Thermal harmonic mean

In which Atfe 1/2 is the thermal conductivity at temperature f 1//2- This requires a suitable mean value to be chosen, the arithmetic, geometric or harmonic mean of the thermal conductivities at the known temperatures and tA+1 or nd The type of mean value formation does not play a decisive role if A is only weakly dependent on d or if the step size Ax is chosen to be very small. D. Marsal [2.53] recommends the use of the harmonic mean, so... 

Sphere-Flat Test Results. Kitscha [47] performed experiments on steady heat conduction through 25.4- and 50.8-mm sphere-flat contacts in an air and argon environment at pressures between 10 5 torr and atmospheric pressure. He obtained vacuum data for the 25.4-mm-diameter smooth sphere in contact with a polished flat having a surface roughness of approximately 0.13 pm RMS. The mechanical load ranged from 16 to 46 N. The mean contact temperature ranged between 321 and 316 K. The harmonic mean thermal conductivity of the sphere-flat contact was found to be 51.5 W/mK. The emissivities of the sphere and flat were estimated to be e, = 0.2 and e2 = 0.8, respectively. 

The interfacial value of thermal conductivity can be best interpolated by invoking a harmonic mean approximation, based on values of the same at the adjacent grid points. The explicit scheme gives rise to a system of discretized algebraic equations that are not mathematically coupled. However, the cost that one might have to pay against this simplification is that the scheme is conditionally stable. On the other hand, the implicit scheme requires a coupled system of linear algebraic equations to be solved but is unconditionally stable (the issues of stability in the context of discretized equations will be elaborated later). 

The thermal conductivity, k, and the dynamic viscosity of the fluid, p., can be modeled using the arithmetic and harmonic means of each component in accordance with the expressions [5] ... 

The interfacial value of thermal conductivity can be best interpolated by invoking a harmonic mean approximation, based on values of the same at the adjacent grid points. 

This relation may be interpreted as the mean-square amplitude of a quantum harmonic oscillator 3 o ) = 2mco) h coth( /iLorentzian distribution of the system s normal modes. In the absence of friction (2.27) describes thermally activated as well as tunneling processes when < 1, or fhcoo > 1, respectively. At first glance it may seem surprising... 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

Assuming isotropic and harmonic vibration, the thermal parameter B becomes the quantity shown in equation 3.6, where u2 is the mean square displacement of the atomic vibration ... 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

X-Ray results provide important information regarding molecular and atomic motions, through determination of the thermal factor (B), which gives a measure of the mean square (harmonic) displacement (u ) of an atom or group from its equilibrium position. The two are related by the Debye—Waller equation B = A highly mobile pro-... 

More sensitive to the level of theory is the vibrational component of the interaction energy. In the first place, the harmonic frequencies typically require rather high levels of theory for accurate evaluation. It has become part of conventional wisdom, for example, that these frequencies are routinely overestimated by 10% or so at the Hartree-Fock level, even with excellent basis sets. A second consideration arises from the weak nature of the H-bond-ing interaction itself. Whereas the harmonic approximation may be quite reasonable for the individual monomers, the high-amplitude intermolecular modes are subject to significant anharmonic effects. On the other hand, some of the errors made in the computation of vibrational frequencies in the separate monomers are likely to be canceled by errors of like magnitude in the complex. Errors of up to 1 kcal/mol might be expected in the combination of zero-point vibrational and thermal population energies under normal circumstances. The most effective means to reduce this error would be a more detailed analysis of the vibration-rotational motion of the complex that includes anharmonicity. 

Thermal vibrations of atoms which have a frequency of about 10 second" are slow compared to the X-ray frequency, which is about 10 second". Consequently, the atoms appear to be stationary to the X-rays and the diffraction pattern represents a time average of many instantaneous states. If the motion of the atoms is harmonic so that the restoring force is proportional to the distance of the atom from its rest position and if the motion is isotropic so that the mean square displacements of the atom in all directions are the same, then is related to the temperature factor, B, by... 

Displacements may arise not only from thermal motion but also from static disorder when corresponding atoms in different unit cells take up slightly different mean positions. Certain side chains, especially those exposed, may take up a few radically different conformations in different molecules so that separate images of them can be seen with reduced occupancy in electron density maps. The mean square displacement will also include contributions from lattice disorders but these are usually small in protein crystals that diffract well to high resolution [191]. In principle, the thermal vibrations can be distinguished from static disorder by varying temperature. Simple harmonic vibrations are expected to decrease linearly with temperature. 

To remove fast oscillations we work in the interaction picture introducing new operators Aj = fi/(z)exp(—ikjz)- The damping of vibration modes is modelled as coupling of each vibration mode to the broad reservoir of harmonic oscillators in thermal equilibrium [140]. The two important parameters are damping constants jv and mean number of chaotic phonons (nVj),j= 1,2. Finally we arrive at [127] the following ... 

The simplest model of thermal diffuse scattering is one where the crystal is considered to be an Einstein solid . In this approximation all the atoms are vibrating independently as isotropic harmonic oscillators, each atom having the same mean square displacement U. For this model... 

For free particles, the mean square radius of gyration is essentially the thermal wavelength to within a numerical factor, and for a ID harmonic oscillator in the oo limit. 

---