Model lumping

At lower Reynolds numbers, the axial velocity profile will not be flat and it might seem that another correction must be added to Equation (9.14). It turns out, however, that Equation (9.14) remains a good model for real turbulent reactors (and even some laminar ones) given suitable values for D. The model lumps the combined effects of fluctuating velocity components, nonflat velocity profiles, and molecular diffusion into the single parameter D. 

A relatively simple mathematical model composed of 21 or 23 transcendental and rational equations numbered (7.25) to (7.47) was presented to describe the steady-state behavior of type IV FCC units. The model lumps the reactants and products into only three groups. It accounts for the two-phase nature of the reactor and of the regenerator using hydrodynamics principles. It also takes into account the complex interaction between the... 

In this section we examine the primary transient phenomena that are of interest to SOFC analysis, and provide the fundamental model equations for each one. Examples for the use of these models are given in later sections. While the focus is on reduced-order models (lumped and one-dimensional), depending on the needs of the fuel cell designer, this may, or may not be justifiable. Each fuel cell model developer needs to ensure that the solution approach taken will provide the information needed for the problem at hand. For the goal of calculating overall cell performance, however, it is often that one-dimensional methods such as outlined below will be viable. 

Homogeneous and heterogeneous models Lumped and distributed models Rigorous (relatively rigorous) and empirical models Linear and nonlinear models... 

Minimality for nonlinear state-space models Lumped process models with differential index 1 can be described by input-affine state-space models, where the notion of minimality is related to the number of state variables necessary and sufficient to produce a given input-output behaviour. This means, that one naturally uses the single quality index = dim x = n for both linear and nonlinear state-space models and orders the functionally equivalent models accordingly. 

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 °. 

Catalyst poisoning is generally regarded as any process by which the number of catalyst active sites are reduced, and can occur due to sulfur coverage of active nickel sites, sintering of the nickel into fewer active sites, or coke deposition. The model lumps all those effects into the activity parameter. 

In the case of washout of sulfur dioxide (SO2), a precursor of acid rain, the high solubility and the chemical reactivity of aqueous SO2 result in nonattainment of equilibrium. Thus, semiempirical models have been proposed for the SO2 concentration in air beneath the rain-forming cloud. These models lump together all processes affecting SO2 removal from air (i.e., dissolution into water droplets, hydration, oxidation, and ionization). A first-order decay constant for the SO2 concentration. A, varies with the rainfall characteristics (rainfall rate and size of raindrops). Boubel et al. (1994) suggest a value of A equal to... 

Fig. 8.11 Decalin transformations on metal -modified zeolites A) Reaction scheme and B) chemical structures of the starting material and representative components of the main modeled lumps decalin (D), 2-D skeletal isomers of decalin (2-D), 3-D skeletal isomers of decalin (3-D), ring opening products (ROPs) and cracking products (CPs). (Reprinted with permission from M. Kangas, D. Kubicka, T Salmi, D.Yu. Murzin, Reaction routes in selective ring opening of napthenes. Top. Catal. 53 (2010) 1172—1175. Copyright 2010 Springer).

Radial density gradients in FCC and other large-diameter pneumatic transfer risers reflect gas—soHd maldistributions and reduce product yields. Cold-flow units are used to measure the transverse catalyst profiles as functions of gas velocity, catalyst flux, and inlet design. Impacts of measured flow distributions have been evaluated using a simple four lump kinetic model and assuming dispersed catalyst clusters where all the reactions are assumed to occur coupled with a continuous gas phase. A 3 wt % conversion advantage is determined for injection feed around the riser circumference as compared with an axial injection design (28). 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

The reaction network is shown in the paper. The kinetic characteristics of the lumps are proprietary. Originally, the model required 30 person-years of effort on paper and in the laboratory, and it is kept up to date. 

It is not feasible to model the reaction of each hydrocarbon species with oxides of nitrogen. Therefore, hydrocarbon species with similar reactivities are lumped together, e.g., into four groups of reactive hydrocarbons olefins, paraffins, aldehydes, and aromatics (32). 

This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using matliematical expressions show how physical... 

The PC version runs comparatively slow on large problems. FIRAC can perform lumped parameter/control volume-type analysis but is limited in detailed multidimensional modeling of a room or gaa dome space. Diffusion and turbulence within a control volume is not modeled. Multi-gas species are not included in the equations of state. 

EXP AC analyzes an interconnected network of building rooms and ventilation systems. A lumped-parameter formulation is used that includes the effects of inertial and choking flow in rapid gas transienl.s. The latest version is specifically suited to calculation of the detailed effects of explosions in the far field using a parametric representation of the explosive event. A material transport capability models the effects of convection, depletion, entrainment, and filtration of... 

Rawlings etal. (1992) analysed the stability of a eontinuous erystallizer based on the linearization of population and solute balanee. Their model did not depend on a lumped approximation of partial differenee equations and sueeess-fully predieted the oeeurrenee of sustained oseillations. They demonstrated that simple proportional feedbaek eontrol using moments of CSD as measurements ean stabilize the proeess. It was eoneluded that the relatively high levels of error in these measurements require robust design for effeetive eontrol. 

Figure 2-75. Lumped element models of transmission line electrical characteristics.

However, in the case of the electronic orbital model there is no way in which the inter-electronic repulsions can be physically reduced. This form of distinction has not been sufficiently emphasized by philosophers. I believe that the nature of the orbital model shows that not all theoretical models can be lumped together as in the work of Achinstein [1968]. 

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