Mohammad Mehdi Papari*, Jalil Moghadasi, Soodabeh Nikmanesh and Mahmood Reza Dehghan

1 INTRODUCTION

Many attempts have been made to obtain the forces between molecules, as they are important in determining the physical and chemical properties of matters. The results of kinetic and statistical-mechanical theories provide theoretical expression for various equilibrium and non-equilibrium properties in terms of the potential energy of interaction between molecules [1, 2]. Therefore, the evaluation of such quantities from a known pair-potential energy function is not especially difficult. For these reasons, one of the central objectives of chemical physics of gases has been and remains the expression of bulk thermophysical properties in terms of molecular quantities. In such a description, chemical physicist seeks to relate characteristics of the bulk gas, such as viscosity, to the properties of individual molecules that makeup the gas and the intermolecular potential between them.

A very precise extended principle of corresponding states has been formulated for the noble gases and eleven polyatomic gases at low density [3, 4]. It has been proved that it is capable of correlating equilibrium and transport properties of noble gases, eleven polyatomic gases and their multi-component mixture,over a wide temperature range, with an accuracy commensurate to the best measurements [5]. The principle seeks the maximum use of theory and experiment that can be obtained without assumptions about the functional form of the pair interaction potential.

Experimental information on the pair interaction potential can be extracted from a study of any process that involves collisions between molecules. Of particular value is the development of an inversion technique for bulk properties, which are readily available for a wide variety of substances. The inversion scheme is an important method for generating the intermolecular potentials from the bulk properties and their corresponding state correlation. Among the bulk properties, transport properties and especially viscosity are among the important sources for the extraction of information about the intermolecular potential energy.

One can ask two questions about any set of measurements of a bulk property:

(1) What specific information about pair-interaction does it contain?

(2) How can the information be extracted directly?

The inversion technique yields reliable answers to the aforesaid questions [6-20]. The first inversion of viscosity data was due to Dymond [21], who used a method based on the one devised by Hirschfelder and Eliason [22] for calculating approximate transport properties economically.

The direct inversion method should serve two purposes: (1) to illustrate and test theoretical principles of kinetic theory of gases; and (2) to reproduce the reduced collision integrals and their dimensionless ratios, which are necessary and sufficient information to obtain transport properties of gases.

The purpose of this paper is to generate effective pair potential energies using an iterative inversion method for CF4-CO2, CF4-N2, CF4-O2, CF4-CO, CF4-SF6, and CF4-CH4systems from corresponding state correlation of viscosity. The calculated potential energies have been employed to predict low density transport properties of the aforementioned systems. Employing Chapman-Enskog solution of the Boltzman equation[1], we compute the viscosities, diffusion coefficients and thermal diffusivities. Also the thermal conductivities are obtained through Vesovic’s method [23-26].

2 TRANSPORT COEFFICIENTS

The Chapman-Enskog solution of the Boltzmann transport equation [1] supplies expressions for the transport properties of pure gases and their multi-component mixtures at low densities in terms of collision integrals represented by(,)lsΩ . Here, the superscripts l and s appearing in Ω denote weighting factors that account for the transport mechanism by molecular collision. The collision integrals are related to the intermolecular forces by the following relations

where θ is the scattering angle, Q(l)(E) is the transport collision integral, b is the impact parameter, E is the relative kinetic energy of colliding partners, w is the relative velocity of colliding molecules, rmis the closest approach of two molecules, and kBT is the molecular thermal energy. Hence, the potential u(r)would serve as the input information required in calculating collision integrals and, consequently, the transport properties.

Fundamentally, transport coefficients describe the process of relaxation to equilibrium from a state perturbed by application of temperature, pressure, density,and velocity or composition gradients. Its importance revolves around the fact that the knowledge of transport properties of materials is crucial in an immense variety of engineering design calculations.

According to the kinetic theory of gases, the viscosity (η) and diffusion coefficient (D) of single substances in term of collision integrals are

The collision diameter σ is defined as the separation distance when the intermolecular potential function is equal to zero, ρ is the number density and m is the molecular mass. In the above equations the reduced collision integral Ω*(l,s)and the collision integral ratios A*, B*, C*, E*and F*may be defined as

For the viscosity of mixtures

Diffusion in multi-component mixtures is entirely described in terms of the binary diffusion coefficients Dij

where p is the pressure and Δijis a higher order correction term of the binary diffusion coefficient, which can be defined as

The expression for the thermal diffusivity of a binary mixture is

where kTis a higher order correction term for the thermal diffusivity. This term is usually negligible compared with experimental uncertainties in αT. The other quantities in Eq. (21) are

Interchanging subscripts 1 and 2 in the expressions of S1and Q1, we have S2and Q2. The sign convention for αTrequires that subscript 1 denotes the heavier component. In the basic development of the Chapman-Enskog theory, only binary elastic collisions between the molecules are considered and molecules are supposed to be without internal degrees of freedom. Since the internal degrees of freedom of polyatomic molecules involve transporting energy in gases,this theory can not be applied to calculate the thermal conductivity.

The thermal conductivity of a multi-component polyatomic gas mixture at zero density can be expressed in the form analogous to that for a mixture consisting of monatomic species namely [26]:

where xiis the mole fraction of species i and the symbol λ∞indicates the full formal first-order kinetic theory result obtained by means of expansion in Thijsse basis vectors [27]. The resulting expressions for the elements of the determinants, Lij, were first derived by Ross et al. [24] and with complicated functions of the effective cross-sections, so they had little value for practical evaluation of thermal conductivity.It was shown for pure polyatomic gases [23],atom-diatom mixtures [24] and atom-molecule mixtures [25] that accurate and relatively simple expressions can be obtained by means of the Thijsse approximation [27], which identifies the total energy as the dominant factor in determining thermal conductivity.

The Thijsse approximation has been applied to polyatomic systems [27] and the resulting expressions for elements Lijare written in terms of, at least in principle, measurable quantities rather than effective cross-sections. Nevertheless, the resulting expressions still require a knowledge of too many quantities,namely diffusion and relaxation of internal energy for different species, which are not readily available or in some cases impossible to obtain by experimental means alone. Thus, it was felt that in order to provide practical means for calculating thermal conductivity a further set of approximations had to be made. Hence,all the quantities in the expressions for the elements Lijwere replaced by their spherical limits [26]. This rather heuristic approximation was based entirely on the results obtained for an atom-molecule mixture where expressions for thermal conductivity based on the analogous spherical approximation were shown to predict experimental values to within a few percent[26]. Following the application of Thijsse and spherical approximations to the full results, the relevant determinant elements Lijare given by

where λqis the thermal conductivity of pure molecular spec0ies q, λqq′is the interaction thermal conductivity, cpqis the ideal-gas isobaric heat capacity of q,R is the gas constant, and the quantities A*and B*are ratios of effective cross-sections given by Eqs. (9) and(10), respectively. In addition, yqis the mass ratio of species q, given by

where Mqthe relative molecular weight of species q.The interaction thermal conductivity can be related to the more readily available viscosity,qqη′through the following expression

Evaluation of the thermal conductivity of a multicomponent polyatomic gas mixture requires the knowledge of the thermal conductivity and the isobaric heat capacity of each of the pure species. This information is readily available for a large number of fluids as a function of temperature either in terms of correlations or directly from experimental information.Furthermore, three binary interaction parameters, namely,, are also required as a function of temperature. They can be calculated from the foregoing equations. It should be mentioned that because the viscosity and diffusion coefficients are concerned with transporting momentum and mass, respectively, and therefore do not involve an internal degree of freedom,the Chapman-Enskog theory retains in its useful form,but the collision integrals must be averaged over all possible relative orientations occurring in collisions[28]. Assuming that all relative orientations have equal probability, Monchick and Mason [28] have proposed a simplification of this treatment.

3 DETERMINATION OF PAIR POTENTIAL ENERGY BY INVERTING THE VISCOSITY DATA

The degree of success for extraction of information about the force from analyzing the bulk properties depends on the accuracy of both the measurements and theory connecting the force to macroscopic properties, and on the sensitivity of this connection. The inversion procedure is of considerable importance to obtain nonparametric interaction potential energy and transport properties. This scheme relives us of the variation of the selected multi parameter analytic equation parameters for the pair potential function so as to optimize the fit to a wide range of thermophysical data of a material.

For molecules that interact with an inverse power potential, we can write

where Cmis a constant having both positive and negative values, and r is inter-nuclear distance. It has been shown that for molecules that interact with an inverse power law there is a relation between temperature and r as below [29]For realistic potentials it is found that G varies with temperature in a complicated way, since the collisions have different energies and probe different parts of the potential function (which, in terms of the model, have different effective values of m). It is also found that the variation of G with temperature is very similar for all realistic potential functions [29] and we may write

, where G0(T) is calculated by using an approximate potential function0()u r such as a LJ(l2-6) potential.

The inversion technique is initiated by estimating G, an inversion function, from an initial model potential such as the LJ(l2-6). The inversion function is a function of the reduced temperature (T*) alone. We have estimated this function using the LJ(12-6) model as the initial model. Given a set of reduced viscosity coefficient collision integrals,(2,2)*Ω , over a wide range of reduced temperature from the extended law of corresponding states [5] on the one hand, and estimating the G function from initial model potential LJ(12-6) on the other, it is possible to transform a pair of data () to u/ε versus r/σ on the potential energy curve using Eqs. (32) and (35). The details of the inversion procedure, which has been applied on the extended principle of corresponding sates [5], are given as a flow chart in Fig. 1.

4 RESULTS AND DISCUSSION

In this study, an iterative inversion procedure is used to infer the intermolecular pair interaction potential energies of aforementioned mixtures from corresponding state correlation for viscosity. Then, using the inverted pair potential energies along with Chapman-Enskog [1] version of the kinetic theory of gases together with the method proposed by Vesovic et al.[23-26], transport properties of studied gas mixtures with acceptable accuracies are computed. To perform the full inversion procedure, the experimental data should be extended over as wide a temperature range as possible. In this respect, a corresponding state correlation for viscosity collision integral is taken from Ref. [5] to calculate the reduced viscosity collision integral. As already mentioned in previous section, for each mixture, a two-iterative inversion procedure is applied to the calculated reduced viscosity collision integrals to generate isotropic and effective pair potential energies of respective systems. The inversion of viscosity collision integrals, to yield potential energy, requires experimental data over a wide range of temperature. Consequently, to integrate Eqs.(1)-(3) over the given range,u(r) should be extrapolated in the long-range region (low temperature). The long-range part ofu(r) has the following form

Figure 1 Flow chart of the iteration steps in the inversion method

Table 1 Least-squares coefficients, correlation coefficients (R2), and standard errors (Es) for Eq. (37)

Table 2 Least-squares coefficients, correlation coefficients (R2), and standard errors (Es) for Eq. (38)

The effects ofC8andC10on the transport properties are so small that we ignore them in our calculations.The value ofC6is estimated from the low-temperature viscosity data using Eq. (36). Thus the effective potential energies obtained from the inversion method are used to perform the integration over the whole range and, in turn, to evaluate the improved collision integrals over the given range. The calculated collision integrals are correlated with the following polynomial equations

Parameters in the above equations, correlation coefficients,R2, and standard errors,Es, are listed in Tables 1 and 2.

The measurements of the viscosity are more practical and accurate than the measurements of the other transport properties, so the respective collision integrals are expected to be more reliable than the others. In this respect, we use the accurate viscosity to predict other transport properties. The required values of scaling parametersσandεare taken from Ref. [5].

The expressions provided by the Chapman-Enskog version of the kinetic theory together with the calculated collision integrals obtained from the inverted potential energies are employed to calculate viscosities, diffusion coefficients and thermal diffusivities of the aforementioned mixtures.

Figures 2 and 3 demonstrate the deviations of the calculated viscosity values of afore-cited mixtures from those reported in Refs. [30] and [31] at different temperatures and mole fractions. The calculated viscosities agree with experimental values within 1%.Unfortunately, lacking of experimental data, we can not compare the obtained viscosity of CF4-CO mixture with the measured ones.

Figure 2 Deviations of calculated viscosity values of gaseous CF4 mixtures gaseous from those reported in Ref. [30]at different temperatures and mole fractions

In addition, the comparison of the calculated viscosities with those calculated from Davidson’s [32]and Reichenberg’s methods [33] are shown in Fig. 4.The errors of the calculated viscosities coefficients are at most ±1.7% in comparison with those estimated using Davidson’s method [32] and within ±1% when is compared with the ones computed from Rechenberg’s method [33].

Figure 3 Deviations of calculated viscosity valuesof CF4-CH4 and CF4-SF6 gaseous systems from those reported in Refs. [31] and [31], respectively, at different temperatures and mole fractions ◆ 0; ■ 0.2; ▲ 0.4; ● 0.6; ◇ 0.8; □ 16 △ 0.1765; ○ 0.2608; + 0.3499; ☆ 0.398;0.5741;0.5887;0.7487; × 0.7552

Figure 4 The comparison of the calculated viscosities of carbon tetrafluoride mixtures with Refs. [32], [33]compared with Ref. [32]: ◆ +CO2; ■ +N2; ▲ +O2; ● +CO;◇ + SF6; □ + CH4compared with Ref. [33]: △ +CO2; ○ +N2; + +O2; ☆ +CO;+SF6;+CH4

Eventually, we have correlated the calculated interaction viscosities of our mixtures with the following function

where01η= μPa·s and01T= K. Parameters in the above equation are allowed to vary for all the systems using non-linear least squares method and listed in Table 3. The correlation coefficients,R2, and standard errors,Es, for each case are also included.

Figure 5 shows how the calculated diffusioncoefficients of aforesaid systems deviate from those given in [30, 31, 34]. The accuracy of this property is of the order of 4%.

Table 3 Least squares coefficients, correlation coefficients (R2), and standard errors (Es) for Eq. (39)

Figure 5 Deviations for diffusion coefficients at different temperatures for carbon tetrafluoride with Refs. [30], [31], [34]compared with Ref. [30]: ◆ + CO2; ■ + N2; ▲ + O2;◇ + SF6;□ + CH4compared with Ref. [31]: ○ + CH4compared with Ref. [34]: △ + CH4

Also the values of diffusion coefficients are correlated with the following equation

wherep0= 0 .1 MPa, andD0= 1 cm2·2s-1. ConstantsaD,bD,cD, correlation coefficients,R, and standard errors,Es, are shown in Table 4.

Thermal diffusivities for all systems are calculated and fitted into the following equation

The related constants, correlation coefficients and standard errors are reported in Table 5. Unfortunately,lacking of literature data for thermal diffusivity, we can not evaluate the accuracies of our work.

In the case of thermal conductivity, the predicted viscosities obtainedviathe inverted pair potential energies are employed to predict thermal conductivities using Eqs. (25)-(29). The calculated interaction thermal conductivities are correlated with the following polynomial

The parameters of Eq. (42) are listed in Table 6.

Typically, the calculated thermal conductivity ofCF4-CH4mixture is compared with those given in Ref.[35] (Fig. 6). The maximum deviations are within ±5%.

Table 4 Least-squares coefficients, correlation coefficients (R2), and standard errors (Es) for Eq. (40)

Table 5 Least squares coefficients, correlation coefficients and standard errors for Eq. (41)

Table 6 Least squares coefficients, correlation coefficients and standard errors for Eq. (42)

Figure 6 Deviations for the thermal conductivity of CF4-CH4 mixture at temperature 303 K and different mole fractions compared with those given in Ref. [35]

5 CONCLUSIONS

The most important benefit of the present work is that knowing the intermolecular forces from inversion of corresponding states of viscosity, we are able to calculate other useful property of the gas, at any temperature and thereby relieve ourselves of the need to measure it. The inversion method is advancement over the traditional approaches that consider a potential function with several parameters and try to adjust them using experimental results.

The reasonable agreement between the calculated transport properties and those given in literature demonstrates the ability of the inversion scheme.

ACKNOWLEDGEMENTS

The authors express to Research Committees of Shiraz University of Technology and Shiraz University,their sincere thanks due to supporting this project and making computer facilities available

NOMENCLATURE

A*ratio of collision integrals

aD,aα,aη,aλ,a1,a2constant

B*ratio of collision integrals

bimpact factor, m

bD,bα,bη,bλ,b1,b2constant

C6induced dipole-induced dipole dispersion coeffi

cient, J·m6

C8induced quadrupole-induced dipole dispersion coef

ficient, J·m8

C10induced quadrupole-induced quadrupole dispersion

coefficient, J·m10

C*ratio of collision integralscD,cα,cη,c1,c2constant

ideal-gas isobaric heat capacity ofq

Dijbinary diffusion coefficient, m2·s-1

dα,dη,dλ,d1,d2constant

Esstandard error

E*ratio of collision integrals

eαconstant

F*ratio of collision integrals

fDhigher order correction factor for diffusion

fαconstant

fηhigher order correction factor for viscosity

Ginversion function

hPlank’s constant, J·s

kBBoltzman constant, J·K-1

kThigher order correction term for thermal diffusivity

mmolecular mass, kg

ppressure, Pa

Q(l)transport cross-section, m2

Rgas constant, J·mol-1·K-1

R2correlation coefficient

rintermolecular distance, m

rmclosest approach of two molecule, m

Ttemperature, K

T*reduced temperature

u(r) intermolecular potential energy, J

wrelative velocity of colliding molecules

xmole fraction

Δ12higher order correction term for diffusion coefficient

εenergy-scaling factor, J

ηijinteraction viscosity, Pa·s

ηmixmixture viscosity, Pa·s

θscattering angle, rad

interaction thermal conductivity

λ∞mixture thermal conductivity, W·m-1·K-1

μreduced mass

vnumber of components in the mixture

σlength-scaling factor, m

Ω(l,s)collision integral, m2

Ω*(l,s)reduced collision integral

Superscripts

l,sweighting factors related to the mechanism of transport by molecular collisions

* reduced

Subscripts

Ddiffusion coefficient

αthermal diffusivity

ηviscosity

λthermal conductivity

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