Characteristic scales small Peclet number

In the outer region, we must first determine the appropriate characteristic length scale. To do this, we follow the example of the small Peclet number heat transfer problem of the previous section, and introduce a rescaling,... 

An important characteristic of a property distribution is encapsulated in the Peclet number, Pe = ULIk, which is the ratio of diffusive time-scale to advective timescale of the system. In this definition, U and L are the characteristic velocity and length scales of the flow. The Peclet number is a measure of the relative importance of advection versus diffusion, where a large number indicates an advectively dominated distribution, and a small number indicates a diffuse flow. Numerical modeling indicates that certain tracer distributions, in particular tracer-tracer relationships, are significantly affected by the Peclet number, and consequently can be used to determine the nature of the fluid flow (Jenkins, 1988 Musgrave, 1985, 1990). 

To obtain a valid approximate solution for heat transfer from a sphere in a uniform streaming flow at small, but nonzero, Peclet numbers, we must resort to the method of matched (or singular) asymptotic expansions.4 In this method, as we have already seen in Chap. 4, two (or more) asymptotic approximations are proposed for the temperature field at Pe 1, each valid in different portions of the domain but linked in a so-called overlap or matching region where it is required that the two approximations reduce to the same functional form. The approximate forms of (9-1), from which these matched expansions are derived, can be obtained by nondimensionalization by use of characteristic length scales that are appropriate to each subdomain. 

Diffusive flow for neutrals The importance of convective vs. diffusive flow of neutrals is determined by the Peclet number Pe = uL/D, where L is a characteristic dimension of the system. Away from inlet and exit ports, the characteristic length will be on the order of the reactor dimension. The system will be primarily diffusive when Pe 1. For CI2 gas in a reactor with L 0.1 m and a neutral species diffusivity of D 5m s at 20mtorr, the Peclet number will be Pe 1 when M = 50ms. Convective gas velocities are not likely to be that high, except for a small region near the gas inlet ports. It follows that gas flow can be approximated as diffusive this obviates the need for solving the full Navier-Stokes equations which adds to the computational burden. It should be noted that both the diffusivity and the convective velocity scale inversely with gas pressure, so the Pe number is independent of pressure. However, as the pressure is lowered to the point of free molecular flow, the gas diffusion coefficient has no meaning any more. Direct Simulation Monte Carlo (DSMC) [41, 143] can then be applied to solve for the fluid velocity profiles. 

It can be noted that in practice, the Poisson-Boltzmann equation can be used to a good effect even in the presence of thick EDLs, provided that the Peclet number based on the EDL thickness (i.e., Pckd = Uj [k/D, where ref is the characteristic velocity scale along the axial direction and D is the diffusion coefficient of the solute) is small [5]. In such cases, a closed-form analytical expression for ij/ can be obtained from Eq. 27 as... 

---