Lumped mechanical models

A good understanding of the detailed chemistry of oxidant formation makes it possible to construct more compact chemical models. These generalized or lumped mechanism models reduce the number of individual chemical reactions by combining similar or sequential reactions and ig-... 

Lumped mechanical models have been used to analyze impact dynamics. Generally, muscle is represented by a combination of a spring and a dashpot, whereas a ligament is modeled by a spring. Human body vibrations can be analyzed by similar models (Fritton et al., 1997). 

In the case of viscoelastic loaded QCM two approaches have been followed one methodology is to treat the device as an acoustic transmission line with one driven piezo-electric quartz layer and one or more surface mechanical load (TLM) [50, 51]. A simpler approach is to use a lumped-element model (LEM) that represents mechanical inter-actions by their equivalent electrical BVD circuit components [52, 53]. 

One can show [42] that, when the surface mechanical impedance is not large, the distributed model in the vicinity of resonance (where we make measurements) can be reduced to the simpler lumped-element model of Fig. 13.8(b). This modified Butterworth-van Dyke (BVD) electrical equivalent circuit comprises parallel static and motional arms. The static... 

N. de Nevers, Fluid Mechanics for Chemical Engineers, McGraw-Hill, New York, 1991, p. 386. E. Tronconi and P. Forzatti, Adequacy of lumped parameter models for SCR reactors with monolith structure, AIChE J. 38(2) 20 (1992). 

Accelerometers can be described using the lumped-parameter mechanical model shown in Fig. 7.1.2. 

Two a priori lumped kinetic models were discussed in Chapter VIII the pH, YH model for pyrolyses and the COX and COLE model for oxidations. These models are basic models as they have the morphology which must be obeyed by all detailed reaction mechanisms of pyrolysis and oxidation. 

As has already been seen, these a priori lumped kinetic models account for the macroscopic properties of the reactions, such as cool flames, the delays of autoignition... but without being able to relate them to the chemical structure of the reactants. In other words, they are incapable of describing both qualitatively and quantitatively the formation of individual molecules, such as the toxic substances (carbon monoxide, aldehydes, butadiene, aromatics, PAHs, soot) or the tropospheric pollutants (nitrogen and sulphur oxides, unburnt hydrocarbons, various oxygenated compounds), produced by the burning of fuels. There is therefore a strong requirement to develop detailed reaction mechanisms, likely to predict both the kinetic and chemical characteristics of these reactions. 

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

This paper presents a brief state of the art review of direct coal liquefaction. The review Includes Important pilot scale processes available for the liquefaction and a brief description of the structure of coal and the chemistry, mechanism and available lumped kinetic models for the liquefaction process. It also Includes some discussions on the role of catalysts during coal liquefaction and on the use of model compounds for the understanding of coal liquefaction kinetics. Reactor design aspects are covered In a separate paper and will not be repeated here. 

It is obvious that in spite of vast efforts being put forth on the development of various processes, the basic understanding needs further work. The areas of most importance are (a) analytical chemistry for the product distribution, (b) hydrogen transfer mechanism and (c) sophisticated lumped kinetic model. Future work needed for the reactor design is discussed by Shah and Gopal (106). 

Burd J, Noetzel V and Tamerius J (1993) Rapid testing of biomaterials for complement activation using in vitro complement immunoassays. 19 Annual Meeting of the Society of Biomaterials, Birmingham, Alabama Maines BH and Brennen CE (2005) Lumped parameter model for computing the minimum pressure during mechanical heart valve closure. J Biomech Eng 127 648-655... 

The structure of lumped process models depend on both the mechanisms taking place in the system and on the choice of input variables. Two practically important different cases are considered. 

Kinetic model is a key point to describe complex FTS reaction. Although much research has been done to deal with FTS, the kinetic model research stiU stay at lumped kinetic model (Anderson, 1956 Derosset et al., 1976 Feimer et al., 1981 Frohning and Comils, 1974 Kam et al., 1960 Thomas and Eckert, 1984), because the reaction mechanism of FTS is very complex containing series of surface reactions. The lumped kinetics can only explain syngas conversion rate, the distribution of products usually be described by the semiempirical model (like ASF model) (Wang et al., 1999). However, if the lumping kinetics use in the reactor simulate, only total conversion and distribution of temperature can be obtained. T able 2 (W ang et al., 1999) shows the lumped kinetics for fix-bed FTS on iron-based catalyst. 

In the following sections, this behavior is described by electrical and mechanical models separately and finally combined to an electromechanical model consisting of lumped parameters. 

Fig. 12 Combined electrical and mechanical lumped parameter model showing the full signal transduction from electrical stimuli to mechanical output. Calculation method for Pe w) may be chosen from Eqs. 38 or 40, according to Sect. 3.2.1 in this chapter...

The description of respiratory mechanical system is based on lumped-parameter model [10] of Fig. 2 for the relation between respiratory neural P(t) and the mechanical outputs V(t), V(t) and the dynamical equation is given by ... 

Figure 2.41a shows a lumped circuit model of an arc horn flashover proposed by Shindo and Suzuki [29]. Inductance and resistance values and closing times of switches are determined from a given voltage wave-form across the arc horn gap based on a theory of a discharge mechanism. 

An early application of this idea was used by Dunker (1986), who applied it to the modelling of tropospheric ozone formation. He started from a lumped mechanism containing 47 species and identified 10 parameterising variables (species concentrations or functions of species concentrations). All in aU, 20,736 grid points... 

The catalytic hydrocracking of heavy oil has been well represented by the five-lump kinetic model shown in Figure 6.15 (Sdnchez et al., 2005). Although catalytic and thermal reactions follow different mechanisms, the same kinetic model was used to represent the NHDC. Hydrocracking of vacuum residue was assumed to follow second order as demonstrated earlier, while first order was considered for the other reactions. The reaction rate (r for each lump as a function of the product composition (y and the corresponding kinetic constant k is as follows ... 

All these steps can influence the overall reaction rate. The reactor models of Chapter 9 are used to predict the bulk, gas-phase concentrations of reactants and products at point (r, z) in the reactor. They directly model only Steps 1 and 9, and the effects of Steps 2 through 8 are lumped into the pseudohomoge-neous rate expression, a, b,. ..), where a,b,. .. are the bulk, gas-phase concentrations. The overall reaction mechanism is complex, and the rate expression is necessarily empirical. Heterogeneous catalysis remains an experimental science. The techniques of this chapter are useful to interpret experimental results. Their predictive value is limited. 

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