Generalised Model Description

The energy and mass balance equations for reacting systems follow the same principles, as described previously in Secs. 1.2.3 to 1.2.5. 

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The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

The model given above provides a description of the overall synthesis process. To reach the next level of detail, we may consider the reaction as comprising three stages based on the concepts of the Lowe equilibrium model [62,92] (section 5.1). For any given system, changes in the solid and solution species can be monitored by chemical and physical methods and are reflected in quite simple properties such as reaction pH [63]. Although there will be considerable variation between one system and another, a reasonable generalised mechanism can be proposed (Fig. 4) [50],... 

The kinetics of adsorption from solutions of surfactant mixtures are described on the basis of a generalised Langmuir isotherm. The simultaneous adsorption leads to the replacement of less surface active compounds by those of higher surface activity, which are usually present in the bulk at much lower concentration. More general descriptions of the process are possible on the basis of the Frumkin and Frumkin-Damaskin isotherms, which include specific interfacial properties of the individual surface active species. Quantitative studies of such very complex models can be performed only numerically. 

In Chapter 2, we introduced the concept of stochastic processes. Most but not all interest-rate models are essentially descriptions of the short-rate models in terms of stochastic process. Financial literature has tended to categorise models into one of up to six different types, but for our purposes we can generalise them into two types. Thus, we introduce some of the main models, according to their categorisation as equilibrium or arbitrage-free models. This chapter looks at the earlier models, including the first ever term structure model presented by Vasicek (1977). The next chapter considers what have been termed whole yield curve models, or the Heath-Jarrow-Morton family, while Chapter 5 reviews considerations in fitting the yield curve. 

A Markovian description can be naturally introduced by generalising deterministic systems modelled by ordinary differential equations. In other words, the stochastic version of a deterministic process without after-effect is a Markov process. 

Data-driven or black-box modelling, where a description of the process is obtained solely by developing models for the available data. This approach can provide very accurate models of the system at a given set of conditions, but the model cannot generalise well to other conditions. Furthermore, developing such models can be difficult, since the selection of appropriate terms and relevant data is a nontrivial task. Unless the correlations are strong, it may be difficult to decide on an appropriate data-driven model. 

We shall refer to the described models as the Curtiss-Bird model (Eq. (53)) and the Doi-Edwards model (Eq. (55)). There are different generalisations of these simple models, which were undertaken for the model to give a more accurate description of experimental evidence [64 - 70]. 

Figure 5.1 shows a generalised domestic and international supply chain model with information, material and financial flows. In this figure, each box with a black border represents supply chain members, and the uncertainty reasons (taken from interview data) are represented by the boxes with white borders. There are three types of flows information (line with square dot), material (solid line) and financial (long dash line) in the model. All information flow related uncertainties are represented by time uncertainties. All material flow related uncertainties are represented by quantity uncertainties. The financial flow related uncertainties are considered in the first objective function (cost). The activities in the early description can be categorised and consolidated into four sub-models that are represented in the SC simulation program. 

For oriented polymers it is shown that the value of n depends also on the drawing ratio [18,27]. Reduction in n with growth in A, is a general tendency. A considerable number of factors influencing the value of n makes its description within the frameworks of structnral and molecular models difficult. Therefore the authors of papers [34-36] generalised the influence of the indicated factors on the value of n with the application of fractal analysis methods in the example of uniaxially stretched PCP. Molecular characteristics of PCP crosslinked networks are adduced in Table 4.2. 

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