Multicomponent diffusion in gases and plasma mixtures

High Temperatures ^ High Pressures, 2002, volume 34, pages 109 ^ 116 15 ECTP Proceedings pages 1337 ^ 1344 DOI:10.1068/htwu73 Multicomponent diffusion in gases and plasma mixtures Irina A Sokolova Institute of Mathematical Modelling, Russian Academy of Sciences, Department of Physical Chemistry, Properties of Matter, Miusskaya pl. 4a, Moscow 125047, Russia; fax: +7 095 972 0723; email: sokolova@imamod.ru Presented at the 15th European Conference on Thermophysical Properties, Wu«rzburg, Germany, 5 ^ 9 September 1999 Abstract. Molecular diffusion fluxes derived from kinetic theory are investigated. New expressions for molecular flux given by Stefan ^ Maxwell relationships derived for any level of Sonine's polynomial expansion are presented with corresponding coefficients, and analysed. The accuracy of the coefficients is investigated. New simplified expressions for multicomponent diffusion coefficients are introduced, and compared with the predictions of kinetic theory for high orders of Sonine's polynomial expansions. 1 Introduction Molecular diffusion processes of chemically reacting gases are important for many applied problems. Most of the plasma processes of industrial relevance use gases of different mixtures: argon and carbon dioxide or argon and oxygen in gas-plasma arc welding, very complicated mixtures in plasma metallurgy and plasma waste destruction, and so on. Nowadays transport properties of multicomponent high-temperature gases and plasmas are not widely available from experimental investigations. Diffusion coefficients used to be derived for binary mixtures near room temperature. Experiments at high temperature do not meet practical requirements. On the other hand, theoretical description by kinetic theory approaches of diffusion processes may be successful in the case of thermal ideal plasmas. The Boltzmann equation is generalised for plasma mixtures on the basis of the Debye ^ Hu«ckel interaction potential for charged particles. It makes it possible to develop the Chapman ^ Enskog method and Sonine's polynomial expansion to deduce gas-dynamic equations of the Navier ^ Stokes (NS) system and expressions for molecular fluxes (Ferziger and Kaper 1972), which complete the NS system. The electrical fields of charged particles are taken into account. The equations for molecular flux are deduced with explicit expressions for all the transport coefficients, which may be calculated accurately (Kolesnikov et al 1985; Sokolova 1997). Nevertheless the Chapman ^ Enskog method and Sonine's polynomial expansion lead to complex expressions for diffusion coefficients. The ordinary Chapman ^ Enskog procedure gives the multicomponent diffusion coefficients as the ratios of two determinants of matrices. Their orders (number of elements in rows and in columns) depend on the number of species in the mixture, N, and the number of polynomials in Sonine's expansion, x, and are given by (Nx 1) and (Nx), respectively. In the case of ionised mixtures, x must be at least 2 or 3, and the matrix orders are too large to include computer codes for diffusion coefficients into program packages for numerical solution of magneto-gas-dynamic equations. Simplified models for diffusion description are being developed (Sokolova 1993; Sokolova and Tirskii 1996; Murphy 1996; Desmeuzes et al 1997; etc). For mixtures of neutral gases the models are compared and estimated for use in practical calculations. The usual main simplification is based on the combination of the binary diffusion coefficients provided that pair diffusion properties are nearly identical. As is evident now,

110 IASokolova 15 ECTP Proceedings page 1338 the simplified expressions of constant Lewis number, Blottner formula, binary diffusion approach, and so on, are not able to represent with good accuracy the diffusion phenomena, especially the mole fraction of species distributions. In addition, the models for diffusion coefficients of neutral particles are not accurate for ionised gases, because the conformation to the second (or third) level of Sonine's polynomial expansion is required. Here, molecular diffusion fluxes derived from kinetic theory are investigated. New expressions for molecular flux in the form of Stefan ^ Maxwell relationships derived for any level of Sonine's polynomial expansion, with corresponding coefficients, are presented and analysed. The accuracy of the coefficients in Stefan ^ Maxwell relationships in x approaches is investigated. The new simplified expressions for multicomponent diffusion coefficients are obtained via binary diffusion coefficients. They are compared with the results of kinetic theory predictions for high orders of Sonine's polynomial expansions. 2 Mass diffusion equation The most accurate way to describe diffusion processes in multicomponent reacting mixtures is to treat molecular species separately, and solve the continuity equation for each mass, i: dr i dt r ih.v H.J i ˆ w i, i ˆ 1,..., N. (1) Here J i ˆ r i V i, the mass diffusion flux; V i is the mass diffusion velocity, r i is density; w i is the net rate of formation of species i through chemical reactions; v is the massaverage velocity of the gas mixture. The usual kinetic theory and the Chapman ^ Enskog method gives the expression for the diffusion flux as an explicit one via the thermodynamic diffusion force, d i, as follows (Devoto 1966): J i ˆ nm i j 6ˆ i m j m D ijd j D Ti H ln T, i ˆ 1,..., N, (2) d i ˆ Hx i x i c i H ln p c i rf p i XN r j F j, (3) where D ij is the multicomponent diffusion coefficient, and D Ti is the thermodiffusion coefficient due to Hirschfelder's formalism; T is temperature, p is pressure, F i is the force reduced to mass m i, x i ˆ n i =n, where x i is the molar fraction, n i is the number density; c i ˆ m i x i =m, where c i is the mass fraction, and m i is the mass of i th species. In the Sonine's x polynomial approximation the diffusion coefficient is as follows: q 00, q 01, q 0, x 1 d hj d hi D ij Š x ˆ 3rn 1=2 i 2pkT 1 q 10, q 11, q 1, x 1 0 2nm j m i det q 0, j 6ˆ i, (4) q x 1 0 q x 1 1 x 1 x 1,,, q d ki 0 0 0 where k is the Boltzmann constant, and q mp, is the matrix of elements q mp, ij ; h, k ˆ 1,..., N; i,,..., N, 0 1 q mp, q mp 11 q mp,, 1N ˆ @ A, q mp, N1 q mp, NN det (q) is the determinant of the matrix of the numerator without the last column and last row.

Multicomponent diffusion in gases and plasma mixtures 111 15 ECTP Proceedings page 1339 The convergence over the x number in the diffusion coefficients has been investigated for various mixtures. For neutral molecules and atoms with similar masses of species, the errors of the first approach, x ˆ 1, are bounded by 1% as compared with higher orders of x. For species mixtures of strongly differing masses the second approach, x ˆ 2, improves the values to about 2% ^ 5% (Sokolova 1993). Thus, even if a neutral mixture is treated, matrices of order N N are needed. The problem is the worst in the case of partially ionised gases. At least the second approach, x ˆ 2, to the diffusion coefficients is required. The x ˆ 2 approximation improves the values of the diffusion coefficients up to 30% as compared to x ˆ 1. The corresponding matrices in the ratio of the diffusion coefficients are 2N (2N 1) in the numerator and 2N 2N in the denominator. 3 Stefan ^ Maxwell relationships In applied gas-dynamics calculation, a useful way to determine the diffusion fluxes, J i, is by the Stefan ^ Maxwell relationships. The explicit expression of the thermodiffusion forces via diffusion flux makes it possible to solve the direct Cauchy task for the given boundary conditions. In addition (N 1) equations are needed instead of the N diffusion equations in equations (2). 3.1 Neutral gases In the case of a neutral gas mixture, the resolution of diffusion flux equations (2) via thermodynamic diffusion forces, d i, equation (3), with the first approach (x ˆ 1) to many-component diffusion coefficients and the second approach (x ˆ 2) to thermodiffusion coefficients is as follows: d i ˆ XN i ˆ 1 i 6ˆ 1 n i n j n 2 D ij Š 1 V j V i XN i ˆ 1 i 6ˆ 1 n i n j n 2 D ij Š 1 DTi D T i H ln T, i ˆ 1,..., N 1, (5) n i m i n j m j where [D ij ] 1 are the binary diffusion coefficients in the first approach (x ˆ 1), D 12 Š 1 ˆ 3 1=2 2pk m1 m 2 T 1, 16n m 1 m 2 Q 11, 12 Q ls, ei are the reduced collision integrals, and D Ti are the thermodiffusion coefficients. In the second (x ˆ 2) approach, D Ti are the determinant ratios of matrices of order 2N (2N 1) and 2N 2N. 3.2 Ionised gases The exact resolutions of the Stefan ^ Maxwell equations for any Sonine's polynomial expansion mixtures have been published (Kolesnikov et al 1985). They are, are follows: d i ˆ D i J 0 i x i d i ˆ H ni n ni n r i r D ij J 0 j k Ti H ln T, H ln p r i ei XN p m i n j e j E, (6) r J 0 i ˆ J=m i is the reduced diffusion flux, D ij are the inverted binary diffusion coefficients with the corrections, j ij, due to Sonine's x-approximation, D i j ij, D n D ij Š i ˆ XN x j D ij, i ˆ 1,..., N 1, 1

112 IASokolova 15 ECTP Proceedings page 1340 and j ij Š x ˆ 2 0 q 01, js K q 0, x 1 js D ij Š 1 1 q 10, 3 x i x j det q ri q 11, rs K q 1, x 1 rs q x 1, 0,, rs K qrs ri q x 1 1 x 1 x 1, x52. (7) Here k Ti are the coefficients of the thermodiffusion relationship. They are direct ratios of determinants, contrary to the common expression (Ferziger and Kaper 1972) used to derive k Ti through the many-component diffusion coefficients, D ij, and the thermodiffusion coefficient, D Ti, as follows: D ij k Ti ˆ D Ti, i ˆ 1,..., N. We have k Ti Š x ˆ 5 0 x s 0 K 0 1 q 10, ri q 11, rs q 11, rs K q 1, x 1 rs 2 det q q x 1, 0, rs q x 1, 2, rs K qrs ri q x 1 1 x 1 x 1, x52. (8) For any x-approximation the matrix orders of k Ti are lower for N mixture species, N (x 1), as compared to thermodiffusion coefficients and many-component diffusion coefficients, which are of order N x. In the first approach, x ˆ 1, the corrections j ij and coefficients k Ti are zero, and expression (6) becomes identical to expression (5). 4 Accuracy of the n approaches As is known, high orders of x-approximations are needed for ionised mixtures. For partially ionised plasma of the mixture CO 2 Ar (0:5 0:5 of molar fractions) at chemical equilibrium, corrections j ij are shown in figures 1 and 2 for pairs of species including neutral particles and heavy ionised ones, respectively. As one can see the corrections are small and do not exceed 5% in the first case, and are 4% for the 0:000 j ij 0.04 O ^Ar Corrections 0:004 0:008 0:012 O^C Ar ^ C Ar ^ O Corrections 0.02 0.00 O ^O 0:016 0 1 2 3 4 0:02 1 2 3 4 Figure 1. Corrections to binary diffusion coefficients in Stefan ^ Maxwell relationships in an ionised mixture of CO 2 Ar at pressure p ˆ 10 5 Pa at chemical equilibrium. Figure 2. Corrections to binary diffusion coefficients in Stefan ^ Maxwell relationships in an ionised mixture of CO 2 Ar at pressure p ˆ 10 5 Pa at chemical equilibrium. Calculations: x ˆ 2 ( ) is the second approach, x ˆ 3 ( ) is the third approach, x ˆ 4 (ö)is the fourth approach by Sonine's polynomials.

Multicomponent diffusion in gases and plasma mixtures 113 15 ECTP Proceedings page 1341 O ^ Ar pair. Therefore, the corrections to binary coefficients in the Stefan ^ Maxwell relationships for pairs of heavy species (neutral or ionised) may be neglected, on the whole. For pairs including electrons (subscript E), the corrections j Ei are shown in figures 3 and 4 for the same mixture, CO 2 Ar. The corrections j Ek for pairs of electrons and neutral particles reach 20%. Corrections j Ei for pairs of electrons and ions reach 40%. 0:40 j ij 0:20 e^c 0:20 e^ar 0:10 e^o Corrections 0:00 0:20 0:40 1 2 3 Figure 3. Corrections to binary diffusion coefficients in Stefan ^ Maxwell relationships in an ionised mixture of CO 2 Ar at pressure p ˆ 10 5 Pa at chemical equilibrium. Calculations: as in figure 2. The errors of x-approximations (x ˆ 2) and (x ˆ 3) to the corrections j ij (x) for pairs of various species in the mixture CO 2 Ar (0:5 0:5) at chemical equilibrium are shown in figure 5. The errors of the second approach (x ˆ 2) with respect to approach (x ˆ 4) can reach 20% ^ 40% for electron ^ neutral pairs and 60% for electron ^ ion pairs at low ionisation. At high ionisation the difference does not exceed 20%. The approach (x ˆ 3) on the whole differs from approach (x ˆ 4) by some percent except when the ionisation is low. However, we must say that convergence of the x approaches is poor in the narrow range of temperature at low ionisation, when the correction values j ij are small (see figures 1 ^ 4). The errors of corrections to j ij are not important. Thus, the second approach is good enough to compute the diffusion coefficients in the Stefan ^ Maxwell relationships with accuracy of a few percent (2% ^ 3%) and when ionisation is low. 5 Generalisation of the ambipolar diffusion expression If we assume the absence of local charge everywhere in the flow and negligible electrical current of charged particles, the electrical field E of charged particles can be excluded from the diffusion equations [F i ˆ (e i E i =m i ), where e i is the electrical charge of species i]. Taking the Stefan ^ Maxwell equation (6) for the above conditions, the final expression is as follows: Hx i k a T i e^o H ln T k a H ln p ˆ D a i D i J 0 i x i p i Corrections 0:00 0:10 0:20 1 2 3 4 Figure 4. Corrections to binary diffusion coefficients in Stefan ^ Maxwell relationships in an ionised mixture of CO 2 Ar at pressure p ˆ 10 5 Pa at chemical equilibrium. Calculations: as in figure 2. jˆ1 D a ij J 0 j, i ˆ 1,..., N 2, (9) e^c

114 IASokolova 15 ECTP Proceedings page 1342 Error Error (a) 1.0 0.8 0.6 0.4 1.2 1.0 j(x)=j(4) x ˆ 4 x ˆ 3 x ˆ 2 j(x)=j(4) 1 2 3 1 2 3 (b) x ˆ 4 x ˆ 3 x ˆ 2 e^o e^ar 1.2 1.0 0.8 1.2 1.0 j(x)=j(4) x ˆ 4 j(x)=j(4) x ˆ 4 x ˆ 3 x ˆ 3 x ˆ 2 x ˆ 2 e^c e^o 0.8 0.8 (c) where D a ij 0.6 1 2 3 1 2 3 (d) Figure 5. The errors of x-approximations of the corrections j ij (x) in an ionised mixture of CO 2 Ar at pressure p ˆ 10 5 Pa at chemical equilibrium. ˆ D ij x i e i s ˆ 1 k a T i ˆ k Ti x i e i s ˆ 1 k a p i ˆ k pi x i e i s ˆ 1 x s e j e s D kj e 2, e 2 ˆ XN e s k Ts e 2, e s k ps e 2. s ˆ 1 x s e 2 s, Equation (9) is written in the form for neutral gas mixtures, and if all e i ˆ 0, the coefficients are identical. In the case of a quasi-neutral mixture of species, the independent equations (9) number (N 2), and in the case of a neutral species mixture, there are (N 1). The implementation of equation (9) instead of equation (6) in gas-dynamics calculations allow us to take into account the influence of electrical fields of particles on species diffusion, without adding the Maxwell ^ Poisson equations to the NS system. For three-species ionised mixtures (N ˆ 3), containing one neutral component only, electrons, and one type of ion, equation (9) may be regarded as a generalisation of the ambipolar diffusion equation. (The known ambipolar diffusion equations have been derived on the basis of the simplified relationship J i ˆ rd i Hx i.)

Multicomponent diffusion in gases and plasma mixtures 115 15 ECTP Proceedings page 1343 6 Simplified description of the diffusion coefficients 6.1 Stefan ^ Maxwell relationship diffusion coefficients For partially ionised mixtures of gases the coefficients in the Stefan ^ Maxwell relationship (6), D ij and k Ti may be simplified by taking into account the small mass of electrons. Neglecting the terms of order m 1=2 E =m N, the corrections to binary diffusion coefficients are simplified. First, the coefficients D ii of identical species pairs are not included in equation (6). Then, corrections to binary diffusion coefficients of neutral or heavy ionised particle pairs are not important. So, for partially ionised gases, all the binary diffusion coefficients in the Stefan ^ Maxwell relationship may be calculated without x approaches (in other words, assuming j ij ˆ 0), with the exception of the electron ^ heavy particle pairs. In order to meet the 2% ^ 3% accuracy of the Stefan ^ Maxwell relationship, only the second x-approach to correction j Ei is needed. In this case the corrections are the algebraic expressions, as follows: j Ek Š 2 ˆ 1 1 1 q 01, q 10, k E EE, (10) 8 n E n k Q 11, q 11, Ek EE where Q ls, are the reduced collision integrals. 5 q 01 ˆ 8n 1 1 k E En k Q E k 3Q 12 E k, q 10 EE 2 ˆ 8 X,,,, 5 1, 1 n E n j Q E j 3Q 12, E j, 2 p n 2 q 11, ˆ 8 EE 2 EQ 22,, E j 15Q 12,, E j E j EE 8 X j 6ˆ E 25 n E n j 4 Q 1 1 j 6ˆ E 12Q 13. 6.2 Thermodiffusion relationships The values of thermodiffusion coefficients of species in the mixture CO 2 Ar (0:5 0:5) at chemical equilibrium are shown in figures 6 and 7. The thermodiffusion ratios of neutral particles are small and they may be neglected in the Stefan ^ Maxwell equations. For electrons and ions, the thermodiffusion relationships in the second x-approach are simplified and they are as follows: q 01, EE q 11, EE q 01, k TE Š 2 ˆ 5 n E, k 2 n Ti Š 2 ˆ 5 n E Ei, k 2 n q 11 Tk Š x ˆ 0, (11), EE k ˆ 1 where n E is the number density of electrons in the mixture, and n is the whole number density of species. Expressions for q ls, are given above. 0:04 EE Thermodiffusion ratio 0:02 0:00 0:02 O Ar 0:04 0 4 8 12 16 10 3 T K Figure 6. Thermodiffusion ratios for neutral species in chemical equilibrium mixture CO 2 Ar (0:5 0:5) atp ˆ 1 atm. C Thermodiffusion ratio 0:40 0:20 0:00 0:20 C O e Ar 1 2 3 4 Figure 7. Thermodiffusion ratios for ionised species in chemical equilibrium mixture CO 2 Ar (0:5 0:5) atp ˆ 1 atm.

116 IASokolova 15 ECTP Proceedings page 1344 6.3 Three-component mixtures The expressions for multicomponent diffusion coefficients [equation (4)] may be simplified through binary diffusion coefficients with corrections due to x-approximation. By comparing the two expressions (2) and (6) and the corresponding transformation, we have obtained a relationship for a three-component mixture (N ˆ 3): D ik ˆ x k D ik x i D il D lk XN x s D s 6ˆ k ks M x im i =m k x i D ik XN x s D ik D il D s 6ˆ i is, M ˆ XN i ˆ 1 x i m i, (12) where M is the mass of the mixture. The binary diffusion coefficient D il is replaced by the expression D il f il, where f il ˆ (1 j il ) 1, for pairs containing electrons in an ionised mixture. These new expressions for diffusion coefficients are not laborious for practical applications in magneto-gas-dynamic calculations, and do not yield the accuracy of the kinetic theory formulas. The generalisation of equation (12) for many-component mixtures of neutral gases demonstrates an accuracy of 1% ^ 2% as compared to kinetic theory calculation by the use of determinant ratios (Sokolova 1993). 7 Conclusion New expressions for multicomponent diffusion phenomena as Stefan ^ Maxwell relationships in any x-approximation have been derived for ionised gases. Excluding the electrical fields of ionised species from the equations made it possible to account for the influence of electrical fields of species without adding Maxwell ^ Poisson equations into the NS system, and the generalised ambipolar diffusion equation for many-component ionised mixtures in any x-approximation of Sonine's expansion. The careful analysis of various approximations and development of simplified expressions for diffusion phenomena in ionised gas mixtures give the new improved simplified formulas as simple algebraic expressions without determinant ratios. They are sufficiently accurate when compared with kinetic theory foundations and are suitable for practical calculation. Acknowledgements. The work is supported by RFBR, Project number 97-01-00005. References Desmeuzes C, Duffa G, Dubroca B J, 1997 J. Thermophys. Heat Transf. 11 36 ^ 44 Devoto R S, 1966 Phys. Fluids 9 1230 ^ 1240 Ferziger J H, Kaper H G, 1972 Mathematical Theory of Transport Processes in Gases (Amsterdam: North-Holland) Kolesnikov A F, Sokolova I A, Tyrsckii G A, 1985 Rarefied Gas Dynamics. Proceedings of the Thirteenth International Symposium on Rarefied Gas Dynamics ( New York: Plenum) volume 2, pp 1359 ^ 1366 Murphy A B, 1996 J. Phys. D, Appl. Phys. 29 1922 ^ 1932 Sokolova I A, 1993 Mathematical Model. (Rus.) 5 71 ^ 91 Sokolova I A, Tirskii G A, 1996, in Proceedings of the Eighth International Symposium on Transport Phenomena in Combustion Ed. S H Chan (Washington, DC: Taylor and Francis) pp 1013 ^ 1024 Sokolova I A, 1997, in Transport Phenomena in Thermal Science and Engineering Plenary Sessions 1 ^ 8 (Japan, Kyoto) volume 1, pp 103 ^ 108 ß 2002 a Pion publication printed in Great Britain