Detailed modelling purpose

It is fundamentally important that the different COD fractions in wastewater be quantified and determined by direct measurement methods. The number of fractions must be minimized, determined by the details desirable and required, for example, for modeling purposes. 

To finally conclude, it can be stated that the PET market will grow rapidly in the next few years, even without the detailed knowledge we would prefer to have for modelling purposes. However, neither an optimum plant performance nor an optimum economy will be achieved without reliable and predictive process models. 

More detailed modeling exposes some weaknesses, for example, the need to use coarse monolithic catalyst structure to achieve reasonable reheat periods without excessive pressure drops. The cycle time of approximately 4 minutes is somewhat short for practical purposes. The cold spot formation in the reaction phase (Figure 14) and the resultant inability of the reaction to distribute itself over the catalyst to utilize the stored heat optimally is probably a modeling artefact caused by too low literature values for the activation energy, possibly reflecting an incorrectly interpreted film transport limitation. 

In this section, the phenol-formaldehyde reactive system is considered as an example of identification of reduced kinetic models. The kinetic model containing 13 components and 89 reactions, developed in Sect. 2.4 to study the production of 1,3,5-methylolphenol, is too detailed and complex for control and monitoring purposes. Thus, in this section this model is referred to as detailed model, while four reduced kinetic models, based on lumped components and reactions, are developed. 

Depending on the purpose of the model and the status of the molecular knowledge of the network, either simplified kinetic core models or kinetic detailed models can be constructed. Core models are most useful for showing principles of regulation or dynamics [5, 37, 41, 63-66] or to study a network in more phenomenological terms when it is poorly characterized [67-70]. In what follows we only consider detailed kinetic models. 

A detailed model for a gas-liquid column with external recirculation loop has been published by Orejas [11]. The model takes into account the axial dispersion and mass transfer from bubbles. An important conclusion is that the mass-transfer rate is fast compared with the chemical reaction. As a result, a pseudohomoge-neous model for liquid-phase reaction may be applied for design purposes. 

For technical and engineering purposes, a detailed reaction mechanism and a detailed model is usually not required. A formal kinetic model describing the time dependent evolution of the concentration with a mathematical function is fitted to the obtained experimental data. In such cases, the variables k and n do not have a real physical meaning based on the mechanism of the reaction. 

Two level factorial designs are primarily useful for exploratory purposes and calibration designs have special uses in areas such as multivariate calibration where we often expect an independent linear response from each component in a mixture. It is often important, though, to provide a more detailed model of a system. There are two prime reasons. The first is for optimisation - to find the conditions that result in a maximum or minimum as appropriate. An example is when improving die yield of synthetic reaction, or a chromatographic resolution. The second is to produce a detailed quantitative model to predict mathematically how a response relates to die values of various factors. An example may be how the near-infrared spectrum of a manufactured product relates to the nature of the material and processing employed in manufacturing. 

Mathematical modeling attempts to connect constructions made in the experiential world with those made in the world of ideas. Since we cannot perceive with infinite detail, we cannot measure with infinite resolution, and we cannot compute in the experiential world with infinite precision, there exists a gap between any system constructed in terms of experiential distinctions and a mathematical system that attempts to model it. While this gap cannot be fully avoided, it can be at least partially bridged by choosing appropriate mathematical constructions for our modeling purposes. 

The commercial process simulators contain a range of distillation models with different degrees of sophistication. The design engineer must choose a model that is suitable for the purpose, depending on the problem type, the extent of design information available, and the level of detail required in the solution. In some cases, it may make sense to build different versions of the flowsheet, using different levels of detail in the distillation models so that the simpler model can be used to initialize a more detailed model. 

The purpose of the next task is thus to derive an expression for the turbulent energy, k, in terms of v and one empirical parameter only. Again, the mathematical operations involved are outlined briefly in an easy to understand manner, thereafter the detailed model derivation is given. 

The widespread occurrence in process plant of nozzles, intentional and unintentional, means that it is necessary for the control engineer to have an understanding of their physical principles for modelling purposes. We will consider in this chapter the features of an ideal, frictionless nozzle. Results from this simplified model will be put to immediate use in Chapter 6 in order to account for the unavoidable nozzle formed at beginning of a gas pipe. Moreover, the idealized approach provides the groundwork for the more complex models of Chapter 14, where turbine nozzles with friction are considered in some detail. 

The engineer engaged in dynamic modelling will very often find it useful to produce a simplified, analytically linearized model of an important part of the plant in addition to his main detail model. Such a linearized model serves two important purposes ... 

Two of the best-known models used for this purpose are the Elsasser and the statistical Goody models, both of which employ the Lorenz profile for description of individual line shapes. These models give very accurate predictions over a bandwidth of approximately 50 cm"1, which is considered narrow for most practical purposes. (At X = 1 pm, this bandwidth translates to about AX = 0.05 pm.) Because of this, the model is called the narrowband model. Although this technique is significantly simpler than the line-by-line models, it still requires an extensive database about the species considered and significant computational effort. Such a detailed model can be considered useful only if the species concentration distribution is known very accurately, which is usually not the case. 

Appendix B describes in detail the Specific Ion Interaction Theory, which is the model selected in the NEA-TDB review series to describe the ionic interactions between components in aqueous solutions. This allows the general and consistent use of the selected data for modelling purposes, regardless of the type and composition of the ground water. 

The purpose of this annex is to build a detailed model of the measurement process based on the concepts introduced in previous sections of the chapter. The mathematical model improves understanding and more clearly demonstrates how the various concepts relate to each other. In an ideal world all measurements would be obtained free of variation. In the real world, all measurements are perturbed to some degree by a system-of-causes that produces error or variation in the output of the instruments or machines used for the testing. There are two general variation categories for any system. These categories are defined by the character and source of deviations that perturb the observed values compared to what would be obtained under ideal conditions. 

Based on thermodynamical and kinetic descriptions of the individual process steps, a meta-model can be developed which is able to describe and predict the behaviour of a whole chemical production process. Such a process model can be developed for different purposes and at different levels of detail To design a chemical production process, a detailed model of the potential plant(s) necessarily includes the description of the system s dynamics. In contrast, once the production process is designed, a model is necessary to describe the dependency of the system s output w.r.t. certain control parameters. Figure 2.6 depicts a prototypical procedure in chemical process modelling. 

Such a procedure of treating substituent effects has several important features (a) As it is related to a detailed model it reveals the limitations of the proposed methods lucidly and avoids misinterpretations of substituent effects (sets of numerical values) resulting from purely numerical data-fitting processes which start from simple formulas (b) for numerical purposes the method may be extended successively that is, new parameters may be evaluated without the necessity to readjust the old ones (the set of numerical values of the parameters is not related to a restricted set of molecules which when subjected to a least-squares fit shall reproduce the experimental values for this given set). 

Diffusion is the mass transfer caused by molecular movement, while convection is the mass transfer caused by bulk movement of mass. Large diffusion rates often cause convection. Because mass transfer can become intricate, at least five different analysis techniques have been developed to analyze it. Since they all look at the same phenomena, their ultimate predictions of the mass-transfer rates and the concentration profiles should be similar. However, each of the five has its place they are useful in different situations and for different purposes. We start in Section 15.1 with a nonmathematical molecular picture of mass transfer (the first model) that is useful to understand the basic concepts, and a more detailed model based on the kinetic theory of gases is presented in Section 15.7.1. For robust correlation of mass-transfer rates with different materials, we need a parameter, the diffusivity that is a fundamental measure of the ability of solutes to transfer in different fluids or solids. To define and measure this parameter, we need a model for mass transfer. In Section 15.2. we discuss the second model, the Fickian model, which is the most common diffusion model. This is the diffusivity model usually discussed in chemical engineering courses. Typical values and correlations for the Fickian diffusivity are discussed in Section 15.3. Fickian diffusivity is convenient for binary mass transfer but has limitations for nonideal systems and for multicomponent mass transfer. 

For particular cases, it maybe required to add more complex phenomena with additional effects or more evolved descriptions of the same mechanisms. In general, however, reduced models are appropriate and desirable. Historically, this stemmed from the shorter computational effort and time required for the numerical solution of such models. Today this is also an advantage for optimization, control, and real-time simulation applications, and reliable simplified models are still used for almost all purposes due to the lower number of dimensionless parameters requiring estimation and to the success found in the description of experimental results. On the other hand, complex detailed models fulfill the most generic purpose of reactor simulation, which is related to the prediction of the actual behavior from fundamental, independently measured parameters. Therefore, it is important to understand the equivalence and agreement between both detailed and reduced models, so as to take advantage of their predictive power without unnecessary effort. 

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