An ellipsoidal BGK model is proposed for a binary mixture of rarefied gases in the frame of kinetic theory. It fulfils the crucial properties of the actual Boltzmann equation (collision invariants, equilibria, entropy dissipation), and introduces a further constraint on velocity equalization of the two species. The model features two disposable relaxation parameters which can be used to fit exactly, in the continuum limit, Fick's law for diffusion velocities and Newton's law for the viscous stress in the relevant set of Navier-Stokes equations. Positivity of temperature fields is guaranteed by a physically meaningful restriction on the parameters themselves.

In this paper, we consider the problem of locating a target node (TN) moving along a corridor in a large industrial environment by means of ultrawide band signaling from fixed anchor nodes (ANs) uniformly positioned at the same height on both sides of the corridor. For a representative geometry of a large indoor (industrial) scenario, we formulate an analytical approach to the optimized placement (in terms of internode distance) of ANs using the criterion of minimizing the average mean square error (MSE) in the time-difference-of-arrival-based estimated positions of the TN.Under the assumption of a fixed variance of the range estimation error, we derive a simple closed-form expression for the optimal inter-AN distance in terms of the corridor width and the height of the ANs. The effectiveness of the analytical approach is confirmed by simulations. We also show that the proposed approach allows the MSE in the TN position estimates to reach the Cramer Rao lower bound.

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