(591e) Transport Properties for Lennard-Jones Particles | AIChE

(591e) Transport Properties for Lennard-Jones Particles

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University of Michigan
University of Michigan

Classical kinetic theory1-6 provides a reliable method to obtain transport properties of multicomponent dilute gases that saves time and cost compared to experiments. In the theory, transport coefficients, including thermal conductivity, viscosity, and diffusivity, are given as algebraic functions of the collision integral Ωij(l,s) arising from Chapman and Enskog's solution of the Boltzmann gas equation. For example, binary diffusion coefficients can be evaluated by Dij = 3kT/(16nμijΩij(1,1)). The particles are modeled as a group of colliding pairs, which obey the classical momentum and energy conservation laws with the following parameters: (1) the separation distance r, which is initially infinity; (2) the distance of closest approach rm; (3) the initial relative speed g, which follows a Maxwell-Boltzmann distribution; (4) the Lennard-Jones (6-12) potential3 φ(r), which embeds the characteristics of gas molecules; and (5) the impact parameter b, which is rm when φ(r) is zero.

 Fig. 1. Trajectory map for binary Lennard-Jones interactions. I: collision with inflection; II: collision without inflection; III: deflection without inflection; IV: deflection with inflection; V:  loop with inflection; and VI: loop without inflection.

For the evaluation of transport properties, the following three factors should be calculated in consecutive order: (1) the deflection angle χ(b,g); (2) the reduced collision cross section Q(l)(g); and (3) the collision integral Ωij(l,s). First, the deflection angle is given as χ(b,g) = π - 2θ(rm), where

.               (1)

Particle trajectories during a collision event can be drawn with Eq. 1. Integration of χ(b,g) is difficult, however, because the integrand includes singularities. These can be removed analytically, allowing calculation of the deflection angle χ to high accuracy.

The trajectory map shown in Fig. 1 arises from a contour plot of the deflection angle χ(b,g) in terms of dimensionless b and g values. Four lines, demarcating deflections (χ = 0), loops (χ = π), orbits  (χ = -∞), and inflections, divide the map into six regions. Representative trajectories are also included in each region, showing how Fig. 1 helps to understand the topology of binary Lennard-Jones interactions.

The trajectory map informs the numerical procedures used when integrating Eq. 1. By comparing the grid independent χ value to the recent previously reported value6, we could see that our computer program is working properly as well as there are critical regions on the map. In regions IV, V and VI, to estimate χ with the same accuracy as other regions, the number of mesh points should be increased to guarantee the accuracy of the collision integral Ωij(l,s).

The ongoing process is to evaluate the reduced cross section Q(l)(g), which is necessary to obtain Ωij(l,s), and ultimately the transport coefficients. The reliability will be examined by comparing the numerical results with previously reported experimental values. Once the code is successfully created, a variety of transport coefficients can be evaluated using simple algebraic relations.

References

[1] S. Chapman and T. Cowling, 2nd Ed. Cambridge, 1960.

[2] J. Hirschfelder, R. Bird, E. Spotz, J. Chem. Phys. 16 (1948) 968-981.

[3] J. Hirschfelder, C. Curtiss, R. Bird, Wiley, New York, 1954.

[4] E. Akhmatskaya and L. Pozhar, USSR Comput. Maths. Math. Phys. 26(2) (1986) 185-190.

[5] U. Storck, ZAMM 78 8 (1998) 555-563.

[6] F. Sharipov and G. Bertoldo, J. Comp. Phys. 228 (2009) 3345-3357.

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