Thermophysical time dependence

As a result of the time-dependent voidage variations near the heating surface, the thermophysical properties of the packet differ from those in the bed, and this difference has not been included in the packet model. The limitation of this model lies in not taking into account the nonuniformity of the solids concentration near the heating surface. Thus, the packet model under this condition is accurate only for large values of Fourier number, in general agreement with the discussion in 4.3.3. 

The basic idea behind an atomistic-level simulation is quite simple. Given an accurate description of the energetic interactions between a collection of atoms and a set of initial atomic coordinates (and in some cases, velocities), the positions (velocities) of these atoms are advanced subject to a set of thermodynamic constraints. If the positions are advanced stochastically, we call the simulation method Monte Carlo or MG [10]. No velocities are required for this technique. If the positions and velocities are advanced deterministically, we call the method molecular dynamics or MD [10]. Other methods exist which are part stochastic and part deterministic, but we need not concern ourselves with these details here. The important point is that statistical mechanics teUs us that the collection of atomic positions that are obtained from such a simulation, subject to certain conditions, is enough to enable aU of the thermophysical properties of the system to be determined. If the velocities are also available (as in an MD simulation), then time-dependent properties may also be computed. If done properly, the numerical method that generates the trajectories... 

As described in the previous sections, the changes in the effective thermophysical properties (density, thermal conductivity, and specific heat capacity) are mainly determined by the decomposition process. This process, being kinetic, is not just an univariate function of temperature, but also on time. Therefore, and in contrast to true material properties, effective properties are dependent not only on temperature, but also on time. In order to model the time-dependent physical properties, related kinetic processes must be taken into account, as described by the kinetic equations in Chapter 2. 

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

Analytical solutions for cases of temperature-dependent thermal conductivity are available [22, 23]. In cases where the solid s thermophysical properties vary significantly with temperature, or when phase changes (solid-liquid or solid-vapor) occur, approximate analytical, integral, or numerical solutions are oftentimes used to estimate the material thermal response. In the context of the present discussion, the most common and useful approximation is to utilize transient onedimensional semi-infinite solutions in which the beam impingement time is set equal to the dwell time of the moving solid beneath the beam. The consequences of this approximation have been addressed for the case of a top hat beam, p 1 = K = 0 material without phase change [29] and the ratios of maximum temperatures predicted by the steady-state 2D analysis. Transient ID analyses have also been determined. Specifically, at Pe > 1, the diffusion in the x direction is negligible compared to advection, and the ID analysis yields predictions of Umax to within 10 percent of those associated with the 2D analysis. 

If the vaporization intensity is independent of the thermophysical parameters of the LPC gas and is determined only by the initial temperature of the liquid, one may observe the dependence of the shock intensity on the nature of a test gas in the LPC. For example, as is obvious from q. (8), the use helium instead of air at the same temperature T leads to a decrease in the shock overpressure Ap2 by about 2.5 times. 

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