The Transition Regime

Fuchs Theory The matching of continuum and free molecule fluxes dates back to Fuchs (1964), who suggested that by matching the two fluxes at r = A + Rp, one may obtain a boundary condition on the continuum diffusion equation. This condition is, assuming unity accommodation coefficient 

the steady-state continuum transport equation for a dilute system is 

Relating the binary diffusivity and the mean free path using Dab Aabca = 3 and letting Kn = Ab/Rp, one obtains 

Note that the definition of the mean free path by Dab/AbcA = 5 implies, using (12.26), that, for a = 1 

The value of A used in the expressions above was not specified in the original theory and must be adjusted empirically or estimated by independent theory. Several choices for A have been proposed the simplest, due to Fuchs, is A = 0. Other suggestions include A = Ab and A = 2 AB/cA (Davis 1983). 

Relating the binary diffusivity and the mean free path using and letting 

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Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically statistical methods are used to quantify turbulence. 

The Intermediate Regime (Best known as the Transition Regime Law, for 2 [Pg.273]

Those particles with sizes d > d" at a given set of conditions (v, p, Pp, and a ) will settle only in the turbulent flow regime. For particles with sizes d < d, d" will settle only when the flow around the object is in the transitional regime. Recall that the transitional zone occurs in the Reynolds number range of 0.2 to 500. The sedimentation numbers corresponding to this zone are 3.6 < S, < 82,500 and 0.0022 < S2 < 1,515. 

For the same case of n = 1200 rpm and r = 0.5, we obtain u,/Ug = 800, whereas for the turbulent regime the ratio was only 28. This example demonstrates that the centrifugal process is more effective in the separation of small particles than of large ones. Note that after the radial velocity u, is determined, it is necessary to check whether the laminar condition. Re < 2, is fulfilled. For the transition regime, 2 < Re < 500, the sedimentation velocity in the gravity field is ... 

This ratio represents an average between similar ratios for the laminar and turbulent regimes. In the most general case, u, = f(D, Pp, p, /r, r, w), and hence we may ignore whether the particle displacement is laminar, turbulent or within the transition regime. This enables us to apply the dimensionless Archimedes number (recall the derivation back in Chapter 5) ... 

However, for the lubricants with lower viscosity, e.g.. Polyglycol oil 1 and 2 with the kinetic viscosity of 47 mm /s to 145 mm /s in Table 1, the transition from EHL to TFL can be seen at the speed of 8 mm/sand23 mm/s, i.e., the relationship between film thickness and speed becomes much weaker than that in EHL. The transition regime can be explained when the film reduces to several times the thickness of the molecular size, the effect of solid surface forces on the action of molecules becomes so strong that the lubricant molecules become more ordered or solid like. The thickness of such a film is related to the lubricant viscosity or molecular size. 

In order to obtain a qualitative view of how the transition regime differs from the continuum flow or the slip flow regime, it is instructive to consider a system close to thermodynamic equilibrium. In such a system, small deviations from the equilibrium state, described by thermodynamic forces X, cause thermodynamic fluxes J- which are linear functions of the (see, e.g., [15]) ... 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

Recently, des Cloizeaux has conceived a rubber -like model for the transition regime to local reptation [56]. He considered infinite chains with spatially fixed entanglement points at intermediate times. In between these... 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

Clearly, the solution of this equation at forced-convection electrodes will depend on whether the fluid flow is laminar, in the transition regime, or turbulent. Since virtually all kinetic investigations have been performed in the laminar flow region, no further mention will be made of turbulent flow. The reader interested in mass transport under turbulent flow is recommended to consult refs. 14 and 15. 

The value of T from Eq. 29 is 0.575 for y = 1.4 and doesn t vary by more than 10% for values of y from 1.1 to 1.67. At very large Kn the value of T is 0.399, the value for free molecular diffusion through an orifice [46]. In between the high and low Kn number limits the value of T assumes intermediate values [45]. Thus, it can be seen that using the low Kn limit expression in the transition regime is a conservative assumption when calculating gas flows for the purpose of sizing vacuum pumps. 

Churchill also provided a single equation that may be used for Reynolds numbers in laminar, transitional, and turbulent flow, closely fitting/= 16/Re in the laminar regime, and the Colebrook formula, Eq. (6-38), in the turbulent regime. It also gives unique, reasonable values in the transition regime, where the friction factor is uncertain. 

They observed eg to be reduced by the presence of solids, although this reduction is less pronounced in the churn-turbulent regime than in the transition regime. The relationships of Koida et al. [37] are... 

The above relationship, in particular, has been tested in the transition regime (see Fig. 4). 

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