Page 228 - gas transport in porous media

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2
u(x, y, z). Typically, the flow reaches steady-state at time > L /ν, where ν is the
kinematic viscosity of the carrier fluid, and L is a characteristic length scale perpen-
dicular to the flow direction (e.g., the channel width). Then the memory used for the
19-vector carrier fluid can be reclaimed, and reallocated for additional repeat units of
solid geometry (and the lattice used for the tracers) in the x direction. The velocity field
at x, y, z is then u((x MOD N x ), y, z). The lattice Boltzmann updates for the tracers
are still carried out at each step, but rather than recalculate the velocity at each point,
the saved velocity field is used. We refer to these additional repeat units as “clones.”
As mentioned, the viscosity of the carrier fluid, and the diffusion coefficients of
the tracers in the fluid, are entirely dependent on the collision parameter τ. In the 3D
model above,

1 1
τ 0 − τ s −
2 2
ν = and D ms = (13.7)
3 3
2
where D ms is the molecular diffusion coefficient of tracer s, in units of lu /ts.
In one important way, the LB method is better suited to model dispersion in gas
transport, than dispersion in liquids such as water. It is difficult to set up a dispersion
calculation wherein the τ 0 of the bulk fluid, or carrier, is vastly different than the
τ s of the tracers, either for reasons of numerical stability or convenient scaling. In
2
2
water, the ν ∼ 0.01 cm /s, but D m ∼ 10 −5 cm /s, so there is a three-order of
magnitude difference that must be spanned. But in gases, kinetic theory leads us
to expect the kinematic viscosity and diffusion coefficients (for similar molecules)
2
will be approximately the same. Thus ν for air is ∼0.14 cm /s at 18 C and 1 atm,
2
and D m for CO 2 ,O 2 and H 2 O in air are approximately ∼0.14, 0.18 and 0.24 cm /s,
respectively, at the same temperature and pressure. Hence, one is more likely to find
a suitable ν/D m for gases.

13.2.3 Boundaries
The equations and calculation method above describe the update of the free fluid. The
processingatcalculationboundariesandsolidwallsrequirespecialattention, toensure
mathematical “closure;” i.e., that the total mass of each component is conserved in the
absence of intentional sinks. The calculation domain is finite in size; thus during the
translation stage, the distributions that point out of the domain would be lost, unless
a special effort were made to preserve that mass. A simple way to solve this dilemma
is to use “wrap” boundaries, so that a vector leaving the right side of the automaton
at x max is reinjected at 0. Similarly, a vector that leaves the left side of the automaton
at x = 0, is reinjected at x max , the last site in the x-direction. This wrap is performed
automatically in the translation step, and applies to the y- and z-directions as well.
Onemustalsodecidehowtotreatsolidsduringthecollisionstep. Thebasicproblem
is this: before a collision step, the preceding translation step moves vectors from free
fluid to solid sites, and these vectors point into the solid. Yet there may be no vectors
pointing out of the wall into the fluid, since there is no fluid inside the solid before