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Molecular Versus Turbulent Mixing

The relative importance of advection and diffusion in the distribution or mixing of a chemical species like hydrogen may be derived through non-dimensioning the general advection-diffusion equation of transport. This dominance is a function of flow velocity u, species diffusivity D and time t, and may be expressed in terms of the non-dimensional Péclet number:

 $$Pe = {D \over {u^2 t}} = {D \over {uL}}$$ (1)

where D is the diffusion coefficient, u is the velocity, t is time and L is a characteristic length scale. Diffusion is the dominant mechanism when Whereby diffusion is dominant for Pe>>1 and transport by advection transport dominates for Pe<<1. It is important to note that, whenever large times, t, or characteristictravelled lengthscale, L = ut, are considered, the advection transport would always dominate. The Péclet number expresses the ratio between the characteristic times of advection and diffusion. The length travelled by a particle is proportional to t for advection and to t1/2 for diffusion.

Characteristic length and time scales for advection and diffusion transport may be expressed by

 $$L_{advection} = ut$$ (2) $$t_{advection} = {L \over u}$$ (3) $$L_{diffusion} = \sqrt {Dt}$$ (4) $$t_{diffusion} = {{L^2 } \over D}$$ (5)

These expressions are useful as rules of thumb.

References

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