Layer models

The necessity to calculate the electrostatic contribution to both the ion-electrode attraction and the ion-ion repulsion energies, bearing in mind that there are at least two dielectric ftmction discontinuities hr the simple double-layer model above. 

CATALYSTS - REGENERATION - FLUID CATALYTIC CRAC KING UNITS] (Vol 5) Turbulent boundary layer model... 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

Various proposed values for the constants can be found in the literature [8]. Despite double-layer model predictions [148,149] that exponents Jt and y are both unity, and a dimensional analysis model [204] giving x as 1.88 andy as 0.88, test work on a practical scale [202,203] has indicated that both exponents are approximately equal to 2. This implies that a is roughly independent of pipe diameter and that the ratio //3 s 4/jt s 1. 

The structure refinement program for disordered carbons, which was recently developed by Shi et al [14,15] is ideally suited to studies of the powder diffraction patterns of graphitic carbons. By performing a least squares fit between the measured diffraction pattern and a theoretical calculation, parameters of the model structure are optimized. For graphitic carbon, the structure is well described by the two-layer model which was carefully described in section 2.1.3. 

Zukowski, E. and T. Kubota, 1980, Two Layer Modeling of Smoke Movement in Building Pire.s, Fire Material 4. 

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [120], Gouy and Chapman [121,122], and Stern [123], and in most analytic studies, the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g. as a simple dipolar fluid. Computer simulations... 

In a three-layer model, we represent one part of the system by MM, one part by a semi-empirical Hamiltonian and one part by an ab initio Hamiltonian. 

From the difference of these morphological characteristics, a simplified three-layer model could be set up and the inhomogeneous structure could be classified into these three layers. Figure 23 illustrates the relative thickness of the core, transition, and skin layer for the four sample groups, depending on the processing condition. 

Fig. 20.8 Gouy-Chapman diffuse layer model of the double layer...

The Three-Layer Model Based on Values of Thermal Expansion.155... 

For the N-layer model in particulates, which is an improvement of the three-sphere model, it has been shown5), by taking into consideration the boundary conditions between phases, that ... 

In order to simplify the procedure of evaluating the extent of mesophase and its mechanical and thermal properties, a simple but effective three-layer model may be used, which is based on measurements of the thermal expansions of the phases and the composite, below and above the transition zone of the composite, lying around its glass transition temperature Tgc. 

Then, the three-layer model provides an easy method for evaluating the characteristics of the mesophase, by introducing a significant flexibility in the study of the physical behaviour of particulates. The drawback of the model is its instability to the values of the thermal expansions and the moduli of the composite, which must be evaluated with very high accuracy, fact which is a difficult task. Small deviations in measuring the a s and the E s may vary considerably the balance of characteristic values of the composite. However, the introduction of the influence of the mesophase to the physical behaviour of the composite, made in this model, is a certain advancement in the knowledge of the behaviour of these complicated substances. 

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

Indeed, the multi-layered model, applied to fiber reinforced composites, presented a basic inconsistency, as it appeared in previous publications17). This was its incompatibility with the assumption that the boundary layer, constituting the mesophase between inclusions and matrix, should extent to a thickness well defined by thermodynamic measurements, yielding jumps in the heat capacity values at the glass-transition temperature region of the composites. By leaving this layer in the first models to extent freely and tend, in an asymptotic manner, to its limiting value of Em, it was allowed to the mesophase layer to extend several times further, than the peel anticipated from thermodynamic measurements, fact which does not happen in its new versions. 

A three-layer model for fiber composites may be developed, based on the theory of self-consistent models and adapting this theory to a three-layered cylinder, delineating the representative volume element for the fiber composite. 

The three-layer model, as previously mentioned, as well as the multi-layer model, were previously applied to study the behaviour, especially of fiber composites 3A). The three-layer model, based on the self-consistency of phases, gave relationships between stresses and displacements between phases, which, when solved, may give... 

A series of models were introduced in this study, which take care of the existence of this boundary layer. The first model, the so-called three-layer, or N-layer model, introduces the mesophase layer as an extra pseudophase, and calculates the thickness of this layer in particulates and fiber composites by applying the self-consistent technique and the boundary- and equilibrium-conditions between phases, when the respective representative volume element of the composite is submitted to a thermal potential, concretized by an increase AT of the temperature of the model. 

Thermally spiked 204 Thermal spikes 195 Three-layer models 152, 174, 175 Three-term models 149 Three-term unfolding model 176,185 Tie molecules 124 Tie-taut molecules 141 Time-dependent hardness 121 Tire cord 6... 

An alternative approach to the representation of results for solid-liquid flow is to use the two-layer model which will be described in the following section. It will be seen that the coefficient of friction between the particles and the wall of the pipe is an important parameter in the model. It is suggested that its complete absence in equation 5.24 may be an important reason for the extent of the scatter. Unfortunately, it is a quantity which has been measured in only a very few investigations. It is interesting to note that the form of equation 5.19 was obtained by NEWITT et alP2) using a force balance similar to that... 

The two-layer model is being progressively updated as fresh experimental results and correlations become available. The most satisfactory starting-point for anyone wishing to use the model to calculate pressure gradients for flow of solids-liquid mixtures in a pipeline is the text of SHOOK and Roc.o(52) which includes a worked example. However, there are many pitfalls to be avoided in this area, and there is no substitute for pracucal experience gained by working in the field. 

Figure 5.12. Comparison of predicted and experimental results for simplified two-layer model (3.5 mm...

Various pc electrode models have been tested.827 Using the independent diffuse layer electrode model74,262 the value of E n = -0.88 V (SCE) can be simulated for Cd + Pb alloys with 63% Pb if bulk and surface compositions coincide. However, large deviations of calculated and experimental C,E curves are observed at a 0. Better correspondence between experimental and calculated C,E curves was obtained with the common diffuse-layer electrode model,262 if the Pb percentage in the solid phase is taken as 20%. However, the calculated C, at a Ois noticeably lower than the experimental one. It has been concluded that Pb is the surface-active component in Cd + Pb alloys, but there are noticeable deviations from electrical double-layer models for composite electrodes.827... 

In general Figures 6.18 to 6.25, and in particular figures 6.18, 6.19, 6.20, 6.24 and 6.25 show, beyond any reasonable doubt, that the effective double layer model of promotion, expressed mathematically by Equations 6.65 and 6.66, grasps the essence of promotional kinetics. 

As already discussed in Chapter 6 (Figure 6.25) the observed complex rate dependence of CO oxidation on pco, P02 and UWR (O) (Figs. 4.16, 4.31, 9.6 and 9.7) can be described in a semiquantitalive fashion by the effective double layer model presented in Chapter 6. The system provides an excellent paradigm of the promotional rules Gl, G2 and G3 which are summarized by the general inequalities (6.11) and (6.12) written specifically here for the CO oxidation system ... 

INTERRELATION OF PROMOTION, ELECTROCHEMICAL PROMOTION AND METAL-SUPPORT INTERACTIONS THE DOUBLE-LAYER MODEL OF CATALYSIS... 

The process of equilibration of the atmosphere with the ocean is called gas exchange. Several models are available, however, the simplest model for most practical problems is the one-layer stagnant boundary-layer model (Fig. 10-18). This model assumes that a well-mixed atmosphere and a well-mixed surface ocean are... 

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