Predicting diffusivities in dense fluid mixtures

Predicting diffusivities in dense fluid mixtures

C. DARIVA, L.A.F. COELHO and J. VLADIMIR OLIVEIRA

Programa de Engenharia Química, PEQ/COPPE/UFRJ, 21945-970, Rio de Janeiro - RJ, Brazil,

E-mail: Vladimir@PEQ/COPPE/UFRJ.BR

(Received: September 8, 1998; Accepted: December 8, 1998)

Abstract - In this work the Enskog solution of the Boltzmann equation, as corrected by Speedy, together with the Weeks-Chandler-Andersen (WCA) perturbation theory of liquids is employed in correlating and predicting self-diffusivities of dense fluids. Afterwards this theory is used to estimate mutual diffusion coefficients of solutes at infinite dilution in sub and supercritical solvents. We have also investigated the behavior of Fick diffusion coefficients in the proximity of a binary vapor-liquid critical point since this subject is of great interest for extraction purposes. The approach presented here, which makes use of a density and temperature dependent hard-sphere diameter, is shown to be excellent for predicting diffusivities in dense pure fluids and fluid mixtures. The calculations involved highly nonideal mixtures as well as systems with high molecular asymmetry. The predicted diffusivities are in good agreement with the experimental data for the pure and binary systems. The methodology proposed here makes only use of pure component information and density of mixtures. The simple algebraic relations are proposed without any binary adjustable parameters and can be readily used for estimating diffusivities in multicomponent mixtures.

Keywords: Statistical mechanics, diffusion coefficient, mixture.

INTRODUCTION

The design of separation processes requires the accurate determination of diffusion coefficients and besides conventional techniques many applications can be envisaged such as tertiary-oil recovery, dense fluid cleaning of solid surfaces and the extraction of natural products from vegetables matrices. As a consequence, in recent years there has been a growing interest in correlating and predicting diffusivities in dense fluids and numerous experimental data concerning this subject have been reported in the literature. However, there is not a rigorous and successful theory for diffusion of dense fluids due to their complex microscopic structures.

In a general sense, the existing methods to correlate or predict diffusivities are based on molecular dynamics results and on the hydrodynamic and statistical mechanical theories of fluids. The hydrodynamic theory, as expressed by the Stokes-Einstein relation, does not correlate the diffusivities well when the solute particles are no longer much larger in size compared to the solvent molecules (Evans et al., 1981). Nonetheless, some empirical correlations based on the Stokes-Einstein equation have been proposed in the literature (Hayduk and Minhas, 1982; Liu and Ruckenstein, 1997; Umesi and Danner, 1981). The free-volume theory has also provided a number of empirical correlations (Guo and Kee, 1991; He et al., 1998 ).

In terms of practical applications and theoretical investigations, the Enskog’s kinetic theory of hard-sphere fluids has been the most fruitful. In this theory, the transport properties are calculated by very simple equations relating the particle mass, temperature, fluid density, particle size and the pair correlation function at the contact point. Based on the smooth or rough-hard-sphere theory, several empirical methods have been developed (Erkey et al., 1990; Sun et al., 1994; Eaton and Akgerman, 1997).

Most of the above mentioned correlations are devoted to pure fluids or infinite dilution diffusion coefficients and hence just a few methods for predicting the concentration dependence of mutual diffusion coefficients have been proposed (He, 1995; Wang et al., 1996). Bandrowski and Kubaczka (1982) pointed out that it is practically impossible to determine exactly from the theory the dependence of mutual diffusion coefficients on concentration. Despite the drawbacks mentioned by Taylor and Krishna (1993), empirical relations such as Vignes or Darken and their modifications (see, for example, Reid et al., 1987 and Pertler et al., 1996), which are based, respectively, on infinite dilution and intradiffusion diffusivities, have been used for this purpose.

In this work our main objective is to show that the Enskog based equations work well for estimating diffusivities of dense pure fluids and fluid mixtures when the hard-sphere diameter is obtained from the Weeks et al. (WCA) (1971) perturbation theory of liquids. Experimental self-diffusion data were used to fit the hard-sphere diameters and these diameters were employed in predicting diffusivities in binary mixtures without any cross parameter. Further, calculations along the gas-liquid boundary were also accomplished in order to investigate the behavior of Fick diffusion coefficients in the proximity of a binary vapor-liquid critical point. We then show that with a very simple equation it is possible to predict diffusivities in pure as well as in multicomponent mixtures without introducing empirical relations for the concentration dependence of mutual diffusion coefficients.

THEORY

Enskog’s solution of the Boltzmann equation for a dense pure hard-sphere fluid is given by (Chapman and Cowling, 1970; Reed and Gubbins, 1973):

(1)

where r * (º r s ³) is the reduced density, g(s ) is the pair correlation function at the contact point for hard-spheres (Carnahan and Starling, 1969) and,

(2)

where, m is the particle mass, T the absolute temperature, r is the number density, k the Boltzmann constant and s the hard-sphere diameter.

The density correction proposed by Speedy (1987) has been used to more accurately account for the density dependence of D, namely,

(3)

where,

(4)

where D_SHS is the so-called diffusion coefficient for the smooth hard-sphere (SHS) fluid.

To account for the coupling between translational and rotational motions, Chandler (1975) has proposed the following equation for the self-diffusion coefficient:

(5)

where D_RHS is the so-called diffusion coefficient for a rough hard-sphere (RHS) fluid and A is a coupling factor bounded between zero and one (0<A<1), and rigorously independent of density. It is unity for the perfect smooth hard-sphere fluid and should also be independent of temperature if only binary collisions occur. However, the literature data reveal that the coupling parameter is a strong function of temperature, affecting sensibly the diffusion coefficient and, therefore, Eq. (5) may not be a good basis for predictive methods (Rocha et al., 1997). Furthermore, the rough hard-sphere theory is not capable of predicting diffusivities in mixtures using only pure component parameters. For instance, for binary mixtures, another coupling parameter, A₁₂ , which is estimated from the binary data is needed and it is an undesirable feature for applications in engineering design.

Very recently Rocha et al. (1997), based on the idea of Schrodt and Davis (1974), showed that the WCA (Weeks et al., 1971; Andersen et al., 1971) perturbation theory of liquids coupled with the Enskog solution of the Boltzmann equation is a successful theory for estimating diffusivities in liquids and liquid mixtures. Basically, the method presented by Rocha et al. consists in making the coupling parameter, A, unity through the use of an effective hard-sphere diameter, d, that comes from the first-order WCA perturbation theory of liquids; i.e., by solving the following equation:

(6)

where Y^hs(r,r,d) is the cavity distribution function for a hard-sphere molecule of diameter d at density r , the hard-sphere potential, , is:

(7a)

(7b)

and,

(8)

(9)

(10)

where f ^ref(r) and f ^P(r) are the reference (repulsive) and perturbation (attractive) potentials, respectively; r_m is the distance where the actual potential, f (r), is a minimum and it is given by the Lennard-Jones (6,12) pair-potential model:

(11)

Instead of solving Eq. (6) at a given temperature and density, we have adopted the recent and accurate proposal of Souza and Ben-Amotz (1993),

(12)

where T* (º T/(e /k)) is the reduced temperature, e is the Lennard-Jones energy parameter and the constants a₁=1.5001, a₂=-0.03367, a₃=0.0003935, a₄=-0.09835, a₅=0.04937, a₆=-0.1415 (Souza and Ben-Amotz, 1993).

By Eq. (11) one can see that the hard-sphere diameter, d, is calculated as a function of temperature and the number density for given values of s and e . In this way, all the s in the diffusion coefficient equations are replaced by the WCA hard-sphere diameter, d.

RESULTS AND DISCUSSION

In this work we follow the approach of Rocha et al. (1997) to estimate diffusion coefficients of dense pure fluids and fluid mixtures using the smooth-hard-sphere theory (A is made unity in the rough hard-sphere model) with the effective hard-sphere diameters calculated from the WCA perturbation theory, Eq. (11). Of course, the coupling between rotational angular and translational momenta plays an important role in diffusion but it is shown that the smooth hard-sphere model is sufficient to estimate diffusivities in dense pure fluids and fluid mixtures. As previous mentioned, the coupling parameter is a strong function of temperature, but in our case the variation with temperature is given by the equilibrium perturbation theory and the resulting value of A is, in most cases, nearly unity.

In Table 1 we present the results and conditions of fitting of the self-diffusion coefficients using the density correction of Speedy. When densities for pure fluids were not given along with the diffusion coefficients, they were estimated using the modified Racket equation (Reid et al., 1987). The Lennard-Jones parameters of distance, shown in Table 1, were estimated using the Maximum Likelihood Method (Anderson et al., 1978) weighing experimental self-diffusion data according to its precision. We have found that the energy parameters are insensitive to the experimental self-diffusivity data, therefore, for simplicity, they were fixed in this work equal to those presented in the literature (Wilhelm and Battino, 1971; Reid et al., 1987; Hirschfelder et al., 1954). Thus, the model makes use of only one adjustable parameter, s , which is obtained from pure component information. Note that the components figuring in Table 1 are the most commonly used solvents for supercritical fluid processing purposes and once the size parameters have been obtained safe predictions can be accomplished.

Table 1: Average absolute deviations of the fitted self-diffusion coefficient and Lennard-Jones size and energy parameters.

Substance	Range of T(K)	NP+	AAD%*	s (10-10m)++	e /k (K)**	Data Source***
Carbon dioxide	298	7	3.7	3.6283	195.2	a
Ethane	294	5	2.5	4.1047	215.7	b
Propane	294	5	4.5	4.6398	237.1	b
n-hexane	232-333	10	10.4	5.5617	517.0	c
Cyclohexane	313.0	7	2.5	5.4050	573.0	d
Ethylene	298	19	9.9	3.9671	224.0	e
Benzene	298	10	1.7	5.0081	531.0	f
Toluene	298	13	2.6	5.2829	575.0	g
Sulfur hexafluoride	296	6	4.8	4.7808	201.0	h
Carbon tetrachloride	298	6	5.7	5.1761	536.0	i
Acetone	237-333	4	5.6	4.6132	362.0	j
Overall deviation (%)			5.5

* ; ⁺number of data points

⁺⁺size parameters estimated from self-diffusion data in this work

^**Literature values (Reid et al., 1987; Wilhelm and Battino, 1971; Hirschfelder et al., 1954).

^***aEtesse et al. (1992); ^bGreiner-Schmid et al. (1991); ^cDouglas and McCall (1958); ^dJonas et al. (1980); ^eArends et al. (1981); ^fMcColl et al. (1972); ^gHarris et al. (1993); ^hDeZwaan and Jonas (1975); ⁱChandler (1975); ^jKruger and Weiss (1970).

As an example, Figure 1 depicts experimental and predicted values in a wide range of density for the self-diffusion coefficient of carbon dioxide, outside the temperature range where the size parameter was fitted. It can be seen from this figure a reasonable agreement between experiment and theory showing that the hard-sphere diameter is capable of accounting very well for changes in temperature and density. Note also that there is some divergence in the experimental data obtained by different authors at similar conditions.

Figure 1:
Experimental and predicted self-diffusion coefficient for carbon dioxide at 323.15K (Overall deviation (%)=12.4)

Binary Systems

For a binary mixture, the mutual diffusion coefficient is given by (Chapman and Cowling, 1970):

(13)

where r_m is the number density of the mixture, d₁₂ the cross hard-sphere diameter, and m₁₂ (º m₁m₂/(m₁+m₂)) the reduced mass of solute 1 and solvent 2. The parameter d₁₂ is calculated as the arithmetic mean of the pure component hard-sphere diameters, d₁ and d₂, following the approach of Lee and Levesque (1973), at the temperature of the mixture and density of the pure substances:

(14)

Considering the importance of some natural products in the field of food industries, soil decontamination as well as for pharmaceutical applications and trying to test the present approach for a greater number of systems, we have predicted mutual diffusivities of a variety of compounds of prominent interest. For these substances, not appearing in Table 1, the Lennard-Jones energy parameters were estimated from the simple relation:

(15)

and the Lennard-Jones size parameters from:

(16)

and the critical properties were estimated from the Joback method (Reid et al., 1987).

For the density correction of Speedy, we have assumed the van der Waals mixing rule (Grundke and Henderson, 1972) so that the reduced density of the mixture, r^*_m, is given by:

(17)

In Table 2 and from Figures 2 to 6 the predicted mutual diffusion coefficients at infinite dilution are compared with the experimental data. Though these systems incorporate both great differences in size (asymmetries) and in intermolecular forces (chemical nature), making them difficult to predict, the agreement between experiment and theory can be considered quite good, with an overall average absolute error of 16.9%. Notice that no binary adjustable parameters were employed in the calculations. It should also be noted that good predictions, when compared to those obtained from the most recent methods available in the literature (Eaton et al., 1997; Liu and Ruckenstein, 1997), were achieved for those systems whose solute-size parameters were estimated from Eq. (16) and this may be a useful characteristic of the present model.

Table 2: Average absolute deviations of the predicted values for the mutual diffusion coefficient of species 1 at infinite dilution in component 2 (D

Solvent	Solute	Range of T(K)	NP	AAD%	Data Source*
Carbon dioxide	Acetone	303-333	7	5.6	a
	Benzene	308-328 313 303-333 313-333	17 02 11 15	17.7 9.8 16.6 6.2	b c a d
	Toluene	313-333 308-328	15 18	13.5 26.8	d e
	Ethylene	298-348	23	16.3	f
	Naphthalene	313 303-333	02 11	32.2 20.9	c a
	Phenanthrene	303-333 308-328	07 14	23.0 22.1	a g
	1,3,5 trimethylbenzene	303-333	07	20.7	a
	Ethylbenzene	313-333	15	18.6	d
	Hexachlorobenzene	313-333	14	20.7	g
	i-propylbenzene	313-333	15	21.3	d
	n-propylbenzene	313-333	15	21.8	d
	benzoic acid	298-318	07	17.1	h
	behenic acid ethyl ester	308-318	17	23.8	i
	butyric acid ethyl ester	308-318	16	15.6	i
	capric acid ethyl ester	308-318	16	19.4	i
	caprylic acid ethyl ester	308-318	16	19.4	i
	cis-11-eicosenoic acid methyl ester	313	02	19.3	c
	docosahexenoic acid ethyl ester	308-318	17	21.6	i

Table 2 (cont.)

Solvent Solute Range of T(K) NP AAD% Data Source docosahexenoic acid methyl ester 308-318 17 20.9 i erucic acid methyl ester 313 02 21.2 c linoleic acid methyl ester 308-328 19 20.8 j myristic acid ethyl ester 308-318 16 20.7 i myristoleic acid methyl ester 313 02 19.8 c stearic acid ethyl ester 308-318 17 19.2 i caffeine 308-328 21 12.3 e chrysene 303-333 04 24.2 a d1-limonene 313 02 23.4 c indole 313 02 6.5 c oleic acid 308 02 15.7 h pyrene 303-333 03 24.6 a vitamin A acetate 313 02 10.1 c vitamin E 313 02 22.3 c ethyl acetate 308-328 15 26.5 e phenol 308-328 21 13.3 e Ethylene carbon dioxide 298-348 27 9.2 f Ethane 1-octene 296-322 06 5.8 k 1-tetradecene 293-322 09 12.6 k Propane 1-octene 296-337 08 7.7 k 1-tetradecene 292-337 08 14.8 k Sulfur hexafluoride 1,3,5 trimethylbenzene 328 11 27.1 l 1,4 dimethylbenzene 328 11 21.9 l benzene 328 09 28.6 l toluene 328 11 30.6 l carbon tetrachloride 328 07 31.1 l Cyclohexane benzene 298-523 08 4.8 m toluene 298-523 08 3.9 m naphthalene 298-523 08 4.5 m phenanthrene 298-523 08 4.6 m p-xylene 298-523 08 5.7 m n-hexane benzene 333-543 15 12.5 n toluene 333-543 15 11.2 n naphthalene 333-543 15 9.1 n phenanthrene 333-543 15 8.3 n p-xylene 333-543 15 9.6 n Overall deviation (%) 16.9

^*

^a

Sassiat et al. (1987);

^b

Chen (1983);

^c

Funazukuri et al. (1992);

^d

Suarez et al. (1993);

^e

Lai and Tan (1995);

^f

Takahashi and Hongo (1982);

^g

Akgerman et al. (1996);

^h

Catchpole and King (1994);

ⁱ

Liong et al. (1992);

^j

Funazukuri et al. (1991);

^k

Noel et al. (1994);

^l

Kopner et al. (1987);

^m

Sun and Chen (1985a);

ⁿ

Sun and Chen (1985b)

Figure 2:
Experimental and predicted values for the mutual diffusion coefficient of benzene (1) at infinite dilution in carbon dioxide (2) – see Table 2.

Figure 3:
Comparison of experimental and predicted diffusion coefficients of aromatics at infinite dilution in carbon dioxide – see Table 2.

These results show that very good predictions can be obtained even when the temperature of the system is out of the range where the size parameters were fitted. For instance, diffusivities of solutes in n-hexane were predicted in a wide range of temperature, from sub to the supercritical solvent region. Thus, as found by Rocha et al. (1997), it is not necessary to attribute any reduction in diffusion to inelastic effects as proposed in the rough-hard-sphere theory when the hard-sphere diameter is adequately estimated. Since the WCA theory overestimates the hard-sphere diameter (Kang et al., 1985; Saumon et al., 1989) and based on the Enskog original solution of the Boltzmann equation for hard-sphere fluids, Eqs. (1) and (2), it would imply in an appropriate treatment for interpreting diffusion, as pointed out by Speedy et al. (1989).

As in the case of self-diffusivities of carbon dioxide, one can observe from Figure 2 some divergence in the experimental data leading to some of the scatter in the predictions shown in Figures 3 to 6. It is also worth noticing from Figures 1 to 6 that as the density increases the agreement between experimental and calculated values for both self- and mutual diffusivities is much improved. This fact might be explained in terms of the WCA perturbation theory of liquids which works very well at higher densities where the structure is primarily determined by repulsive forces.

Concerning the infinite dilution diffusivities, some recent developments should be commented such as those presented by Eaton et al. (1997) and Liu and Ruckenstein (1997). Both methods contain two adjustable parameters obtained from experimental mutual diffusion coefficient of solutes at infinite dilution in some supercritical solvents and they cannot be applied to estimating the concentration dependence of mutual diffusivities.

Considering that the present approach is not limited to infinite dilution case and, in fact, has been successfully used to predict mutual diffusion coefficients in liquid mixtures (Rocha et al., 1997), the behavior of Fick diffusivities was investigated in the vicinity of a binary vapor-liquid critical point. Firstly, the vapor-liquid equilibrium data for carbon dioxide-heptane from Kalra et al. (1978) were employed to calculate mutual diffusivities in the liquid phase along the gas-liquid coexistence curve. To do so, Eqs. (12), (13) and (17) were used together with the reported temperature, molar fraction and density (an indirect measurement).

In order to compare the predicted and experimental mutual diffusion values it is important to mention that the phase equilibrium data from Kalra et al. are in the temperature range of 311-477K while the diffusion data from Saad and Gulari (1984) for this system are in the range of 283-323K. We have not attempted to use an equation of state to fit the phase equilibrium data at 311K and perform extrapolations since the model is very sensitive to the mixture density values.

Consequently, we present in Figure 7 the experimental mutual diffusion of Saad and Gulari from 283 to 313K and our predictions at 311K. Though at higher pressures large deviations are observed, the predicted results can be considered encouraging since a qualitative analysis is possible and, to the best of our knowledge, there is no other methodology available in the literature to perform such predictions. It should be stressed again that no binary interaction parameters or empirical mixing rules were employed in the calculations.

Figure 7:
Experimental (Saad and Gulari, 1984) and predicted mutual diffusion coefficient for the system carbon dioxide-heptane in the liquid phase along the gas-liquid boundary. Vapor-liquid equilibrium data from Kalra et al. (1978).

According to Saad and Gulari experimental data, two interesting phenomena can be observed: crossover points and the appearance of minima. As the crossover dictates in most cases a compromise between two process variables, namely temperature and pressure, thus affecting the yield and selectivity of the process, an investigation whether the present model is able to predict the occurrence of crossover points was carried out. For this purpose, vapor-liquid equilibrium data for carbon dioxide-acetone from Adrian and Maurer (1997) were used to generate the mutual diffusion coefficients in the liquid phase along the gas-liquid boundary. It can be noticed from Figure 8 a crossover point showing the competition between kinetic (temperature) and structural effects, as given by the density of the mixture (in molecular units). The second effect, the occurrence of minimum in the diffusion curves, is now under investigation.

Figure 8:
Predicted mutual diffusion coefficient for the system carbon dioxide-acetone in the liquid phase along the gas-liquid boundary. Vapor-liquid equilibrium data from Adrian and Maurer (1997).

CONCLUSIONS

We have predicted diffusion coefficients in dense pure fluids and fluid mixtures using the Enskog kinetic and WCA theories. Mutual diffusion coefficients at infinite dilution and as a function of concentration were also predicted making use of the smooth-hard-sphere theory with a temperature and density dependent hard-sphere diameter, obtained from the WCA theory of liquids. Mutual diffusion coefficients were predicted using only pure component information and density of mixtures. The proposed method with the Speedy density correction correlates the pure component diffusivities well and predicts the binary diffusion coefficients in good agreement with experimental values. The method applied here needs, in fact, only one adjustable parameter that is obtained from self-diffusion data of pure substances. We have achieved good results thanks to the well-established and meaningful effective hard-sphere diameter.

NOMENCLATURE

A coupling factor of the rough hard-sphere theory

a_i constants of the Souza and Ben-Amotz correlation for the effective hard-sphere diameter

d effective hard-sphere diameter, m

D self-diffusion coefficient for dense hard-sphere fluid, m²/s

D₀ defined by Eq. (2), m²/s

D₁₂ mutual diffusion coefficient of 1 in binary mixture of 1 and 2, m²/s

D_RHS self-diffusion coefficient for the rough hard-sphere theory, m²/s

D_SHS self-diffusion coefficient for the smooth hard-sphere theory, m²/s

g(s) pair correlation function at the contact point for hard-spheres

k Boltzmann constant, J/(molec.K)

m particle mass, Kg

N number of components

NP number of data points

P correlation of reduced density

r distance center-to-center of two particles

r_m distance where the pair potential function is a minimum, m

T temperature, K

T* reduced temperature

v volume, Kgmol/m³

x_i mole fraction of component i

Y^hs cavity distribution function for hard-sphere fluids

Greek Letters

b

º (kT)

^-1 , J

^-1

e

energy parameter of the Lennard-Jones model, J

f

actual pair potential function, J

hard-sphere potential, J

f

^p perturbation pair potential function, J

f

^ref reference pair potential function, J

r

number density, number of molecules/m

³

r

^* reduced density

number density of the mixture, number of molecules/m³

reduced density of the mixture, Eq. (17)

s

hard-sphere diameter, m

Subscripts

RHS rough hard-sphere

SHS smooth hard-sphere

c critical value

Superscripts

SP Speedy

ACKNOWLEDGEMENT

This work was supported by PRONEX, grant no. 124/96 (Minitério de Ciência e Tecnologia - Brazil). The authors would like to thank CNPq for the financial support.

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Publication Dates

• Publication in this collection
  16 Dec 1999
• Date of issue
  Sept 1999

History

• Received
  09 Aug 1998
• Accepted
  12 Aug 1998