in solutions of sodium chloride (NaCl) and/or magnesium chloride (MgClJ. Geologists and geochemists are interested because of gypsum and anhydrite conversion that occurs in nature (Bock, 1961; Zen, 1965). Engineers involved in the production of petroleum and in water desalination are interested because of deposition of calcium sulfate scales. These scales also are a problem in the operation of both boilers and cooling towers (Denman, 1961). Much experimentalwork has been required in order to understand the complex solubility behavior of calcium sulfate in water and in electrolyte solutions (Marshall et al., 1964; Power et al., 1964; Ostroff and Metler, 1966; Yeatts and Marshall, 1969). However, these experiments are costly and time consuming. It is desirable to have a model that could give reliable predictions for such systems (Gering and Lee, 1989). In contrast to the extensive amount of modeling given in the literature for aqueous solutions containing single electrolytes, only limited reports have been made on multisalt systems. Generally, models used for multisalt systems are built upon the calculation of activity coefficients of single salt electrolyte solutions together with the application of certain "mixing rules", All these equations are empirical or semiempirical and contain a number of adjustable binary and ternary parameters. For example, the models developed by Guggenheim (1967), Bromley (19731, and Meissner and Kusik (1972) can be applied to *Author to whom correspondence should be addressed. Current address: Benger Laboratory, Dupont Company, Waynesboro, VA 22980.

multisalt systems with one adjustable parameter per ion pair. Pitzer's model (1973,1974, 1979) uses three to four parameters per ion pair, one parameter for each like charged ion pair and one ternary interaction parameter. Chen (1980) developed a model that contains two parameters per water-salt pair and two parameters for each salt-salt pair. In all these models, the user must regress experimental data to determine the values of the parameters in the equations. Recently, a model for electrolyte solutions has been developed by using perturbation theory (Jin and Donohue, 1988a,b; Jin, 1989). The only adjustable parameters used in the model are for the sizes of the ions. By use of an equation of state from this model, predictions of thermodynamic properties for a large number of aqueous solutions containing single strong or weak electrolytes were made. In previous calculations, one parameter was used for each electrolyte (ion pair). This introduced an inconsistency when using the model for multisalt calculations because, for example, a chloride ion would have two different sizes in a mixture containing sodium chloride and potassium chloride. Here, we refit experimental data to obtain radii for individual ions and then apply the equation of state to aqueous solutions containing multiple salts. In the following sections, we will give a new physical description of this model and then discuss the use of multiple sets of single salt data to determine the ionic radii. Finally, we present calculations of thermodynamic properties of multiple-salt aqueous solutions. 11. Physical Description of the Model While there are hundreds of different models for nonelectrolyte systems, there are relatively few for mixtures containing electrolytes because little is known about the structure of electrolyte solutions. Some of the questions 0 1991 American Chemical Society

Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 241

00

interactions (Guggenheim, 1967; Meissner and Kusik, 1972; Bromley, 1973; Pitzer and Kim, 1974; Pitzer, 1979; Chen, 1980). Because they all treat the solvent as a continuous medium with a dielectric constant (4 in the charge-charge interaction term, they also are considered primitive models. Although the forms of these models differ, good representation of mean ionic activity coefficients and osmotic coefficients can be obtained by fitting the parameters in these equations to experimental data. A general form of this type of model can be presented as (2) pj = ~ j + ’ ~(y(t) + &j(Vee, Ves, qas, **e)

0

0

@@

0 I

Ions Shielded ions Dipolar solvent molecules

I (C)

Figure 1. Schematic illustrations of three types of models describing electrolyte solutions: (a) Debye-Huckel and primitive models; (b) a theoretical nonprimitive model; (c) our model.

that still are unresolved are as follows: What is the nature of the interactions between ions and neutral molecules very near them (i.e., hydrated water molecules)? How many solvent molecules are in the sheath surrounding an ion? What is the size of an ion in solution? In this paper, we show what we think are the important phenomena in electrolyte solutions and give a physical picture of how this model differs from previous models. This is done in Figure 1. Plate a in Figure 1 illustrates what are called primitive models. These models are based on the work of Debye and Huckel (1923a,b) in which ions are treated as charged particles immersed in a continuous medium with a dielectric constant, t. There are no terms in these models explicitly for molecule-molecule and charge-molecule interactions. It is presumed that these interactions all are taken into account through the dielectric constant in the charge-charge interaction terms. Because Debye-Huckel theory is valid only at infinite dilution, two different classes of models have been developed which attempt to extend Debye-Huckel theory to higher concentrations. Stell and Lebowitz (1968), Henderson and Blum (1980, 1981), and Henderson (1983) developed restricted primitive models by adding higher order ion-ion interaction terms to an expansion series based on Debye-Huckel theory. For this type of model, the chemical potential of component j in the system can be written conceptually in simple form: pj

= pj”

+ py(€)

(1)

where pP is the chemical potential of species j at a standard state and the superscript cc represents charge-charge interactions. Although the restricted primitive models are relatively simple, they work only for very dilute solutions, not for electrolytes with concentrations greater than about 0.01 m. The other class of models that have been developed to correct Debye-Huckel theory is more empirical. The models include additional terms that attempt to account for ion-solvent molecule and ion-molecular electrolyte

where Apj presents additional terms that correct DebyeHuckel theory for higher concentrations and 7’s are empirical parameters for electrolyte-electrolyte, electrolytesolvent, and solvent-solvent interactions. It should be noted that in these equations the parameters are defined for each electrolyte (ion pair) and not for each individual ion. The fact that several empirical parameters are necessary to fit experimental data implies that a single factor, the dielectric constant, is not adequate to account for ion-molecule interactions. Although it is difficult to give a clear physical picture for some of these models, because of their utility and simplicity, they are used widely in industry. Plate b in Figure 1 depicts a recent theoretical model where the ion-molecule and molecule-molecule interactions are calculated explicitly according to the potentials (Henderson et al., 1986). This type of model is referred to as nonprimitive, and the dielectric constant is not used. The chemical potential from Henderson’s nonprimitive model can be written as (3) pj = pp py pfd p y

+

+

+

where superscripts cd and dd indicate charge-dipole and dipole-dipole interactions. In this model, an aqueous electrolyte solution is treated as an ensemble of charged spheres (ions) and dipolar spheres (water molecules). Further, this model suggests that the motions of all particles (ions and molecules) in the system are equally free, that the probability of a particle (either ion or molecule) interacting with another particle in any region of the system is the same, and that this probability is given by the hard-sphere radial distribution function. Unfortunately, nonprimitive models cannot be used yet for actual calculations of electrolyte behavior in real solutions. There are several possible reasons for this: (1)Ions in solution are not as mobile as hard spheres. Like charges repel strongly and are not likely to come into contact. (2) Ions in solution are solvated. Moreover, the interactions of ions with solvent molecules that are solvated to them are fundamentally different from the interactions of ions with solvent molecules that are freely moving in the bulk of the solution. Nonprimitive models neglect this difference and assume that charge-molecule interactions are the same throughout. A comparison between the nonprimitive model and a primitive model (without electrolyte-molecule interaction terms) had been made in our previous work. As shown in Figure 2, it has been observed that both models give poor agreement with experimental data for real electrolyte solutions. One can see that the long-range interactions between ions in the nonprimitive model are so strong that the mean ionic activity coefficient drops to zero very fast. On the other hand, the ion-molecule or electrolyte-molecule interactions must be added to a primitive model to bring the values of mean ionic activity coefficients down. A physical picture of our model is illustrated in plate c of Figure 1. In this model, an electrolyte system is

CD

. . (

KC1

- T =

+* e

- IH,O 25.0'C, P Exp. Data

This Model

Table I. Fitted Values of Ionic Radii, rhmr the Literature Values of Crystal Ionic Radii, r,, Mean Polarizabilities of Ions, a,and the Number of Electrons on Ions, n ion rion, xlos cm re, ~ 1 0 cm 8 rion/rc a X Wem3 ne Na+ 0.950b 1.0669 1.123 0.210d 10 K+ 0.9470 1.330* 0.712 0.870d 18 Rb+ 0.8658 1.480b 0.585 1.81od 36 cs+ 0.7639 1.690b 0.452 2.790d 54 NH,+ 0.9265 1.480" 0.626 1.709' 10 Mg2+ 3.1980 0.780e 4.100 0.120d 10 CaZ+ 1.3865 1.060e 1.308 0.531d 18 SrZ+ 1.2767 1.038 0.843e 36 1.230e Ba2+ 0.4686 1.420e 0.330 1.630e 54 CI1.6869 1.810b 0.932 3.020d 18 Br1.6965 0.870 4.170d 36 1.950b I1.7336 2.200e 0.788 6.480e 54 NOB1.4894 2.060e 0.723 4.5We 29 SO?' 1.7309 1.934e 0.895 2.920' 50

Model (t = 1) Primitive Model without Ion-Molecule Terms

nNightingale (1959). *Pauling (1960). 'Xu and Hu (1986). dKortumand Bockris (1951). eHorvath (1985).

I

I

I

/

= Lobar

~

,

(Robinson and Stokes, 1955)

Z Y w -

I

,#'

-

L .

2

-

Er, Er, W

c

4

_ _ _ _Non-primitive

________

q--__------1

0

1

-

/

0

v z -

I

I

I

I

I

nelectrolyte components and their mixtures. However, any other equation of state could be used. In eq 5, the charge-charge interaction term, Zcc,is calculated by using third-order perturbation expansion based on a reference system of hard spheres (Henderson, 1983; Jin and Donohue, 1988a,b). The last two terms, a third-order perturbation expansion for charge-dipole interactions, Zcd,and a first-order perturbation expansion for charge-induced dipole interactions, Zcid,were derived for interactions of ions with molecules in the bulk of solution (Jin and Donohue, 1988a,b; Jin, 1989). The equation of state contains four parameters: s, the number of segments per particle; q, the normalized surface area per particle; C, a characteristic energy per unit external surface area of a particle; and c, one-third the number of external degrees of freedom. For neutral molecules (water), these four parameters have been evaluated for PACT by fitting simultaneously experimental vapor-pressure and liquid-density data. Their values have been given by Jin and Donohue (1988a,b). Since the ions considered in this work are spherical and not very large, we chose the parameter c to be unity for all ions. Three of the other parameters for ions can be estimated from their ionic radii and mean polarizabilities by using the following equations suggested by Jin and Donohue (1988a,b) 3 rion / ~ C H ;

Sion

=

qion

= rion2/rCH:

(6)

(7)

and 312

,

0.5

Cion - - 356 '-%on e,ion k r,6

where rc is the crystal ionic radius in cm, rCH is the radius of the CH2 group which is chosen as a single segment in PACT (=1.46435 X lo-* cm), ne,ion is the number of electrons on an ion, and aionis the mean polarizability an ion in cm3 (see Table I). This leaves one parameter for each ion, the ionic radius, pion, in eqs 6 and 7. The values of ionic radii for 50 ion pairs were given in our previous papers (Jin and Donohue, 1988a,b). However, to extend this work to multiple-salt systems, it is more appropriate to have one adjustable parameter for each ion rather than one for each ion pair. 111. Determination of Ionic Radii

Analysis shows that calculations of thermodynamic properties using the equation of state are very sensitive

Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 243 Y)

v)

I

I

I

? e

I

1

c1rcl-/rc,cl-=0.932 rBr-/re,Br-=0.870 r,-/r,,,.=0.788

v)

? e

rN,./rc.Na.=l.123

In

9.-

c1Br-

-

0

L

-

1 Na' In

S '

0

res./ r,,c,.=0.452

2

1

I 0.90

1

0.80

00.io

ranion/rc.anion

I 1.10

I 1.00

1.20

rci-/rc,cr

Figure 3. Curvea roltion/rww versus r-,,/recnion obtained by fitting mean ionic activity coefficient data for single-salt aqueous solutions.

I

I

I

I

0.70

0.80

0.90

1.00

I 1.10

raJrc.ar-

to the sizes of ions. Hence, in order to minimize the number of adjustable parameters in the model, the ionic size is treated as the only adjustable parameter in this model. The effectiveness of using one parameter has been shown by Jin and Donohue (1988a,b)for a large number of binary aqueous systems containing single strong electrolytes. In that work, different values for the radius of an ion were obtained depending on the system because experimental data were fitted with one parameter for each ion pair. For example, the ionic radius of chlorine ion in a solution of sodium chloride was different from the radius of chlorine ion in a mixture containing potassium chloride. This made it difficult to apply the model to systems of multiple electrolytes containing the same ion. Therefore, in this work, we still use the ionic radius as the only adjustable parameter but determine it separately for each ion by fitting simultaneously data for several ion pairs. Our study shows that the optimal value for the radius of a cation is strongly affected by the radius used for the anion in solution. It also has been found that the best-fit values for ionic radii of an ion pair in an aqueous solution containing a single electrolyte are not unique. For example, by fitting the experimental data of the mean ionic activity coefficient of NaCl in water, values of 1.5566, 1.6869, and 1.7919 are obtained for rcI- as the ionic radius of cation Na+ are set as 1.444, 1.0669, and 0.5748, respectively. Figure 3 shows the anion radii dependence of cation radii in 12 electrolyte solutions: NaC1-H20, KClH20, RbC1-H20, CsC1-H20, NaBr-H20, KBr-H20, RbBr-H20, CsBr-H20, NaI-H20, KI-H20, RbI-H20, and CsI-H20. In Figure 3, all the ionic radii are normalized by dividing by crystal radii from literature sources. By shifting the rmion/rc,,,,ion axis for the anions, Br- and I-, the curves for each cation can be matched almost one on top of another (see Figure 4). If the curves are perfectly parallel, a range of the best-fit values for each ion can be obtained. However, one can see that the curves of ionic radii of a cation versus ionic radii of different anions, while similar in shape, are not parallel. Therefore, as shown in Figure 4, a unique set of values for the four cations, Na+,

I

I 0.80

0.70

I

I

0.90

1.00

rJrc.1Figure 4. Shifting the ranion/re,anion axis for Br- and I- to determine a single set of values of ionic radii. 0

1

4

i'

t 0

0

1

..N

1

'K

I

Rb'

1

C.*

1

NH:

I

l

Y&" Ca"

I

SI'*

B.'

1

1

1

CI-

Br-

I-

1 NO;

1 SO:-

Figure 5. Comparison of the ionic radii used in our model with the literature values of crystal ionic radii, Stokes ionic radii, and hydrated ionic radii (Pauling, 1960; Horvath, 1985).

K+, Rb+, and Cs+, and three anions, C1-, Br-, and I-, can be determined by choosing a set of cross points for the curves. Values of rionfor other ions can be obtained by using the same method. The rionvalues for nine cations and five anions are listed in Table I. Figure 5 gives a comparison of our ionic radii with crystal ionic radii, Stokes ionic radii, and hydrated ionic radii (Pauling, 1960; Horvath, 1985). Because the hydration number of an ion is inversely proportional to ionic size, the crystal radii and the hydrated radii change in opposite directions. Since the Stokes radii are related to the sizes of bare ions affected by hydration in solution, their change is consistent with the change of hydrated sizes, but their values are smaller than the hydrated values. Our values of ionic radii have the same dependence with atomic

244 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 Table 11. Comparison between Average Absolute Percentage Errors, E,+, Obtained from Mean Ionic Activity Coefficient Calculations Using a Single Set of Values for rionand Average Absolute Percentage Errors, Obtained from Our Previous Calculations (Jin and Donohue, 1988a,b) Using a Size Parameter for Each Ion PaiP

h"+

system NaCI-H,O KCI-H~~ RbCl-HZO CsCl-HZO CaCl2-H20 SrClZ-H20 BaCIz-H20 NH4CI-H20 NaBr-HzO K Br-H 0 RbBr-HzO CsBr-H20 CaBrz-HzO SrBr2-H20 BaBrz-H20 NaI-H20 KI-HZO RbI-HZO CSI-H~O CaIz-H20 Sr12-H20 Ba12-H20 NaN03-H20 KNOB-HZO RbNOs-HzO NHdN03-HZO Na2S04-Hz0 KZSOd-HzO (NH4)2SO,-H20

me 0.1-6.0 0.1-5.0 0.1-5.0 0.1-6.0 0.1-3.0 0.1-3.0 0.1-1.8 0.1-6.0 0.1-4.0 0.1-5.0 0.1-5.0 0.1-5.0 0.1-2.0 0.1-2.0 0.1-2.0 0.1-3.5 0.1-4.5 0.1-5.0 0.1-3.0 0.1-2.0 0.1-2.0 0.1-2.0 0.1-6.0 0.1-3.5 0.1-4.5 0.1-6.0 0.1-5.0 0.1-0.8 0.1-4.0

E,+ 2.52 2.84 1.79 0.36 6.36 4.88 1.77 3.85 3.46 4.47 2.63 2.03 7.59 6.85 7.23 4.45 4.13 5.60 4.32 6.33 7.76 7.04 3.82 0.68 4.37 2.50 2.22 5.14 1.30

Eo,,

'

i-

-

=

I I I 25.0'C, P = 1.0 bar

I

Exp. Data (Robinson and Stokes, 1955, 1970): NaCI-H,O

0 A

KCI-H,O RbCl-H,O

- +

CsC1-H20

22t? 2

T I

E

E*

2.25 2.38 1.18 2.52 3.76 2.43 1.52 2.37 1.99 1.43 1.69 2.87 4.77 4.09 5.81 1.55 2.00 3.40 3.97 8.63 8.21 6.04 1.20 3.17

t

-

/

_.

"t tl

I

1.2

00.0

I

2.4

1

3.6

J

I 4.8

6.0

MOLALITY OF ELECTROLYTE (mol/kgH,O) Figure 6. Comparison of the calculated mean ionic activity coefficients of 1:l strong electrolytes (NaCI, KCI, RbCI, and CsCl) in water using our model with a single set of ionic radii with the experimental data of Robinson and Stokes (1955, 1970).

3.55 9.08 1.08 3.91

I

I

I

T = 25.0'C. P = 1.0 bar

All the errors are given based on the experimental data (Robinson and Stokes, 1970) a t 25 OC and 1 bar.

number as Stokes radii and hydrated radii although the slopes are different. Though it may be an overgeneralization, this suggests that our values of ionic radii depend primarily on hydration effects. By use of values of ionic radii listed in Table I, mean ionic activity coefficient calculations were carried out for a large number of aqueous systems containing single strong electrolytes. Table I1 gives a comparison between the average absolute errors obtained from these calculations, E,*, and the errors obtained from our previous calculations by using ionic radii defined for each ion pair, Eo,,. It is understandable that the errors given by using single set of ionic radii are larger than the errors obtained by using ionic radii for each ion pair. Nevertheless, E+'s for the most systems are less than 570,and the largest error is 7.8%. Figures 6 and 7 show the accuracy of using only a single set of ionic radii.

IV. Calculations for Systems Containing Multiple Salts The solubility of a solid electrolyte in an aqueous electrolyte solution is a strong function of the inter-species forces between molecules and ions. Calculations of the solubility are made by equating chemical potentials, or equivalently fugacities, in the two phases. The solubility depends not only on the activity coefficient of the solute, which is a function of interaction forces between solute and solvent, but also on the standard-state fugacities. The equation for solid-liquid equilibrium is

f p= r i x i f f

t "

(9)

where xi is the solubility (mole fraction) of electrolyte i i? the solution, yi is the liquid-phase activity coefficient, f

I

1

Exp. Data (Robinson and Stokes, 1955. 1970):

-

I

Na$O,-H,O

0

K$O,-H,O

A

(NH,),SO,-H,O

-

-Calculated -

MOLALITY OF ELECTROLYTE (mol/kgH,O) Figure 7. Comparison of the calculated mean ionic activity coefficients of 1:2 strong electrolytes (Na2S04,K2S0,, and (NH4),S04)in water using our model with a single set of ionic radii with the experimental data of Robinson and Stokes (1955, 1970).

is the standard-state fugacity for the electrolyte in the solution, and f ?lid is the fugacity of the electrolyte in its solid phase. From eq 9, the solubility can be calculated by (10)

The standard-state fugacity, 6,can be defined arbitrarily, the only thermodynamic requirement being that it must be at the same temperature, T, as the solution.

Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 245 Table 111. Values of f p''d/f for Electrolyte Solute i in Multisalt Aqueous Solutions at Temperature T solute i (T,oC) solute i fPbd/f; (T,"C) KCl 8.0218 X (25) 2.6591 X lo-' (60) 2.5533 X lo-' (70) NaCl 9.9929 X (25) CaS04 2.7517 X lo4 (28) 2.4078 X lo-' (80) 2.8311 X lo4 (38) 2.3682 X lo-' (90) 2.7517 X lo-' (50)

0

2

I

frIid/f;

'

I

'

I

'

I

'

I

'

NaCI-KCl-H,O (T=ZS"C. P = l bar) I

- -- - --

Exp. Data Bromley prediction

Here, we define the ratio of two fugacities of electrolyte solute i in multisalt aqueous solutions as f ?lid

y(T) = Xi(infinite dilution)(T)

(11)

fi

h

.-

4

or Yi(infinite dilution)(T) =

(12)

1

With the definition given by eq 11 or eq 12, the values of f P1ld/f; can be obtained from the experimental data of solubility of i a t infinite dilution a t temperature T (see Table 111). The activity coefficient yi in eq 10 can be determined by using the definition of a mean ionic activity coefficient given as yi = ( y 2 y y v

9

(13)

uyxi Yj

=

-

u(l)

,

I

/

l

1

%

I

'

I

'

I

'

I

'

-p

RT

I

I

I

I

'

I

'

I

Exp. Data Bromley prediction Pitzer prediction Our prediction

I Y

( X i + 0)

I

bar)

Y

0)

)' j = cation, anion (14)

where u(l)is the molar volume of liquid phase and pi(') is the residual chemical potential of ion j in liquid phase, which is defined as

C(: = pj(T,V,xj) - p$(T,V,xj)

(15)

where the superscript IG denotes the ideal gas state. As mentioned before, the dielectric constant, t, is used in our model to account for the screening caused by solvation of the sheath of water molecules very near each ion. As in previous work, a temperature-dependent dielectric constant for pure water, tw,has been used in our calculations. It has been shown that this treatment gives good accuracy in the calculations for single-salt aqueous solutions (Jin and Donohue, 1988a,b). In this work, the temperature dependence oft, given by Malmberg and Maryott (1956) is used t,

I

0

+

where the subscripts and - denote the cation and anion dissociated from electrolyte i, and u is a stoichiometric coefficient (=u+ + v-). The ionic activity coefficients y+ and y- can be calculated with our equation of state by

\*

0

= 87.74 - 0.40008t + 0.0009398t2- 0.00000141t3 (16)

where t is the temperature in "C. Other than being temperature dependent, tWis treated as a constant. The values of rionlisted in Table I were determined by fitting multiple sets of experimental data for single-salt systems. They are not adjusted in the calculations for systems containing multiple salts.

V. Results and Discussion Using our equation of state, we predicted the solubilities of electrolytes in seven double- and triple-salt electrolyte systems. The systems studied here are KCl-NaC1-H20,

s 2 0 L '9 N 0

r,

.-

&

e

B O

5 0

0

9

00.00

1.50

3.00

4.50

6.00

7.50

9.00

Molality of CaC1, (mole/kgH,O)

Figure 9. Comparison of the calculated solubility of NaCl in an aqueous solution containing CaCl, using Bromley's model (1973) with one mixture parameter, Pitzer's model (1974,1979) with five mixture parameters, and our model without any mixture parameter with the experimental data (Linke and Seidell, 1965) at 25 "C.

CaC12-NaC1-H20, MgCl2-NaC1-H20, CaS04-NaC1-H20, CaS04-MgC12-H20, and CaS04-MgC12-NaCl-H20. A comparison of our predictions with experimental data over wide ranges of temperatures (from 0 to 90 "C) and concentrations of electrolytes (from 0 to 5.5 M) shows the usefulness of this approach (see Tables IV and V). The results of our calculationsfor multisalt systems have been compared with the results obtained by using two other well-known models: Bromley's model (1973) with one binary adjustable parameter and Pitzer's model (1974, 1979) with four binary and one ternary parameters. Parameters for these equations were taken from Zemaitis et al. (1986). Shown in Figures 8 and 9 are calculations of the solubility of NaCl in aqueous solutions containing KCl and CaC1, a t 25 "C and 1 bar. In these double-salt systems,

246 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 Table IV. Average Absolute Percentage Errors Obtained from a Comparison of the Calculated Solubility of Calcium Sulfate with Experimental Data (Ostroff and Metler, 1966) mNaClr

system CaS0,-MgCIz-H20 CaS04-NaCI-Hz0 CaS0,-NaC1-MgC12H2O

T,OC 50-70 28-90 28-90

mMaCli?

mol/kg H,O 0 0-5.50 0-5.50

mol/kg H,O 0.34 0 0.32

(error), ?&

2.92 15.32 17.40 0. v1

Table V. Average Absolute Percentage Errors Obtained from a Comparison of the Calculated Solubility of Strong Electrolvte ( 1 ) with ExDerimental Data at 25 OC and 1 bar ~~

~

m

u o L l l1 o

0

x

.*

i

3

m(2b

system KCI(I)-NaCl (2)-H20 NaCl(l)-KCl (2)-H20 NaCI(l)-CaCIo (2)1Hz0 NaCI()I)-MgCI (2)-H2O

-

'

I

mol/kg H20 9-5.12

(error), 3.19

data source Linke and Seidell (1965)

0-2.25

5.19

Linke and Seidell (1965)

0-4.38

2.00

Linke and Seidell (1965)

0-3.32

5.78

Linke and Seidell (1965)

'

%

I

'

I

'

I

'

I

'

1

-----

NaCI-CaS0,-H20

(T=25"C, P = l b a r ) /

/,,, ~

I

0

1.10

~

-_ _ - - _ ~

,

Exp. Data Bromley prediction Pitzer prediction Our prediction

- - --.

,

l

,

3.30

---.

[ 4.40

l

l

l

---

l

l 0.12

0.08

0.04

t

I 0.16

,

I 0.20

, 024

Molality of MgC1, (mole/kgH,O)

Figure 11. Comparison of the calculated solubility of CaSO, in an aqueous solution containing MgC12 using Bromley's model (1973) with one mixture parameter, Pitzer's model (1974, 1979) with five mixture parameters, and our model without any mixture parameter with the experimental data (Ostroff and Metler, 1966) at 50 and 70 "C. 0

1

N

1

'

1

'

1

'

"

1

'

CaSO,-NaC1-MgC1,-HPO

~ _*

1

-

- _- - - _ _

I 2.20

;o.oo

l -

- - ---- -.- - - - -

,

5:

1

-1

I '

-

Exp. Data: TISOT, P=lbar T=70'C, P=lber Our prediction Pitrer prediction

/

-I

5.50

-

-. ,

6.60

Molality of NaCl

Figure 10. Comparison of the calculated solubility of CaSOl in an aqueous solution containing NaCl using Bromley's model (1973) with one mixture parameter, Pitzer's model (1974,1979) with five mixture parameters, and our model without any mixture parameter with the experimental data (Ostroff and Metler, 1966) at 25 OC.

chlorine, C1-, is a common anion contained in both salts in the mixture. By using only a single value of rionfor each ion, C1-, Na+, K+, and ea2+,our results give good agreement with the experimental data over a wide range of electrolyte concentrations. As shown in these figures, similarly accurate results are obtained by using Bromley's model and Pitzer's model. In Figures 10 and 11, calculated solubilities of CaSO, dissolved in an aqueous solution of NaCl and in an aqueous solution of MgC12using Bromley's model, Pitzer's model, and our model are compared with the experimental data given by Ostroff and Metler (1966). The figures show that our model without any binary or ternary adjustable parameters gives better results than the other models which have a number of mixture parameters. Figure 12 shows calculations for a triple-salt system. Calculations for the solubility of CaSOl in aqueous solu-

1

iZp-/T> M 0 - 0

9

00.00

1.00

2.00

3.00

4.00

5.00

6.00

Molality of NaCl (mole/kgH,O)

Figure 12. Comparison of the calculated solubility of CaSO, in an aqueous solution containing both NaCl and MgClz using Bromley's model (1973) with one mixture parameter, Pitzer's model (1974, 1979) with five mixture parameters, and our model without any mixture parameter with the experimental data (Ostroff and Metler, 1966) at 28 and 50 "C.

tions containing both NaCl and MgClz are carried out at different temperatures. The good agreement between calculations and experimental data shows the potential of this equation for design calculations for systems containing multiple salts. Although the calculations presented in this work are preliminary, they have demonstrated that this equation of state has potential for predicting thermodynamic properties of multiple-salt systems using only a single set of ionic radii and without any additional adjustable binary or ternary parameters. To extend this work to multiple solvent electrolyte systems, it is necessary to develop a method that can give a good estimation of the dielectric constant, e, for a mixed solvent. Future work also will

Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 247

include electrolytic mixtures containing large molecules and polyions. Acknowledgment The support of this research by the Division of Chemical, Biological and Thermal Engineering of National Science Foundation under Grant CBT-8513434 is gratefully acknowledged.

Nomenclature 3c = total number of external degrees of freedom per molecule f = fugacity (bar) k = Boltzmann's constant (=1.380662 X erg/K) ne = number of electrons on a n ion P = pressure (bar) q i = normalized surface area of particle i R = gas constant [(bar-L)/(mol.K)] r, = crystal ionic radius (cm) rion = ionic radius (cm) rCH2 = radius of CH2 segment (cm) si = number of segments of species i T = absolute temperature (K) t = temperature ("C) u = molar volume (L/mol) V = total volume of system (L) x = mole fraction in liquid phase 2 = compressibility factor (=Pu/RT)

Greek Letters = average polarizability (cm3)

CY

y = activity coefficient

= characteristic energy per u n i t external surface area = dielectric constant tl = adjustable parameter t t

p

= chemical potential

Y

= stoichiometric coefficient

Superscripts O

= standard state

c = charge d = dipole id = induced dipole

IG = ideal gas LJ = Lennard-Jones q = quadrupole r = residual rep = repulsion (1) = liquid phase

Subscripts = cation

+

- = anion i, j = component indices w = water

Literature Cited Bock, E. On the Solubility of Anhydrous Calcium Sulphate and of Gypsum in Concentrated Solutions of Sodium Chloride a t 25"C, 30°C, 40°C and 50°C. Can. J . Chem. 1961,39, 1746-1751. Bromley, L. A. Thermodynamic Properties of Strong Electrolytes in Aqueous Solutions. AIChE J. 1973, 19, 313-320. Chen, C. C. Computer Simulation of Chemical Process with Electrolytes. Sc.D. Thesis, Department of Chemical Engineering, MIT, 1980. Debye, P.; Huckel, E. Zur Theorie der Elektrolyte. Phys. 2.19238, 24 (9), 185-206. Debye, P.; Hockel, E. Zur Theorie der Elektrolyte 11. Phys. Z. 192319, 24 (15), 305-325. Based on Denman. W.L. Maximum Re-use of Cooling Water Gypsum Content and Solubility. Ind. Eng. Chem. 1961, 53, 811-822.

....

Gering, K. L.; Lee, L. L. A Amolecular Approach to Electrolyte Solutions: Phase Behavior and Activity Coefficients for MixedSalt and Multisolvent Systems. Fluid Phase Equilib. 1989, 48, 111-139. Guggenheim, E. A. Thermodynumics, 5th ed.; North-Holland Publishing Company: Amsterdam, 1967. Henderson, D. Perturbation Theory, Ionic Fluids and Electric Double Layer; Advances in Chemistry Series 204; American Chemical Society: Washington, DC, 1983; Vol. 3, pp 47-71. Henderson, D.; Blum, L. Perturbation Theory for Charged Hard Spheres. Mol. Phys. 1980,40, 1509-1511. Henderson, D.; Blum, L. Perturbation Theory, The Mean Spherical Approximation, and The Electrical Double Layer. Can. J. Chem. 1981,59, 1903-1905. Henderson, D.; Blum, L.; Tani, A. In Equation of State of Ionic Fluids; Chao, K. C., Robinson, R. L., Jr., Ed.; Advances in Chemistry Series 13; American Chemical Society: Washington DC, 1986; pp 281-296. Horvath, A. L. Handbook of Aqueous Electrolyte Solutions; Ellis Horwood Limited: New York, 1985. Jin, G. Thermodynamics of Electrolyte Solutions. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD, 1989. Jin, G.; Donohue, M. D. An Equation of State for Electrolyte Solutions 1. Aqueous Systems Containing Strong Electrolytes. Ind. Eng. Chem. Res. 1988a, 27, 1073-1086. Jin, G.; Donohue, M. D. An Equation of State for Electrolyte Solutions 2. Single Volatile Weak Electrolytes in Water. Ind. Eng. Chem. Res. 1988b, 27, 1737-1743. Kortum, G.; Bockris, J. O'M. Textbook of Electrochemistry II; Elsevier Publishing Company: London, 1951. Linke, W. F.; Seidell, A. Solubilities of Inorganic and Metal-organic Compounds; American Chemical Society: Washington, DC, 1965; VOl. 11. Malmberg, C. C.; Maryott, A. A. Dielectric Constant of Water from 0 to 100°C. J. Res. Natl. Bur. Stand. 1956,56, 1-8. Marshall, W. L.; Slusher, R.; Jones, E. V. Aqueous Systems a t High Temperature. XIV. Solubility and Thermodynamic Relationships for CaS04 in NaC1-H20 Solutions from 40" to 200 "C, 0 to 4 Molal NaC1. J. Chem. Eng. Data 1964,9, 187-191. Meissner, H. P.; Kusik, C. L. Activity Coefficients of Strong Electrolytes in Multicomponent Aqueous Solutions. AIChE J. 1972, 18, 294. Nightingale, Jr., E. R. Phenomenological Theory of Ion Solution, Effective Radii of Hydrated Ions. J. Phys. Chem. 1959, 63, 1381-1387. Ostroff, A. G.; Metler, A. V. Solubility of Calcium Sulfate Dihydrate in the System NaCl-MgC12-H,0 - - from 28°C to 70°. d. Chem. Eng. Data 1966,11, 346-350. Pauling. L. The Nature of The Chemical Bond. 3rd ed.: Cornel1 University Press: Ithaca, NY, 1960. Pitzer, K. S.Thermodynamics of Electrolytes. J. Phys. Chem. 1973, 77, 268-277. Pitzer, K. S. Theory: Ion Interaction Approach. In Actiuity Coefficients in Electrolyte Solutions; Pytkowitz, R. M., Ed.; CRC Press: Boca Raton, FL, 1979. Pitzer, K. S.;Kim, J. J. Thermodynamics of Electrolytes IV. Activity and Osmotic Coefficients for Mixed Electrolytes. J. Am. Chem. SOC. 1974,96, 5701. Power, W. H.; Fabuss, B. M.; Satterfield, C. N. Transient Solubilities in the Calcium Sulfate-Water system. J.Chem. Eng. Data 1964, 9,437-442. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth: London, 1955. Robinson, R. A.; Stokes R. H. Electrolyte Solutions, 3rd ed.; Butterworth: London, 1970. Stell, G.; Lebowitz, J. L. Equilibrium Properties of a System of Charged Particles. J. Chem. Phys. 1968,48,3706-3717. Vimalchand, P. Thermodynamics of Multi-polar Molecules. Ph.D. Dissertation, The Johns Hopkins University, Baltimore, MD, 1986. Vimalchand, P.; Donohue, M. D.; Celmins, I. Equations of State: Theories and Applications; In Thermodynamics of dipolar molecules: The Perturbed-Anisotropic-Chain Theory; Chao, K. C., Robinson, R. L., Jr., Ed; American Chemical Society: Washington, DC, 1985; pp 297-313. Xu, Y. N.; Hu, Y. Prediction of Henry's Constants of Gases in Electrolyte Solutions. Fluid Phase Equilib. 1986, 30, 221-228. Yeatts, L. B.; Marshall, W. L. Apparent Invariance of Activity Coefficients of Calcium Sulfate at Constant Ionic Strength and Temperature in the System CaSO,-Na2SO,-NaNO3-H20 to the

248

Ind. Eng. Chem. Res. 1991, 30, 248-254

Critical Temperature of Water. J. Phys. Chem. 1969, 73,81-90. Zemaitis, J. F., Jr.; Clark, D. M.; Mal,M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; DIPP, Sponsored by AIChE: New York, 1986. Zen, E.-An. Solubility Measurements in the System CaSO4-NaC1-

H20 a t 35O, 50°,and 7OoC and One Atmosphere Pressure. J.

Petrol. 1965, 6, 124-164.

Received for reuiew February 12, 1990 Accepted July 19, 1990

Prediction of Thermodynamic Properties of Oil and Gas Condensate Mixtures Kim Aasberg-Petersen and Erling Stenby* Institut for Kemiteknik, Building 229, Danmarks Tekniske H@jskole,2800 Lyngby, Denmark

This paper presents a new method for the prediction of phase behavior in reservoir fluids using a four-parameter cubic equation of state. T h e parameters in the employed C7+-characterization procedure are calculated directly from measured molecular weight and specific gravity data for the hydrocarbon fractions. I t is shown that PVT properties (e.g., liquid dropout) and phase equilibria can be accurately predicted even in the near-critical region by using the new method. Saturation points for a number of fluids containing significant amounts of non-hydrocarbons (COPor N,) are calculated. The predicted saturation points agree well with experimental data. Finally, it is shown that the new model is able to accurately predict liquid volumes of mixtures including those with a considerable content of N2.

Introduction Accurate determination of phase behavior is essential in reservoir simulation and in design of transport and separator equipment. As the need for enhanced oil recovery methods such as gas injection is increasing, the demand for accurate phase behavior predictions is growing. The phase behavior model must take into account not only the effects caused by the complicated composition of the reservoir fluids but also the effects caused by the presence of injection gases such as COPor N z in significant amounts. Much effort has been given to develop models suitable for the prediction of phase behavior in oil and gas condensate mixtures. However, most models in the literature are not preaictive, as experimental phase equilibrium data are needed to tune one or more parameters in the model. Recently, Pedersen et al. (1988) developed a characterization procedure for the SRK equation of state coupled with the volume correction term of Peneloux et al. (1982). This procedure does not need experimental data and generally gives good predictions of saturation points and vapor-liquid equilibria. However, the model frequently calculates a too large liquid precipitation for gas condensates when simulating constant composition expansion experiments. In addition, predicted liquid density values are sometimes inaccurate. It is the purpose of this paper to describe a new model that preserves the good qualities of the model of Pedersen et al. with respect to phase equilibrium predictions and at the same time improves liquid dropout and liquid density calculations. The model should also be able to accurately predict the phase behavior of fluids with a considerable content of COPor N2. This is of special importance in the simulation of miscible or immiscible gas injection processes. The Model Models for the prediction of the phase behavior of reservoir fluids are extensively used in compositional reservoir simulation studies. It seems natural to choose a cubic equation of state (EOS) as the thermodynamic basis for the model to be developed, since cubic EOS's are simple and fast models and easy to implement in any reservoir

simulation program. Jensen (1987) found the ALS EOS (Adachi et al., 1983) to be the most accurate for prediction of the phase behavior of well-defined hydrocarbon mixtures with and without a considerable content of C 0 2 or N z . The ALS EOS is given below:

For pure components, the four parameters are calculated as follows:

a ( T ) = (1 + m(1 - ( T / T c ) 1 / 2 ) ) 2 @k

=

RbkRTc 7

k = 1, 2, 3

(3) (4)

Expressions for the calculation of Q,, Qbi, and m are given in the Appendix. The expressions for Qb2 and Qb3 are derived such that the critical point criteria are satisfied. The ALS EOS seems to be well suited for calculation of the phase equilibria of reservoir fluids but often proves to give inaccurate predictions of the densities of hydrocarbon mixtures (Aasberg-Petersen, 1989). It was therefore decided to incorporate the volume translation principle of Peneloux et al. (1982) into the ALS equation, resulting in the following new equation of state: (5)

The b parameters are calculated as follows: (6) bl = @1 - C bz = 62 - C (7) (8) b3 = 03 + C For each component, the Peneloux-type parameter, C, is in this work determined such that the equation of state gives the correct value of the liquid density at atmospheric pressure and at a temperature where the component is a liquid. The C parameters of the C7+fractions are deter-

0888-5885/91/2630-0248$02.50/00 1991 American Chemical Society