Accommodation coefficients

Let us look into some average parameter for defining gas-solid interaction from macroscopic point of view. Tangential momentum accommodation coefficient ( t ) is a measure of the tangential momentum exchange of gas molecules with the solid surface defined as 

and are the tangential momentums of the incoming and reflected molecules, respectively. The tangential momentum of re-emitted molecules corresponding to that of surface is termed as and is equal to 0 for stationary surface. 

Thermal accommodation coefficient oj) is a measure of the energy exchange between gas molecules and solid surfaces defined as 

and are the energy fluxes of incoming and reflected molecules per unit time, respectively. The energy flux of all the incoming molecules that are re-emitted with the energy flux corresponding to the surface temperature is termed as For the gas, the energy flux may be defined as 

The perfect energy exchange between gas molecules and solid molecules corresponds to (7f = 1. The thermal accommodation coefficient is a measure of the fraction of heat transferred between the wall and the gas molecule. If the gas at temperature 600 K interacts with the wall at 300 K, the wall heats up and the gas molecule cools down by 300 K. Thus, the gas molecule adjacent to the solid surface satisfies the constant temperature boundary condition similar to that of the wall. 

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In the case of polyatomic molecules, one may consider separately the accommodation coefficients for translational and for vibrational energy. Values of the latter, civ, are discussed by Nilsson and Rabinovitch [7]. 

Adsorption nil specular reflection accommodation coefficient zero... 

Tables 2,3, and 4 outline many of the physical and thermodynamic properties ofpara- and normal hydrogen in the sohd, hquid, and gaseous states, respectively. Extensive tabulations of all the thermodynamic and transport properties hsted in these tables from the triple point to 3000 K and at 0.01—100 MPa (1—14,500 psi) are available (5,39). Additional properties, including accommodation coefficients, thermal diffusivity, virial coefficients, index of refraction, Joule-Thorns on coefficients, Prandti numbers, vapor pressures, infrared absorption, and heat transfer and thermal transpiration parameters are also available (5,40). Thermodynamic properties for hydrogen at 300—20,000 K and 10 Pa to 10.4 MPa (lO " -103 atm) (41) and transport properties at 1,000—30,000 K and 0.1—3.0 MPa (1—30 atm) (42) have been compiled. Enthalpy—entropy tabulations for hydrogen over the range 3—100,000 K and 0.001—101.3 MPa (0.01—1000 atm) have been made (43). Many physical properties for the other isotopes of hydrogen (deuterium and tritium) have also been compiled (44).

Figure 5 shows conduction heat transfer as a function of the projected radius of a 6-mm diameter sphere. Assuming an accommodation coefficient of 0.8, h 0) = 3370 W/(m -K) the average coefficient for the entire sphere is 72 W/(m -K). This variation in heat transfer over the spherical surface causes extreme non-uniformities in local vaporization rates and if contact time is too long, wet spherical surface near the contact point dries. The temperature profile penetrates the sphere and it becomes a continuum to which Fourier s law of nonsteady-state conduction appfies. 

Mitsuya, Y, Modified Reynoids Equation for Uitra-Thin Fiim Gas Lubrication Using 1.5-Order Slip-Flow Model and Considering Surface Accommodation Coefficient," ASME J. Tri-bol, Voi. 115,1993, pp. 289-294. 

When the fractions of molecules reflected specularly and diffusively are known, the slip length can be determined, as shovm by Maxwell. Maxwell introduced a tangential momentum accommodation coefficient defined as... 

Based on the accommodation coefficient, the slip length is given by... 

If the accommodation coefficient CA is equal or close to unity for liquid metals, as appears most likely for clean systems, then bubble growth in such liquids is little affected by mass transfer effects. It has been illustrated that the growth rate curves for CA = 1 and CA = are not very far apart. 

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... 

The pressure dependence of effective viscosity obviously depends upon the value of the momentum accommodation coefficient. Momentum accommodation data are relatively rare, but some representative data are given in Table 1. Note that all values are relatively close to unity. Because of this observation, momentum accommodation coefficients are normally assumed to be unity in applications... 

The energy of an incident molecule will not normally be the same as that of the molecule when it is scattered from the surface, i.e., ZsP Ef There will be an accommodation to the surface and an exchange of energy with the surface. Complete accommodation or equilibration with the surface would imply that the scattered molecules have the same temperature as the surface. The energy accommodation coefficient, ac, is defined for each surface involved in the problem by the expression... 

For convenience in this derivation, we assume that the energy accommodation coefficient is the same for collisions at surfaces 1 and 2. Thus, we may write... 

Thus, as observed (Fig. 24), the heat transfer rate is directly proportional to pressure at low pressure. Note also that the heat transfer rate is independent of the distance separating the two surfaces, as expected by intuition. It was assumed that the accommodation coefficient was the same on each surface. If this assumption is not valid, the expression for a must be modified to the form... 

Table 4 Experimental Data for Energy Accommodation Coefficient...

As the pressure increases from low values, the pressure-dependent term in the denominator of Eq. (101) becomes significant, and the heat transfer is reduced from what is predicted from the free molecular flow heat transfer equation. Physically, this reduction in heat flow is a result of gas-gas collisions interfering with direct energy transfer between the gas molecules and the surfaces. If we use the heat conductivity parameters for water vapor and assume that the energy accommodation coefficient is unity, (aA0/X)dP — 150 I d cm- Thus, at a typical pressure for freeze drying of 0.1 torr, this term is unity at d 0.7 mm. Thus, gas-gas collisions reduce free molecular flow heat transfer by at least a factor of 2 for surfaces separated by less than 1 mm. Most heat transfer processes in freeze drying involve separation distances of at least a few tenths of a millimeter, so transition flow heat transfer is the most important mode of heat transfer through the gas. 

Figure 34 Experimental vial heat transfer parameters. Accommodation coefficient = 0.67 vial top emissivity = 0.84. (Data from Ref. 5.)...

The diode laser is scanned up and down in frequency by a triangle wave, so that the scan should be linear in time and have the same rate in both directions. In the thermal accommodation coefficient experiments, the external beam heats the microsphere to a few K above room temperature and is then turned off. The diode laser is kept at fairly low power ( 7 pW) so that it does not appreciably heat the microsphere. Displacement of a WGM s throughput dip from one scan trace to the next is analyzed to find the relaxation time constant as the microsphere returns to room temperature. Results from the two scan directions are averaged to reduce error due to residual scan nonlinearity. This is done over a wide range of pressures (about four orders of magnitude). The time constant provides the measured thermal conductivity of the surrounding air, and fitting the thermal conductivity vs. pressure curve determines the thermal accommodation coefficient, as described in Sect. 5.5.2. 

Fig. 5.6 Pressure dependence of thermal conductivity of air, measured using a PDDA coated microsphere of effective radius 298 pm. The fit to (5.11), shown as the curve, gives a thermal accommodation coefficient of 0.92 for air on PDDA. Reprinted from Ref. 5 with permission. 2008 International Society for Optical Engineering...

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