Methods infinite slab

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

Stefan gave an exact solution for the constant-velocity melting of a semi-infinite slab initially at the fusion temperature. This was extended by Pekeris and Slichter (P2) to freezing on a cylinder of arbitrary surface temperature and Kreith and Romie (K6) to constant-velocity melting of cylinders and spheres by a perturbation method, in which the temperature is assumed to be expressible in terms of a convergent series of unknown functions. To make the method clear, consider the freezing of an infinite cylinder of liquid, of radius r0, at constant surface heat flux. For this geometry the heat equation is... 

The second method is based on the rapid convergence of the series solutions. For example, after about 10% of the time for the center temperature to reach 99% of the surface temperature has elapsed, the second term in the series is about 1 2 i> of the first term, Thus after a certain time, only the first term is relevant. Now the dimensionless temperature is always an exponential function of the time for example, con.sider an infinite slab of thickness 2a. The first term of the series for the dimensionless temperature at the center is given by... 

Critical thicknesses of infinite slabs of plutonium nitrate solution have been derived from data on spheres and cylinders, but no data appear in the open literature on the direct measurement of critical thicknesses for thin slabs of plutonium nitrate solution. Experiments have been conducted at the Critical Mass Laboratory of the Pacific Northwest Laboratory using a variable-thickness Blab-type vessel to determine critical thicknesses of bare and reflected infinite slabs of p lutonlum nitrate solution. The esmerlments provide date lor nuclear criticality safety applications in handling and processing plutonium and for checking computational methods. 

There are several surface density techniques, > some of which meet the criteria for a model with appropriately defined areas of applicability. One of interest parallels the solid-angle method in its proposed application by specifying conservative but unknown subcriticality. Only two criticality data points appear necessary that for the geometries of the infinite slab and of the unit considered for storage. The situation is aptly presented in Fig. 1 where the ratio of surface densities, that of the storage array to the infinite slab for the same reflector condition, is shown as a function of the ratio of storage-unit mass to. the critical mass in the same geometry, expressed as a fraction. The systems presented are described in the... 

The method of relativistic transformation of coordinates is evaluated to obtain the exact solution for the transient temperature. Consider a semi-infinite slab at initial concentration Co, imposed by a constant wall concentration Q for times greater than zero at one of the ends. The transient concentration as a function of time and space in one dimension is obtained, yielding the dimensionless variables... 

A method for estimating the cooldown of the concentrated masses in the system is shown in Figure 9. This method involves the division of a given piece of concentrated mass into a series of slabs and the calculation of the cooldown history from unidirectional transient conduction theory. The model chosen for study was a semi-infinite slab, initially at ambient temperature, and having one face perfectly insulated and its uninsulated face suddenly subjected to a step change in temperature of AT... 

Plane-source Method. In a semi-infinite slab where 0 < a < oo, if at f = 0 a constant heating rate q per unit area is generated at the surface a = 0, it can be shown (27) that the temperature at some later time t at an interior point x, that is, T x, t), is given by... 

The considerations in the preceding section make it worthwhile to discuss reflection and transmission at plane boundaries first, one plane boundary separating infinite media, then in the next section two successive plane boundaries forming a slab. In addition to providing useful results for bulk materials, these relatively simple boundary-value problems illustrate methods used in more complicated small-particle problems. Also, the optical properties of slabs often will be compared to those of small particles—both similarities and differences—to develop intuitive thinking about particles by way of the more familiar properties of bulk matter. 

Baxter (B3) uses an enthalpy-flow temperature method, due originally to Dusinberre (D5, D6) and Eyres et al. (E4), whereby the movingboundary effect is reduced to a property variation. To begin with, the melting of a slab of finite thickness initially at the fusion temperature is considered. At the surface of the melt, which is of the same density as the solid, a heat transfer boundary condition is applied. The technique takes into account latent heat effects by allowing the specific heat to become infinite at the fusion temperature in such a way that... 

Another standing topic during the last two decades has been to evaluate the electronic structure of solids, surfaces and adsorbates on surfaces. This can be done using standard band structure methods [107] or in more recent years slab codes for studies of surfaces. An alternative and very popular approach has been to model the infinite solid or surface with a finite cluster, where the choice of the form and size of the cluster has been determined by the local geometry. These clusters have in more advanced calculations been embedded in some type of external potential as discussed above. It should be noted that these types of cluster have in general quite different geometries compared with... 

A more general interface geometry is shown in Fig. 3C. Physically this corresponds to an infinite array of M/C thin film couples separated by vacuum. A salient point is that the vacuum layers should be thick enough that adjacent M/C slabs do not interact. Interaction is possible in two ways either via electronic wavefunction overlap in the vacuum or via Coulombic multipoles. The former interaction is usually vanishing, if more than 1 10 A of vacuum is present. The latter interaction is rather long ranged, but fortunately methods have been devised to electrostatically decouple the slabs.Of course, it is required that both the metal and ceramic layers are thick enough that the interface and surfaces do not interact. 

Whatever the choice of the quantum mechanical method, two representations are currently used to simulate semi-infinite surfaces slabs or clusters. Only the Cl method is restricted to cluster geometries. 

In the late 1980s, Feibelman developed his Green s function scattering method using LDA with pseudopotentials to describe adsorption on two-dimensionally infinite metal slabs [175]. based on earlier work by Williams et al [176]. The physical basis for the technique is that the adsorbate may be considered a defect off which the Bloch waves of the perfect substrate scatter. The interaction region is short-range because of screening by the electron gas of the metal. Feibelman has used this technique to study, for example, the chemisorption of an H2 molecule on Rh(OOl) [177]. S adatoms on Al(331) [178] and Ag adatoms on Pt(l 11)... 

Thermal diffusivity has usually been measured using a quenching method, i.e.. the. solid sample at a uniform temperature is immersed in a temperature-controlled bath at a different temperature. The rate of change of temperature at the center is then monitored with an embedded thermocouple. The sample dimensions are usually chosen so that lateral heat flow can be ignored and regular sample geometries, i.e., "infinite flat slabs, "infinite" cylinders, or spheres, are used. 

The second boundary condition to be considered is that in which the sample surface is subjected to a linear rate of temperature rise. A method based on this has been developed by Shoulberg [47] for diffusivity measurements on polymer melt.s. He used two discs of his material with a thermocouple sandwiched between them the diameter-to-thickness ratio was such that the sample sandwich could be regarded as an infinite flat slab. The sample completely filled the cavity in an aluminum block and was melted in the apparatus. The aluminum block was heated electrically, and the power was adjusted to give an approximate linear rate of temperature rise. Under his experimental conditions this Lusted for about 30 C. 

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