Accommodation coefficient Knudsen

The average incident tangential momentum is muh while the average scattered tangential momentum is muf. If the gas molecule equilibrates with the surface and the scattered momentum is zero, we have Knudsen cosine scattering and complete accommodation of the incident gas molecule with the surface. On the other extreme, if specular reflection occurs, the incident momentum is retained upon scattering and mut = muf. The momentum accommodation coefficient, / is introduced to describe the type of scattering that does occur, and it is defined by... 

For particles whose accommodation coefficient is known, Eq. (10-56) appears to give the most accurate estimate for drag. Since ctr is rarely known to sufficient accuracy, C may instead be estimated for spheres over the whole range of Kn by a semiempirical expression whose form was first proposed by Knudsen and Weber (K6). With the numerical values due to Davies (D2) ... 

Another important factor to recognize is that the net uptake coefficient determined using Knudsen cells may not represent the true uptake or trapping of the gas by the surface if reevaporation into the gas phase occurs, which must be taken into account in such cases. In principle, the mass accommodation coefficient is the... 

The energy exchange between an impinging gas atom or molecule and a surface can be represented by a coefficient, the accommodation coefficient, as defined by Knudsen,11... 

We derive a numerical model for the effect of a spatial temperature gradient on the local equilibrium of a chemical reaction in a low--density gas (Knudsen regime). The gas consists of two constituents and the chemical reaction is assumed to take place at the walls of the container. The numerical results are compared with experimental results on the equilibrium 2Na - Na2. From the comparison it follows that the chemical accommodation coefficient for a Na2 wall collision is essentially equal to 1. 

All these measurements depend on the assumption that the accommodation coefficient / in Knudsen s equation ( 2.VII J) or analogous equations is unity. The value of (2—/)//=a is sometimes called the coefficient of condensation ( 12.VII F), and a value a=l implies that every vapour molecule striking the surface of the liquid or solid form condenses. In other cases, a[Pg.244]

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. 

Ft, Thermal accommodation coefficient K, Thermal conductivity Kn, Knudsen number M, Mass of the fluid Ma Mach number m, Mass flow rate... 

Fig. 31 shows the heat transfer coefficient for various wafer-electrode gap widths and He backside pressures, assuming an accommodation coefficient of aa . = 0.4 [163]. As mentioned above, the heat transfer coefficient is independent of gap width and increases in proportion to pressure in the Knudsen regime. At high pressures, for which gas conduction predominates, the heat transfer coefficient is independent of pressure and varies inversely proportionately with the gap width. The dots on the curves in Fig. 31 signify the point at which the gap width is equal to the molecule mean free path at the corresponding pressure. 

Keywords Accommodation coefficients Condensation Evaporation Free-molecule regime Gas phase transport Knudsen regime Non-continnum regime... 

Figure 5 a shows the variation of the inverse slip coefficient with the outlet Knudsen number considering pressure ratio of II = 1.8 and accommodation coefficient of cr = 0.93. It is observed that the IP-based model concur exactly with the experimental data [6]. This is where the analytical formula of Aubert and Colin [7] departs from the given data for Kn > 0.25. 

Overall the results show that the wall force field penetration depth is an additional length scale for gas flows in nano-channels, breaking dynamic similarity between rarefied and nanoscale gas flows solely based on the Knudsen and Mach numbers. Hence, one should define a new dimensionless parameter as the ratio of the force field penetration depth to the characteristic channel dimension, where wall effects cannot be neglected for large values of this dimensionless parameter. Additionally, the calculated tangential momentum accommodation coefficients for a specific gas-surface couple were found to be constant regardless of different base pressure, channel height, wall velocity, and Knudsen number. Results of different gas-surface couples reveal that TMAC is only dependent on the gas-surface couple properties and independent of the Knudsen number. 

In this equation, a is the tangential momentum accommodation coefficient, equal to unity for perfectly diffuse molecular reflection and zero for purely specular reflection. In Maxwell s model, MsUp overestimates the real velocity at the wall but leads to a rather good prediction of the velocity out of the Knudsen layer, as represented in Fig. 2. After non-dimensionalization with the characteristic length L, a reference velocity uo, and a reference temperature Tq, Eq. 10 is written as follows ... 

The a. have been termed the Knudsen accommodation coefficients [2.88],... 

Transfer processes in the free-molecular regime can be expressed exactly in terms of these Knudsen accommodation coefficients [2.88]. However, this is not the case in the slip-flow regime as has been demonstrated by Kuscer. The situation in the transition regime has not been investigated yet. 

Knudsen accommodation coefficients of normal momentum, tangential momentum, energy, and thermal creep. The additional coefficients a y, a y, and ayy are... 

For various assumed models of surface-gas interactions it is possible to establish the correspondence between the and the Knudsen accommodation coefficients, a. However, it is clear that these various models do not deal with the complexity... 

By use of Knudsen s accommodation coefficients (2.53) for this problem in the same limits, the following result is found for the free-molecular thermal force [2.1281 ... 

PHILLIPS [2.128] has derived an expression corresponding to linearization of (2.85) and introduction of Knudsen s accommodation coefficients. 

It can be shown that the intrinsic growth or evaporation rate associated with a given organic volatility is given by vdC where the characteristic velocity is 226 nm h V(pg m ) [75]. This is modified by three important terms - the mass accommodation coefficient, a, the surface-energy (Kelvin) term for particles smaller than 50 nm or so, and the Fuchs term for gas-phase diffusion limitations in the boundary layer around a particle for particles larger than 50 nm or so (with Knudsen... 

The accommodation coefficients and uj- depend on the gas and surface temperatures and local pressure. Table 3.1 shows the typical accommodation coefficient for some gas-surface combinations. Accommodation coefficient also depends on the Knudsen number of the flow. Figure 3.3 shows the variation of accommodation coefficient with respect to the Knudsen number for nitrogen gas. 

Figure 3.3 Momentum accommodation coefficient of nitrogen gas as a function of Knudsen number...

The above-mentioned velocity solution indicates not only a velocity slip on the wall but also a correction to the slope of the profile (see Figure 3.13) for the Couette flow with slip flow boundary condition. The velocity profile as a function of Knudsen number and momentum accommodation coefficient has been shown in Figure 3.14. There is an increase in slip velocity... 

Figure 3.14 Velocity profile in a Couette flow as a function of (a) Knudsen number with fixed cr = 0.8 and (b) tangential accommodation coefficient with fixed Kn = 0.15...

Figure 3.16(a,b) shows the velocity profile as a function of momentum accommodation coefficient and Knudsen number for the flow between two parallel plates. Both increase in Knudsen number for constant accommodation coefficient and increase in accommodation coefficient... 

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