Number of lumps

Only digital simulation solutions for ordinary differential equations are presented. To present anything more than a very superficial treatment of simulation techniques for partial differential equations would require more space than is available in this book. This subject is covered in severd texts. In many practical problems, distributed systems are often broken up into a number of lumps which can then be handled by ordinary differential equations. 

Al number of lumped chemical reactions Am number of measured variables... 

Effect Of Number Of Lumps If the number of plotting points in Aspen Plus is set at 10 (the default), the resulting exit temperature from the reactors under steady-state conditions in Aspen Dynamics is 578 K. Remember that it should be 583 K from the rigorous integration of the ordinary differential equations describing the steady-state tubular reactor that are used in Aspen Plus. Changing the number of points to 20 produces an exit temperature of 580 K. Changing the number of points to 50 produces an exit temperature of 582 K, which is very close to the correct value. Therefore a 50-lump model should be used. 

However, some numerical difficulties are sometimes encountered in running the simulation in Aspen Dynamics when a large number of lumps are used. The 50-lump case runs very slowly or not at all in the adiabatic reactor cases. In this situation the number of lumps is reduced to get reasonable computing times. 

Reactor and Gas Loop Model The number of lumps used is determined by comparing the rigorous steady-state results with models using different numbers of lumps. The rigorous steady-state reactor exit temperature is 500 K with a 460 K inlet reactor temperature. When a 10-lump model is used with the same amount of catalyst, the steady-state exit temperature is 505 K. This occurs because of the numerical diffusion or effective back-mixing that is inherent with a lumped model. A 50-lump reactor model is used in all the simulations. It gives a steady-state reactor exit temperature of 501 K. The differences between the dynamic responses of the 10-lump and 50-lump reactors are shown in the next section. 

The heat transfer area in each lump is the total area divided by the number of lumps. The overall heat transfer coefficient used in the steady-state design calculations was... 

The problem when trying to make an electrical model of the physical or chemical processes in tissue is often that it is not possible to mimic the electrical behavior with ordinary lumped, physically realisable components such as resistors (R), capacitors (C), inductors, semiconductor components, and batteries. Let us mention three examples 1) The constant phase element (CPE), not realizable with a finite number of ideal resistors and capacitors. 2) The double layer in the electrolyte in contact with a metal surface. Such a layer has capacitive properties, but perhaps with a capacitance that is voltage or frequency dependent. 3) Diffusion-controlled processes (see Section 2.4). Distributed components such as a CPE can be considered composed of an infinite number of lumped components, even if the mathematical expression for a CPE is simple. 

Fill a clear ass bowl nearly full of water, and place on the surface a number of lumps of camphor of nneqnal size, so arranged as to take the form of some grotesque... 

It is hardly surprising that model developers have been compelled to drastically simplify model development along two lines. One is what may be called partition-based lumping, while the other total lumping. In the former case, the reaction mixture is represented by a finite number of lumps and the reactions among them are tracked. The lumped system aims to capture essential features of the real system so that it has sufficient predictive power and robustness over ever-changing feeds and catalysts. In the latter case, the... 

This traditional approach starts with a number of preselected, measurable kinetic lumps and determines the best reaction network and kinetics through experimental design and parameter estimation. The number of lumps depends on the level of detail desired. The lumps, satisfying the conservation law and stoichiometric constraints, are usually selected based on known chemistry, measurability and physicochemical properties (boiling range, solubility, etc.). 

Mosby et al. reported a seven-lump residue hydroconversion model. Lumped models for steam cracking of naphtha and gas oils can be found in Dente and Ranzi s review. The literature aboimds with FCC kinetic models, with the number of lumps being three four °, five, six , eight, ten °. 

We classify kinetic models according to the chemical entities that makeup the model. Typically, the entities or lumps are boiling point lumps or yield lumps, grouped chemical lumps and full chemical lumps. Early kinetic models consist entirely of yield lumps, which represent the products that refiner collects from the main fractionator following the FCC unit Figure 4.4 shows a typical kinetic model based on yield lumps by Takatsuka et al. [9]. Many similar models have appeared in the literature. The models differentiate themselves based on their number of lumps. Models may contain as few as two [10] or three lumps [11] and as many as fifty lumps [12]. We note that models with more lumps do not necessarily have more predictive capabilities than models with fewer lumps [6]. 

Several kinetic models based on Inmping technique have been reported in the literature, which have been derived from different feeds, experimental setups, and reaction conditions. The approaches involve parallel reaction models and parallel-consecutive reaction models. The kinetic models can be classified according to the number of lumps involved, as follows (Joshi et al., 2008) ... 

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