PREDICTING DIFFUSIVITIES IN DENSE FLUIDS

PREDICTING DIFFUSIVITIES IN DENSE FLUIDS¹

Cláudio DARIVA², Luiz A. F. COELHO², José V. OLIVEIRA², ^*

SUMMARY

In this work the Weeks-Chandler-Andersen (WCA) perturbation theory coupled with the Enskog’s solution of the Boltzmann equation for dense hard-sphere fluids is employed for estimating diffusion coefficients in compressed pure liquids and fluids and dense fluid mixtures. The effect of density correction on the estimation of diffusivities is analyzed using the Carnahan-Starling pair correlation function and the correlation of Speedy and Harris which have been proposed as models of self-diffusion coefficient of hard-sphere fluids. The approach presented here is based on the smooth hard-sphere theory without any binary adjustable parameters and can be readily used for estimating diffusivities in multicomponent fluid mixtures. It is shown that the correlated and the predicted diffusivities are in good agreement with the experimental data and much better than estimates of Wilke-Chang equation.

Keywords: Diffusivity, perturbation theory, dense liquids

RESUMO

PREDIÇÃO DE DIFUSIDADE EM FLUIDOS DENSOS. A teoria das perturbações de Weeks-Chandler-anderson (WCA) juntamente com a solução de Enskog da equação de Boltzmann para fluidos densos de esferas rígidas é aplicada a correlação e predição de difusividades de misturas a elevadas pressões. O denominado "efeito de densidade" na solução original de Enskog para fluidos densos é analisado empregando-se a função de correlação aos pares de Carnahan-Starling, e as correlações semi-empíircas de Speedy e Harris, correlações estas desenvolvidas através da simulação molecular de difsividade de fluidos de esferas duras. A proposta apresentada neste trabalhop baseia-se na teoria dos fluidos de esferas rígidas "suaves", sem o uso de qualquer parâmetro binário ajustável e pode ser prontamente estendidas para misturas multicompetentes. Os resultados obtidos mostram que as difisividades preditas neste trabalho estão em excelente concordância com os dados experimentais e fornecem melhores resultados que o método de Wilke-Chang.

Palavras-chave: Difusividade, teoria de perturbação, fluidos densos,

1 - INTRODUCTION

The accurate determination of transport properties, especially diffusion coefficients, has received a lot of attention due to its importance in many chemical engineering applications such as in separation processes and also because it is a challenging academic research area. The transport properties are commonly correlated using hydrodynamic, kinetic or statistical mechanical theories of fluids.

While there is a sufficiently well developed theory of transport properties for gases at low pressures, there is no corresponding development for liquids and dense fluids due to their complex microscopic structures. The hydrodynamic theory provides a simple relation between the diffusion and shear viscosity coefficients generally known as the Stokes-Einstein equation. This theory does not correlate the diffusivities well when the solute particles are no longer much larger in size compared to the solvent molecules as the Brownian assumption of the Stokes-Einstein description is inapplicable [8]. In terms of practical applications and theoretical investigations, the Enskog’s kinetic theory of hard-sphere fluids has been the most fruitful.

The Enskog’s solution of the Boltzmann equation is a simplified view of the complex correlated motions of interacting multibody that occurs in dense fluids. As a consequence, 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. In a general sense, the aim of this work is to show that the Enskog based equations work well for estimating diffusivities of dense fluids and fluid mixtures when the hard-sphere diameter is obtained from the Weeks et al. [28] (WCA) perturbation theory of liquids. Experimental self-diffusion data of benzene, toluene and carbon dioxide were used to fit the hard-sphere diameters and these diameters were employed in predicting diffusivities in binary mixtures. The predicted diffusion coefficients at infinite dilution with the present proposal and those obtained from the application of the Wilke-Chang [29] equation are compared with experimental data.

2 — THEORY

Enskog’s theory of dense hard-sphere fluids yields the following well known relation for the self-diffusion coefficient [18]:

, (1)

where r * (ºrs³) is the reduced density, g(s) is the pair correlation function at the contact point for hard-spheres, which is calculated in this work from the Carnahan and Starling [3] equation:

, (2)

and

, (3)

where, x₃ = p r s ³/6, 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 details of Enskog’s work are well described in the literature [5]. Recent computer simulation studies, performed by Speedy [24], show that the pair correlation function in Eq. (1) should be replaced by a more accurate expression to account for the density dependence of D, namely,

, (4)

where

(5)

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

Harris [10] has recently proposed the following expression for D_SHS:

(6)

where

(7)

and the numerical factor 1.01896 comes from the ninth Chapman-Enskog approximation.

Chandler has pointed out that the collisions between particles are capable of producing a variation in the angular and linear momentum [4]. To account for the coupling between translational and rotational motions, the diffusion coefficient is written as:

, (8)

where D_RHS is the diffusion coefficient for the 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 and that the coupling parameter A affects sensibly the diffusion coefficient and, therefore, Eq. (8) may not be a good basis for predictive methods [20]. Furthermore, the rough hard-sphere theory is not capable of predicting diffusivities in fluid 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.

Although great efforts have been made in evaluation of the A and s parameters for several systems, many approximate theories have been suggested in the literature [7,30].

The present approach is based on the idea of Schrodt and Davis [22], in which the requirement that the pair-potential function is a hard-sphere core cutoff, is eliminated. Their results for the excess pressure are essentially the same as those of Weeks-Chandler-Andersen [1] perturbation theory of liquids. Therefore they postulated that the properties of a real system obey the kinetic equation of a hard-sphere system of diameter d, where d can be determined by solving the following equation:

, (9)

where Y^hs(r,r ,d) is the cavity distribution function for a hard-sphere molecule of diameter d at density r , , is defined by Eq. (13) and the hard-sphere potential, , is:

(10a)

(10b)

The WCA theory predicts very well the equilibrium structure as given by the pair correlation function, especially for liquids and dense fluids. In this way, all the s in the diffusion coefficient equations are replaced by the WCA hard-sphere diameters, d. Some analytical expressions for the WCA hard-sphere diameter have been proposed in the literature [16,17,27] and we have chosen the accurate and recent proposal of Souza and Ben-Amotz [23],

(11)

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 .

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 the WCA theory the potential is divided as follows:

, (12)

, (13)

, (14)

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:

. (15)

3 — RESULTS AND DISCUSSION

In this work the diffusion coefficients of pure compressed fluids and fluid mixtures are estimated using the smooth-hard-sphere theory with the diameters calculated from the WCA perturbation theory.

In Table 1 we present the results of the fitting procedure for the self-diffusion coefficients of pure fluids using the following density corrections: Carnahan-Starling, Speedy and Harris. In order to check the prediction power of the present approach, self-diffusion data were correlated at 298.2K for benzene and toluene and at 323.2 K for carbon dioxide and then estimates were accomplished at other temperatures, as shown in Table 2.

TABLE 1.
Absolute average deviations*of the fitted self-diffusion coefficients

Substance	T(K)	Range of P [bar]	Carnahan-Starling	Speedy	Harris	Data Source
benzene	313.2	1-1000	9.29	2.46	2.83	McCool et al. [15]
toluene	323.2	1-2500	17.66	5.55	5.59	Harris et al. [11]
carbon dioxide	308.2 323.2	70-100 71-240	16.46 19.44	13.69 15.59	- -	Chen [6]

^*

TABLE 2.
Absolute average deviations of the predicted self-diffusion coefficients

Substance	T(K)	Range of P [bar]	Carnahan- Starling	Speedy	Harris	Data Source
benzene	288.20 313.20	1-285 1-975	9.44 10.66	1.98 7.77	0.96 8.26	McCool et al. [15]
toluene	273.20 323.20	1-2000 250-2500	59.29 16.66	8.27 16.73	25.19 5.69	Harris et al. [11]
carbon dioxide	308.20 348.20	70-100 68-250	15.90 11.31	12.80 8.78	- -	Chen [6]

We can observe that, in general, the correlation of Harris and Speedy are almost equivalent and produce much better results than the Carnahan-Starling, as one could expect considering that the Speedy and Harris correlations were specifically developed for understanding diffusivities. The correlation proposed by Harris cannot be applied for carbon dioxide at the conditions presented in these tables since it is not valid outside the reduced density range of 0.471 to 0.943.

It can be seen from Figures 1 to 3 a good agreement between correlated and experimental self-diffusion coefficients in the whole density ranges.

FIGURE 1.
Self-diffusion coefficient for benzene at 313.20 K. Experimetal data from [15].

FIGURE 2. Self-diffusion coefficient for toluene at 323.20 K. Experimetal data from [11].

FIGURE 3. Self-diffusion coefficient for carbon dioxide at 308.20 K. Experimetal data from [6].

The Lennard-Jones parameters of distance were estimated using the Maximum Likelihood Method [2] weighing experimental self-diffusion coefficient data according to its precision and they are shown in Table 3. 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 [19].

TABLE 3.
Estimated Lennard-Jones size and energy parameters.

Substance		s (10-10 m)* e/k (K)+
		Carnahan- Starling	Speedy	Harris
benzene	298.2 313.2	5.3307 5.2789	5.0081 4.9983	4.7858 4.7669	531.0
toluene	298.2 323.2	5.5760 5.5849	5.2890 5.2629	5.0425 5.0301	575.0
carbon dioxide	308.2 323.2	3.5990 3.5866	3.7166 3.6814	- -	195.2

^*

size parameters estimated from self-diffusion data in this work

⁺Literature values [19]

3.1 – Binary systems

For a binary mixture, the mutual diffusion coefficient is obtained straightforwardly by replacing d by d₁₂, the cross hard-sphere diameter, and m by 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 [13], at the temperature of the mixture and density of the pure substances:

(16)

The pair correlation function at the contact point g(d₁₂) for mixtures is calculated using the Carnahan-Starling equation [14]:

(17)

where

(18)

and,

(19)

For the other two density corrections, Speedy (Eq. 5) and Harris (Eq. 7) proposals, we have assumed the van der Waals mixing rule [9] so that the reduced density of the mixture, , is given by:

(20)

For given temperature and density of the pure fluids, the effective hard-sphere diameters are calculated by Eq. (11) using the Lennard-Jones parameters listed in Table 3. The pair correlation function at contact, g(d_ij), is calculated using Eq. (17). The diffusion coefficients are then directly calculated from Eqs. (1)-(3) for Carnahan-Starling correction and from Eqs. (1), (3)-(5) and (20) for Speedy’s correlation. For mixtures, s is replaced by d₁₂ in Eq. (3) and d₁₂ is calculated from Eq. (16) .

In Table 4 the predicted mutual diffusion coefficients at infinite dilution calculated using the two density correction models and those calculated by the Wilke-Chang equation are compared with the experimental data. Comparing experimental and theoretical results, it appears that, as in the case of pure compounds, the coupling factor A of the rough-hard-sphere theory is not essential for estimating diffusivities. It can be also observed that though the density correction suggested by Speedy does not improve significantly the prediction of the original Enskog's kinetic theory, the results obtained here are much better than the Wilke-Chang calculations.

TABLE 4.
Average absolute deviations of the predicted diffusion coefficients at infinite dilution species 1, D^o₁₂.

System	T(K)	Range of P(bar)	Carnahan- Starling	Speedy	Wilke- Chang	Data Source
Benzene(1) - Carbon Dioxide(2)	313.20	80-160	23.07	24.96	79.37	Chen [6]
Toluene(1) - Carbon Dioxide(2)	318.15	76-145	33.89	28.55	111.54	Tan and Lai [26]

It is worth noticing from Figures 4 and 5 that the Wilke-Chang equation overpredicts the diffusion coefficients as could be expected since the assumption of the Stokes-Einstein based equations indicate that the diffusion coefficient is inversely proportional to viscosity does not hold in supercritical fluid systems. It should also be noted from these figures that as density decreases deviations become larger, which might be explained in terms of the series expansion in the perturbation theory.

Figures 4 and 5 depict experimental and predicted values of the diffusion coefficients at infinite dilution. The agreement between experiment and theory can be considered quite good if we take into account that no binary adjustable parameters were employed for describing differences of chemical nature and the asymmetries (difference in size) present in these systems. It is important to note that very good predictions can be achieved even when the temperature of the system is out of the range where the size parameters were fitted. Actually, these extrapolations mean that the effective hard-sphere diameter is capable of accounting very well for changes in temperature and density. Thus, it is not necessary to attribute any reduction in diffusion to inelastic effects such as proposed by Chandler [4] in his rough-hard-sphere theory when the hard-sphere diameter is adequately estimated. Since the WCA theory overestimates the hard-sphere diameter [12, 21] we can observe from Eqs. (1) to (3) that it would imply an increase in the pair correlation function and hence a decrease in the diffusion coefficient, making this treatment appropriate for interpreting diffusion, as pointed out by Speedy et al. [25].

FIGURE 4.
Comparison between experimental [6] and calculated infinite dilution diffusion coefficient D^o₁₂ for the system benzene(1)-carbon dioxide(2) at 313.20 K.

FIGURE 5. Comparison between experimental [26] and calculated infinite dilution diffusion coefficient D^o₁₂ for the system toluene(1)-carbon dioxide(2) at 318.20 K.

4 — CONCLUSIONS

We have predicted self and binary diffusion coefficients using the combined kinetic and liquid theories based on the ideas of Schrodt and Davis. An evaluation of density corrections making use of the smooth-hard-sphere theory with temperature and density dependent hard-sphere diameter, obtained from the WCA theory of liquids, was performed. Mutual diffusion coefficients were also predicted using only pure component information. The proposed method correlates the pure component diffusivities well and predicts the binary diffusion coefficients at infinite dilution in good agreement with experimental values. The method applied here needs, 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.

5 — NOTATION

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. (3), m²/s D₁₂⁰ mutual diffusion coefficient at infinite dilution 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 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 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/m³ r^* reduced density * reduced density of the mixture, eq. (19) s hard-sphere diameter, m density function , eq. (18)

Subscripts

RHS rough hard-sphere SHS smooth hard-sphere

Superscripts

HR Harris SP Speedy

6 — REFERENCES

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[26] TAN, C.S. and LAI, C.C., 1995, Measurements of Molecular Diffusion Coefficients in Supercritical Carbon Dioxide Using a Coated Capillary Column. Ind. Eng. Chem. Res., 34, 674-680

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[29] WILKE, C.K. and CHANG, P., 1955, AIChE J., 1, 264.

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^*

To whom correspondence should be addressed.

¹ Received for publication in 5/8/97. Accepted for publication in 9/12/97.

²

Programa de Engenharia Química/COPPE/UFRJ, Cidade Universitária, Cx. Postal: 68502, CEP: 21945-97 - RJ

Publication Dates

• Publication in this collection
  17 Dec 1998
• Date of issue
  Dec 1997

History

• Accepted
  09 Dec 1997
• Received
  05 Aug 1997