Accommodation coefficient thermal

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...

The sensing methods summarized thus far are intended for absorption detection of molecules in the ambient, but molecules (or indeed thin films) on the microresonator surface can also be detected. In particular, if the surface is covered to such an extent that the optical energy absorbed heats the microresonator, the resulting thermal bistability in the frequency-scan response can be used to determine the absorption and/or thickness of the thin-film coating. This and surface characterization by measurement of the thermal accommodation coefficient were described in Sect. 5.5. These methods offer quite precise measurement, provided that certain reasonable and easily implemented assumptions are satisfied. 

Analogous to the slip velocity between gas and particle at Kn above the continuum flow range discussed in Section A above, a temperature discontinuity exists close to the surface at high Kn. Such a discontinuity represents an additional resistance to transfer. Hence, transfer rates are generally lowered by compressibility and noncontinuum effects. The temperature jump occurs over a distance 1.996kl 2 — a )/Fva k + 1) (K2, Sll) where is the thermal accommodation coefficient, interpreted as the extent to which the thermal energy of reflected molecules has adjusted to the surface temperature. 

Accurate modeling is only possible by the consideration of wavelength-dependent optical and temperature-dependent thermodynamic parameters and the correct application of the thermal accommodation coefficient which is dependent on the ambient particle conditions and is described in detail elsewhere (Schulz et al., 2006 Daun et al., 2007). Moreover, Michelsen (2003) suggested the inclusion of a nonthermal photodesorption mechanism for heat and mass loss, the sublimation of multiple cluster species from the surface, and the influence of annealing on absorption, emission, and sublimation. A more general form of the energy equation including in more detail mass transfer processes has been derived recently by Hiers (2008). For practical use, Equation (1) turns out to be of sufficient physical detail. 

The parameter Tr is the temperature corresponding to the half-range Maxwellian distribution of the particles reflected diffusively. The temperature 7 r is related to the temperature of the reflecting surface through the thermal accommodation coefficient E, defined as... 

As mentioned earlier, the factor a is the thermal accommodation coefficient and am the momentum accommodation or reflection coefficient. From the data of Rosenblatt and LaMer (1946), Schmitt (1959), and Keng and Orr (1966), as a first approximation a value of 1.25 seems reasonable for Cm, whereas for Ct a value of 2 is a good approximation (Brock, 1962b). These numbers then imply values of 0.89 for am and 0.97 for at. 

The colinear collision problem of atom A colliding with a molecule BC was first attempted quantum mechanically by Zener [14,15] and then by Jackson and Mott [28] for the purpose of investigating thermal accommodation coefficients for atoms impinging on solid surfaces. An exponential repulsion was utilized, along with the harmonic-oscillator approximation. The distorted-wave (DW) method was employed to obtain a 1 — 0 transition probability of the form... 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

Fj is the thermal accommodation coefficient. The analysis yields the following fully-developed Nusselt number expression... 

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

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

D.J. Rader, W.M. Trott, J.R. Torczynski, J.N. Castaneda, and T.W. Crasser, Measurements of Thermal Accommodation Coefficients, Report SAND2005-6084, Sandia National Laboratories, Albuquerque (2005). 

J. P. Hartnett, A Survey of Thermal Accommodation Coefficients, Rarefied Gas Dynamics, L. Talbot ed., pp. 1-28, Academic Press, New York, 1961. 

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