Knudsen number values

The effect of the Nusselt number was plotted against the aspect ratio for different Knudsen number values. The results compared well with those of [23], and [24]. 

Transition flow occurs when viscous flow and Knudsen diffusion both play a role, that is in the region with Knudsen number values around unity. Estimates of the value of Kn can be made with the help of the gas kinetic expression for X ... 

Note (1) Figure 3.17 compares the ratio of mass flow rate with slip boundary condition with that of no-slip boundary condition as a function of the pressure ratio and Knudsen number. At Kn 0, the ratio is equal to 1.0, indicating the validity of the no-slip boundary condition. The mass flow rate ratio increases with an increase in the Knudsen number value. 

Depending on the Knudsen number value, different flow regime classifications for gas flow have been proposed as follows ... 

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. 

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. 

Knudsen numbers for EuCI2(g) (estimated diameter 8.70 A) varied for the three orifices between 200 and 0.2 at the extremes of temperatures. These values are within the limits accepted for molecular flow (36). 

For particles of unit density, in air, at a pressure of 1 atm and temperature of 298°K, the upper and lower bounds of the dimensionless coagulation coefficients are plotted as a function of Knudsen number, for different Hamaker constants, in Fig. 3. Obviously, the upper bound for the coagulation coefficient is independent of the Hamaker constant. The upper and the lower bounds tend to the Smoluchowski expression for small Knudsen numbers. For large Knudsen numbers, the upper bound coincides with the free molecular limit, as can be seen from Fig. 3. The lower bound is found to decrease dramatically with a decrease in the Hamaker constant, for large Knudsen numbers. Both the lower and the upper bounds exhibit a maximum at intermediate values of the Knudsen number. 

Knudsen number was varied by varying the pressure of the medium (He) for particles of almost constant size. Their experimental data are compared with different models in Tables II and III. Comparison of the experimental data with the upper bound, the lower bound (for a Hamaker constant of 10-12 erg), and the Fuchs interpolation formula is also shown in Fig. 8. Since the Knudsen number was varied by varying the pressure of the medium for particles of constant size, the upper and the lower bounds approach the limiting values for large Knudsen numbers. The experimental data of Wagner and Kerker (12) lie between the upper bound and the Fuchs for-... 

The mean free path is inversely proportional to the pressure and varies from 30-100 nm at 0.1 MPa to 0.3-1 nm at 10 MPa for normal gases. As previously described, the average size of the pores of commercial catalysts varies over a wide range (1- 200 nm). Therefore, typical values of the Knudsen number are lO-4 - 102. This rough estimate shows that different flow regimes of different complexity occur in practice. 

For small values of the Knudsen number (roughly Kn < 0.3) we operate in the continuum region and y reaches its maximum value ... 

The value of y will be highest at the reactor inlet, where kAj is the highest. Furthermore, high values of y can only be obtained for low values of the Knudsen number Kn. This can be seen from Equation 7.79 where the maximum value for y is given as... 

If both parts of the membrane (the support and the separative layers) can be characterized by values lower than 1 for the Knudsen number (Kn = X/lr where X is the mean free path of species or molecules and rp is the mean pore radius), then all the aspects mentioned here must be taken into consideration. To describe... 

Ceramic membrane is the nanoporous membrane which has the comparatively higher permeability and lower separation fector. And in the case of mixed gases, separation mechanism is mainly concerned with the permeate velocity. The velocity properties of gas flow in nanoporous membranes depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collision. The Knudsen number Kn Xydp is characteristic parameter defining different permeate mechanisms. The value of the mean free path depends on the length of the gas molecule and the characteristic pore diameter. The diffusion of inert and adsorbable gases through porous membrane is concerned with the contributions of gas phase diffusion and sur u e diffusion. 

The applicability of the continuum model is determined by the local value of the non-dimensional Knudsen number (Kn) which is defined as the ratio of the mean free path (lufp) of the heat-carrier medium to the system reference length scale (say thermal diffusion length). Microscale... 

The local value of Knudsen number determines the degree of rarefaction and the degree of validity of the continuum model in a particular flow. The different Knudsen number regimes depicted in Fig. 2 have been determined empirically and are therefore only approximate for a particular flow geometry. The pioneering experiments in rarefied gas dynamics were conducted by Knudsen in 1909 [24]. 

Figure 3. The different zones of applicability of conventional models according to the value of the Knudsen number.

Taking into account the value of V, (Equation (23)) and the definition of tlie Knudsen number, the Darcy coefficient becomes ... 

More reeently, [26] has eonfirmed the need to include the second order slip condition at higher Kn number values. Their work was both theoretical and experimental using nitrogen and helium in a silicon channels. They used the second order slip approximation to obtain the equation for the volumetric flow rate and related it to the ratio of inlet to outlet pressure. It was shown that when using the Navier-Stokes equation, the boundary conditions must be modified to include second order slip terms as the Knudsen number increases. They also studied in depth the accommodation coefficient Fv and verified the need for further study. It was shown that as the Knudsen number increases, the momentum accommodation value deviates further and further from unity for instance Kn -0.5 yields Fv 0.8 for helium. The values found for nitrogen were quite similar. The measurements agreed with past studies such as [11] for lower Kn. 

The Knudsen number (Kn) is used to represent the rarefaction effects. It is the ratio of the molecular mean free path to the characteristic dimension of the flow. For Knudsen numbers close to zero, flow is still assumed to be continuous. As the Knudsen number takes higher values, due to a higher molecular mean free path by reduced pressure or a smaller flow dimension, rarefaction effects become more significant and play an important role in determining the heat transfer coefficient. 

Their results for the non-slip flow ease agreed with [26], who also used the integral transform teehnique to solve for the Nusselt number for flow through a maerosized reetangular ehannel. They did not inelude viscous dissipation in the work, but they did inelude variable thermal aeeommodation eoefficients. Similar to [15], they concluded that the Knudsen number, Prandlt number, aspeet ratio, velocity slip and temperature jump can all cause the Nusselt number to deviate from the eonventional value. 

Depending on the values for the Knudsen number, the Prandtl number, the Brinkman number and the aspeet ratio, heat transfer in mieroehannels can be significantly different from eonventionally sized channels. 

The most appropriate model for mass transfer through air-filled pores can be determined by reference to the Knudsen number, Kn, which is the ratio of the mean free path of the water molecule and the pore diameter [Eq. (8)]. A Knudsen number of 10 or greater is indicative of Knudsen diffusion, whereas a value of 0.01 or less is indicative of Fickian diffusion. Intermediate... 

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