One-dimensional thermal diffusion

Degree of thermal diffusion is an important factor in treating rapid annealing, since it will govern in-depth crystallinity of poly-Si films formed and thermal damage to glass substrates. One-dimensional thermal diffusion coefficient can... 

X. APPENDIX ONE-DIMENSIONAL THERMAL DIFFUSION INTO TWO DIFFERENT PHASES... 

Figure 9. One-dimensional thermal modeling of heat diffusion.

Compared with heat transfer, the process of moisture transport is slower by a factor of approximately 10. For example, moisture equilibration of a 12 mm thick composite, at 350 K, can take 13 years whereas thermal equilibration only takes 15 s. Fick adapted the heat conduction equation of Fourier, and his (Pick s) second law is generally considered to be applicable to the moisture diffusion problem. The one-dimensional Fickian diffusion law, which describes transport through the thickness, and assumes that the moisture flux is proportional to the concentration gradient, is ... 

The periodic temperature fluctuation can be described using the thermal diffusion equation assuming one-dimensional heat diffusion in the sample, good thermal contact between the sample and glass plate via the sputtered gold layer and that the temperature of the glass remains constant. Under these assumptions, the phase lag A6 is given by... 

For the analysis of these assemblies, the BPG code was used extensively along with a four-group, one-dimensional neutron diffusion code. The thermal flux depression factors were obtained from a one-velocity Monte Carlo program. A resonance integral program was used to calculate epithermal absorption parameters for U-238 and U-235,... 

If Dh is indeed time dependent as in eq. (5) it is not obvious that C(x, t) will follow an error function expression as in eq. (3) or that >H will be thermally activated as in eq. (4). We now show that eqs. (3) and (4) still apply with a time dependent diffusion coefficient, by making a coordinate transformation (Kakalios and Jackson, 1988). The one-dimensional diffusion equation... 

The thermal healing has been studied most extensively for one-dimensional gratings. Above roughening, the gratings acquire, for small amplitude to wavelength ratios, a sinusoidal form, as predicted by the classical continuum theory of Mullins and confirmed by experimenf-s and Monte Carlo simulations. - The decay of the amplitude is, asymptotically, exponential in time. This is true for both evaporation dynamics and (experimentally more relevant) surface diffusion. 

In these one-dimensional equations, the independent variable is the spatial coordinate z, and the dependent variables are the temperature T and the species mass fractions Yk. The continuity equation is satisfied exactly by m" = pu, which is a constant. Other variables are the z component of the mass-flux vector jkz, the molar production rate of species by chemical reaction 6)k, the thermal conductivity A, the species enthaplies hk, and the molecular weights Wk. The diffusion fluxes are determined as... 

We note that later too—even in recent years—Ya.B. has turned his attention to this sphere of problems we have here a paper by Ya.B. devoted to diffusion in a one-dimensional fluid flow (6), papers on hydrodynamics and thermal processes in shock waves which are reviewed in the next section, and on the hydrodynamics of the Universe in the next volume in the section devoted to astrophysics and cosmology. Here we consider only Ya.B. s papers on magnetohydrodynamics or, more precisely, on the problem of magnetic... 

Concentration grating Due to the Ludwig-Soret effect, the temperature grating is the driving force for a secondary concentration grating, which starts to build up and is superimposed upon the thermal one. Its temporal and spatial evolution is obtained from the one-dimensional form of the extended diffusion equation... 

The diffusive transport phenomena in nanowires can be described by a semiclassical model based on the Boltzmann transport equation. For carriers in a one-dimensional subband, important transport coefficients, such as the electrical conductivity, a, the Seebeck coefficient, S, and the thermal conductivity, Ke, are derived as (Sun et al., 1999b Ashcroft and Mermin, 1976a)... 

In order to select materials that will maintain acceptable mechanical characteristics and dimensional stability one must be aware of both the normal and extreme thermal operating environments to which a product will be subjected. TS plastics have specific thermal conditions when compared to TPs that have various factors to consider which influence the product s performance and processing capabilities. TPs properties and processes are influenced by their thermal characteristics such as melt temperature (Tm), glass-transition temperature (Tg), dimensional stability, thermal conductivity, specific heat, thermal diffusivity, heat capacity, coefficient of thermal expansion, and decomposition (Td) Table 1.2 also provides some of these data on different plastics. There is a maximum temperature or, to be more precise, a maximum time-to-temperature relationship for all materials preceding loss of performance or decomposition. Data presented for different plastics in Figure 1.5 show 50% retention of mechanical and physical properties obtainable at room temperature, with plastics exposure and testing at elevated temperatures. 

Let us finally also mention here the results of proton nuclear relaxation time 7 measurements on TEA(TCNQ)2 [53,54], From the frequency dependence of 7, it is deduced that the spin motion is a nearly one-dimensional diffusion. Moreover, the temperature dependence of the on-chain spin diffusion rate shows a quite remarkable feature while it is thermally activated below 220 K, it suddenly becomes temperature independent above 220 K. 

If we use Eq. (7.101) instead ofEq. (7.105) and a thermal diffusion coefficient for one-dimensional heat and diffusion flows for a binary mixture, we have the following coupled balance equations ... 

The coupled problem formulation is presented in Sect. 5.1. The loading rates for which thermal diffusion needs to be considered can be estimated for a one-dimensional problem as in [9,57]. This leads to the nondimensional quantity k which compares a characteristic timescale f0 associated with the loading conditions to the time for heat to diffuse over a characteristic length To as ... 

The flow velocities in flame systems are such that transport processes (diffusion and thermal conduction) make appreciable contributions to the overall flows, and must be considered in the analysis of the measured profiles. Indeed, these processes are responsible for the propagation of the flame into the fresh gas supporting it, and the exponential growth zone of the shock tube experiments is replaced by an initial stage of the reaction where active centres are supplied by diffusion from more reacted mixture sightly further downstream. The measured profiles are related to the kinetic reaction rates by means of the continuity equations governing the one-dimensional flowing system. Let Wi represent the concentration (g. cm" ) of any quantity i at distance y and time t, and let F,- represent the overall flux of the quantity (g. cm". sec ). Then continuity considerations require that the sum of the first distance derivative of the flux term and the first time derivative of the concentration term be equal to the mass chemical rate of formation q,- of the quantity, i.e. 

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