Knudsen number

The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = X/D. Molecular flow is characterized by Kn > 1.0 continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. 

Classiflcation on the basis of the Knudsen number, as per Karniadakis and Beskon (2002), is given in Table 2.3. 

Turner et al. (2004) studied the independent variables relative surface roughness, Knudsen number and Mach number and their influence on the friction factor. The micro-channels were etched into silicon wafers, capped with glass, with hydraulic diameters between 5 and 96 pm. Their surface roughness was 0.002 < ks< 0.06 pm for the smooth channels, and 0.33 < / < 1 -6 pm for the glass-capped ones. The surface roughness of the glass micro-channels was measured to be in the range 0.0014 [Pg.39]

The above results show close agreement between the experimental and theoretical friction factor (solid line) in the limiting case of the continuum flow regime. The Knudsen number was varied to determine the influence of rarefaction on the friction factor with ks/H and Ma kept low. The data shows that for Kn < 0.01, the measured friction factor is accurately predicted by the incompressible value. As Kn increased above 0.01, the friction factor was seen to decrease (up to a 50% X as Kn approached 0.15). The experimental friction factor showed agreement within 5% with the first-order slip velocity model. 

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

The Knudsen number is small enough so that the fluid is a continuous medium. 

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

For single-phase gas flow in micro-channels of hydraulic diameter from 101 to 4,010 pm, in the range of Reynolds numbers Re < Recr, the Knudsen number 0.001 < Kn < 0.38, and the Mach number 0.07 < Ma < 0.84, the experimental friction factor agrees quite well with the theoretical one predicted for fully developed laminar flow. 

The main aim of the present chapter is to verify the capacity of conventional theory to predict the hydrodynamic characteristics of laminar Newtonian incompressible flows in micro-channels in the hydraulic diameter range from dh = 15 to db = 4,010 pm, Reynolds number from Re = 10 up to Re = Recr, and Knudsen number from Kn = 0.001 to Kn = 0.4. The following conclusions can be drawn from this study ... 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

Knudsen number Nahme number Nusselt number... 

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

Fukui, S., and Kaneko, R., A Database for Interpoiation of Poiseuiiie Fiow Rates for High Knudsen Number Lubrication Probiems,"ASMR/. Rr/fooZ., Vol. 112,1990,pp. 78-83. 

The applicability of the two different models of gas flow is generally judged from the gas flow regimes according to the magnitude of the local Knudsen number, Kn, defined as... 

Dq = characteristic inverse Knudsen number po o P 2RTq-, Tq = characteristic temperature ... 

In Fig. 2, the flow rate coefficients, Qp, calculated from these different models are plotted in the range of inverse Knudsen number D from 0.01 to 100. We can see that for... 

We can see that the velocity profiles of these models become more and more different from each other with the decrease in the inverse Knudsen number. The difference between the velocity profiles of the first-order and second-order slip models is in the magnitude of the slip velocity, but the shapes of the velocity profiles are almost same. The second-order slip model predicts a larger and larger slip velocity than the first-order slip model when the inverse Knudsen number decreases. The Fukui-Kaneko model gives a medium slip velocity and velocity profile. 

It should be pointed out that the flow rate in the case of the Couette flow is independent of the inverse Knudsen number, and is the same as the prediction of the continuum model, although the velocity profiles predicted by the different flow models are different as shown in Fig. 4. The flow velocity in the case of the plane Couette flow is given as follows (i) Continuum model ... 

Fig. 5—Change of the rates of particle-particle and particle-wall collisions with the inverse Knudsen number.

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

Figure 6 shows the change of with the Knudsen number Kn. When Kn[Pg.101]

Fig. 8—Dependence of the nanoscale effect function on the inverse Knudsen number.

Figure 8 shows the change of Np with the inverse Knudsen number D. When D > 1, the nanoscale effect is weak, and the Xff is slightly less than X./,. When D < 1, however, the difference between and X becomes significant. It is worth to... 

The correction of mean free path, hi by the nanoscale effect function results in a smaller mean free path, or a smaller Knudsen number in other word. As a matter of fact, a similar effect is able to be achieved even if we use the conventional definition of mean free path, / = irSn, and the Chapmann-Enskog viscosity equation, /r = (5/16)... 

XlylmnkT/ird ), or h = [TT[i 2RTI2p), as long as substituting the gap-dependent viscosity rather than the bulk viscosity. Because the effective viscosity decreases as the Knudsen number enters the slip flow and transition flow ranges, and thus the mean free path becomes smaller as discussed by Morris [20] on the dependence of slip length on the Knudsen number. 

To evaluate the intensity of the rarefaction effect, the him thickness h itself is not enough. We have to know the Knudsen number Kn or the inverse Knudsen number D. From the dehnition of Kn, Eq (2), and the relationship between and pressure, ... 

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