Films 3- dimensional models

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

In the film-penetration model (H19), it is assumed that the reactant A penetrates through the surface element by one-dimensional unsteady-state molecular diffusion. Convective transport is assumed to be insignificant. The diffusing stream of the reactant A is depleted along the path of diffusion by its reversible reaction with the reactant B, which is an existing component of the liquid surface element. If such a reaction can be represented as... 

Choo, J. W., Glovnea, R. R, Olver, A. V., and Spikes, H. A., The Effects of Three-Dimensional Model Surface Roughness Fea- [58] tures on Lubricant Film Thickness in EFIL Contacts," ASME J. IHbol,Voi. 125,2003,pp. 533-542. 

Until now, the theoretical discussion has focused on monodisperse two-dimensional model systems. However, some studies have been performed on polydisperse systems, notably by Weaire et al. [68-72] The evolution of a soap froth of random cell sizes and shapes, known as a Voronoi network, was simulated by computer [68] (Fig. 7). The condition that three films must always meet at angles of 120° was again used. Cells with more than six sides were found... 

A reactor for thin-film deposition can have a tubular structure with a flowing chemically active gas. In this case, the PFR model can be used. This one-dimensional model uses the assumption of a uniform component distribution in a reactor section. This means that the boundary layer formation processes are not taken into account. 

The commercial finite element program, Abaqus [17], was used to calculate the stress distribution in an edge delamination sample. A fully three-dimensional model of the combinatorial edge delamination specimen was constructed for the finite element analyses (FEA). For clarity, some of the FEA results and schematics are presented as two-dimensional configurations in this paper (e.g.. Fig. 1). The film and substrate were assumed to be linearly elastic. The ratio of the film stiffness to the substrate stiffness was assumed to be 1/100 to reflect the relative rigidity of the substrate. This ratio also represents a typical organic... 

A numerical simulation of this cell based on a one-dimensional model has been carried out by Ernst (2001), Grasso et al. (2002) and by Burgelman and Grasso (2004). In the work of Ernst and Grasso et al., the spectral response data could be simulated with reasonable accuracy using only a few adjustable parameters. These simulations confirm the electron diffusion length in the p-type CdTe films to be approximately 150 nm. The recombination centre density was found to be lO cm . These data indicate that the nanocrystalline CdTe films are of inferior quality than the material used in the conventional, planar CdTe solar cells, where diffusion lengths of 2 //m and defect densities of lO cm are typical. 

The consideration of thermal effects and non-isothermal conditions is particularly important for reactions for which mass transport through the membrane is activated and, therefore, depends strongly on temperature. This is, typically, the case for dense membranes like, for example, solid oxide membranes, where the molecular transport is due to ionic diffusion. A theoretical study of the partial oxidation of CH4 to synthesis gas in a membrane reactor utilizing a dense solid oxide membrane has been reported by Tsai et al. [5.22, 5.36]. These authors considered the catalytic membrane to consist of three layers a macroporous support layer and a dense perovskite film (Lai.xSrxCoi.yFeyOs.s) permeable only to oxygen on the top of which a porous catalytic layer is placed. To model such a reactor Tsai et al. [5.22, 5.36] developed a two-dimensional model considering the appropriate mass balance equations for the three membrane layers and the two reactor compartments. For the tubeside and shellside the equations were similar to equations (5.1) and... 

A One-Dimensional Analysis for Bo 1. Figure 9.22 depicts the one-dimensional model for evaporation in porous media with heat addition q from the impermeable lower bounding surface maintained at T0> Ts, where T, is the saturation temperature. The vapor-film region has a thickness 8g, and the two-phase region has a length 8g(. 

Ben-Yoseph, E., Hartel, R.W., and Howling, D. (2000). Three-Dimensional Model of Phase Transition of Thin Sucrose Films During Drying, J. Food Eng. 44(1), 13-22. 

The wall heat transfer coefficient can be predicted by a model that is analogous to that outlined here for [72,73]. It should be stressed here that is intrinsically different from the global coefficients discussed in Sec. 11.5.a. Indeed, the latter are obtained when the experimental heat transfer data are analyzed on the basis of a one-dimensional model that does not consider radial gradients in the core of the bed. This comes down to localizing the resistance to heat transfer in radial direction completely in the film along the wall. 

The distinction between conditions in the fluid and on the solid leads to an essential difference with respect to the basic one-dimensional model, that is, the < problem of stability, which is associated with multiple steady states. This aspect was studied first independently by Wicke [90] and by Liu and Amundson [89,91]. They compared the heat produced in the catalyst, which is a sigmoidal curve when plotted as a function of the particle tempoature, with the heat removed by the fluid through the film surrounding the particle, which leads to a straight line. The steady state for the particle is given by the intersection of both lines. It turns out that for a certain range of gas—and particle temperatures—three intersections, therefore three steady states are possible. 

In quite a few of the experiments, there was an extension region in the upper part of the film that was essentially two-dimensional. In particular, this occurred in the early part of the experiments (typically several minutes, at least) and in the upper half to two-thirds of the film for nearly the entire width. Under these circumstances, the interference fringes were essentially horizontal and parallel. These were also the conditions under which the exponent for the drainage rate was measured. This leads us to consider two-dimensional models for film drainage they can be considered as a cross-sectional slice down the centre of the film perpendicular to the surface of the film. All of the theory described in this section will be for this two-dimensional case. 

Further analysis of the chitosan/PVA blend film was done in terms of the simple additivity of the complex moduli of chitosan and PVA. Theoretical calculations were carried out as a function of chitosan content by using the three-dimensional model in Figure 9.7 (Matsuo et al. 1980, 1990, 2001), in which the chitosan layers are surrounded by a PVA plane, so that the strains of the two phases at the boundary in three directions are identical. The parameters 8, n, and p, correspond to the fraction length of chitosan in the three directions, and the volume fraction of chitosan is given as Sop. By using the model system, the complex moduli of the blend may be given by... 

The homogeneous models assttme three phases, i.e., metal, polymer film arrd an electrolyte solutiott. Electrorric, tttixed electrorric (electron or polaron) and iotric charge trarrsport processes are cotrsidered in the metal, within the polymer film and in the solutiott, respectively. The polymer phase itself consists of a polymer matrix with incorporated ions arrd solvent molecrrles. A one-dimensional model is used, i.e., the spatial changes of all qttantities (concerrtrations, potential) within the film are described as a function of a single coordirrate x, which is a good approach when an electrode of usual size is used. The metal Ipolymer and the polymer solution interfacial boundaries are taken as planes. Two intetfacial poterttial differences are considered at the two interfaces, and a potential drop inside the film when crrrrerrt flows. The thicknesses of the electric double layers at the irrterfaces are small in... 

This case is important for thin crystalline films. At first, let us look at the simplest infinite one-dimensional model of the crystal structure. Fig. 5.18, having only zero and first harmonic of density,... 

A more reasonable explanation for the wall-temperature distribution can be developed by considering a simple, one-dimensional model of the flow. If the hydrogen enters the test section at saturation conditions, immediately a nonstable vapor film is established adjacent to the wall. A turbulent exchange between the vapor film and the liquid core takes place. This exchange mechanism is an equilibrium condition between the liquid and gas phases. As the mixture moves through the heated tube, more gas is generated, and continuity considerations demand that the mixture velocity increase. As in any turbulent convective heat-... 

We consider two spheres of fixed radii to have collided to form a film between them of thickness, say, /z- (less than some value to be defined presently). We further assume a random force such as that arising due to turbulent pressure fluctuations that produces a random film drainage process. A positive force is assumed to drain the film while a negative force causes it to thicken by inflow. Although the process is strictly three-dimensional, we shall assume a one-dimensional model, letting the force be always normal to the film. Further, we stipulate that if the film drains to some critical thickness, say h, the film snaps to allow aggregation between the particles. We shall see later how such a model can be formulated mathematically. The instantaneous film thickness H will serve to describe the position of one of the particles relative to the other. 

Fig. 25. (a) Desmeared SAXS curves of PAA and PMDA-ODA polyimide films cured off substrate under different eonditions (1) 90°C/3.5 h + 150°C/8 h + 190°C/4 h + 240°C/5.5 h stepwise (2) 250°C/12 h (3) 350°C/12 h and (4) 430°C/2 h. (b) Schematic diagram of a possible layered morphology based on one-dimensional model analysis. Reprodueed with permission from J Polymer Sci, Polym Phys Ed 1981 19 1293. 1981 John Wiley Sons [72]. (c) Smeared SAXS eurves of PMDA-ODA polyimide films cured off substrate under different temperatures. Reproduced with permission from J Polymer Sci, Polym Phys Ed 1984 22 1105. 1984 John Wiley Sons [73]. 

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