Lumped systems defined

The dimensionless quantities defined above for a plane wall can also be used for a cylinder or sphere by replacing the space variable x by r and the half-thickness L by the outer radius r. Note that the characieristic length in the definition of the Biot number is taken to be the half-thickness L for the plane wall, and the radius for the long cylinder and sphere instead of 1//A used in lumped system analysis. 

Discussion Note that the Biot number in lumped system analysis was defined differenfjy.as Bi = hLJk= h rJ3)lk. However, either definition can be used in determijiifig the applicability of the lumped system analysis unless Bi = 0.1. 

Cutting these explanations in view of the limited space short and excluding distributed capacities, we define a simple, lumped system by the following components ... 

The very basis of the kinetic model is the reaction network, i.e. the stoichiometry of the system. Identification of the reaction network for complex systems may require extensive laboratory investigation. Although complex stoichiometric models, describing elementary steps in detail, are the most appropriate for kinetic modelling, the development of such models is time-consuming and may prove uneconomical. Moreover, in fine chemicals manufacture, very often some components cannot be analysed or not with sufficient accuracy. In most cases, only data for key reactants, major products and some by-products are available. Some components of the reaction mixture must be lumped into pseudocomponents, sometimes with an ill-defined chemical formula. Obviously, methods are needed that allow the development of simple... 

In whole tissue or cell monolayer experiments, transcellular membrane resistance (Rm = Pm1) lumps mucosal to serosal compartment elements in series with aqueous resistance (R = P ). The operational definition of Lm depends on the experimental procedure for solute transport measurement (see Section VII), but its magnitude can be considered relatively constant within any given experimental system. Since the Kp range dwarfs the range of Dm, solute differences in partition coefficient dominate solute differences in transcellular membrane transport. The lumped precellular resistance and lumped membrane resistance add in series to define an effective resistance to solute transport ... 

As described in Chapter 1, the first term on the left-hand side describes the kinetic energy of the electron, V is the potential energy of an electron interacting with the nuclei, VH is the Flartree electron-electron repulsion potential, and Vxc is the exchange-correlation potential. This approach divides electron-electron interactions into a classical part, defined by the Flartree term, and everything else, which is lumped into the exchange-correlation term. The Flartree potential describes the Coulomb repulsion between the electron and the system s total electron density ... 

The term phase defines any homogeneous and physically distinct part of a system which is separated from other parts of the system by definite bounding surfaces. For example, ice, liquid water, and water vapor are three phases. Each is physically distinct and homogeneous, and there are definite boundaries between ice and water, between ice and water vapor, and between liquid water and water vapor. Thus, we say that we have a three-phase system solid, liquid, and gas. One particular phase need not be continuous. For instance, the ice may exist as several lumps in the water. 

A better alternative approach is what will be called the Rudolph method [476], after the person who introduced it into electrochemical simulation. It was known before 1991 under various names, notably block-tridiagonal [280,412,470,471,528,570]. This comes from the fact that if one lumps the large matrix into a matrix of smaller matrices and vectors, the result is a tridiagonal system that is amenable to more efficient methods of solution. In the present context, we define some vectors... 

The description of reaction kinetics for circumstances where an exceptionally large number of species are involved or where well-defined molecular entities cannot be specified poses a particularly difficult challenge. Such situations can arise, for example, in the cracking of petroleum, the pyrolysis of biomass, and the liquefaction of coal. The best way to treat such systems is to either lump classes of reactions or follow the dynamics of functional groups rather than specific molecular species. 

So far, lumping has been defined, but nothing has been said concerning the dynamic behavior of the" system. Now we come to the definition of exact lumping A system is said to be exactly lumpable by the matrix L if there exists an N X N matrix K, enjoying the same properties as K does (i.e., off-diagonal elements of K are nonpositive, = 0, and there exists at least an m" =... 

Now suppose that one still wants to lump the system into a finite, A -dimensional one. This can be achieved by defining an A -dimensional set of Lr(x) distributions, L(t), such that... 

First, we define the necessary and sufficient conditions for exact nonlinear lumping. If we start with the system of equations (4.1) then the new lumped variables are defined by an fl-dimensional non-linear transformation c = h(c) and the new n-dimensional equation system becomes... 

We can see now that there are lumped concentrations of the reaction system and can define a new set of variables c where... 

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