Lumped Analysis

If the step is only equilibrium controlled, a lumped analysis of both fluid and solid phase is possible and the corresponding nonlinear wavefront analysis is straight forward too. Analysis of propagation speed data yield information about the relevant equilibrium sorption mechanism (7). The different form of the transients when a reaction step has been stimulated, is discussed in the Appendix. 

Zero-one law for relaxation modes (eigenvectors) and lumping analysis... 

We summarize results of relaxation analysis and describe the algorithm of approximation of steady state and relaxation in Section 4.3. After that, several examples of networks are analyzed. In Section 5 we illustrate the analysis of dominant systems on a simple example, the reversible triangle of reactions A2<- A3 A. This simplest example became very popular for the lumping analysis case study after the well-known work of Wei and Prater (1962). The most important mathematical proofs are presented in the appendices. 

Many examples. Equations (6), (38), (51), (54), demonstrate that some of z, can include the same concentrations it may be that V n Vy 0 for some i j. Aggregation of some components (possibly with some coefficients) into new group components for simplification of kinetics is the major task of lumping analysis. 

Kinetics of multiscale systems studied in this chapter and developed theory of dynamic limitation demonstrates that in multiscale limit lumping analysis can work (almost) exactly. Lumped concentrations are sums in groups, but these groups can intersect and usually there exist several intersections. 

Wei s later work with J. C. Kuo on lumping and that of Li and Rabitz at Princeton [Chem. Eng. Sci. 44,1413, (1959)] also opened up an important area of formal kinetics. See also Bischoff and Coxson, Lumping strategy. l.E.C. Res. 26,1329 2151, (1983). J. E. Bailey [Lumping analysis of reactions in continuous mixtures. Chem. Eng. J. 3, 5261 (1962)]. forged an important link between continuous mixtures and lumps. 

Aside from the original assumption of a lumped analysis, thus far there have been no other assumptions or approximations to the model. The model relies completely on basic thermodynamic principles, a known cell performance R(I), and rigorous mathematical operations. To solve the model, we need to know the bulk mass and heat capacity of the cell, M and C, respectively the reactant supply flow rate (m = fuel flow + air flow) the inlet temperature and pressure and the change in stream composition due to the electrochemical reaction, AX, so that the change in enthalpy can be calculated the electrical load current, / and the inlet and exit temperatures, Tm and rout. 

The next level of transient modeling improves the spatial accuracy beyond the lumped analysis given in prior sections by breaking up the original control volume that previously spanned the entire cell into numerous additional control volumes, or nodes (see Figure 9.8). However, forpurposes of computational efficiency, it is noted that once the number of computational nodes becomes greater than 10, additional... 

When the Biot number is small, Bi < 0.2, the temperature of the solid is nearly uniform and a lumped analysis is acceptable. The solution to the lumped analysis of (5-18) is... 

When a lumped analysis is not valid (Bi > 0.2), the single-term solutions to (5-18) are convenient ... 

Liu, Y. A., and Lapidus, L., Observer theory for lumping analysis of monomolecular reactions systems. AIChEJ. 19,467 (1973). 

Cotta, R. M., Mikhailov, M. D., Heat Conduetion Lumped Analysis, Integral Transforms, Symbolie Computation, (1997) John Wiley Sons, New York. 

Cotta, R.M., Mid Nhldiailov, M.D. (1997) Heat Conduction Lumped Analysis, Integral Transforms, Symbolic Computation, Wiley-Interscience, Qiichester, UK. 

Dennison JE, Andersen ME, Clewell HJ, Yang RSH. Development of a physiologically based pharmacokinetic model for volatile fractions of gasoline using chemical lumping analysis. Environ Sci Technol 2004 38 5674-81. 

Verhaar HJM, Morroni JR, Reardon KF, et al. 1997. A proposed approach to study the toxicology of complex mixtures of petroleum products The integrated use of QSAR, lumping analysis and PBPK/PD modeling. Environ Health Perspect 105 179-195. 

Assume the radial Biot numbers are small enough to allow a radially lumped analysis. Neglecting the heat loss from the ends and ignoring, for the time being, conduction across the interface, the energy generated at each part is balanced by the convective heat loss to the water and air. Thus,... 

A constant property fluid having velocity V and upstream temperature To flows steadily through an infinitely long tube of cross sectional area A and periphery P. The upstream half of the tube is insulated, while the downstream half either transfers heat with a coefficient h to an ambient at temperature T,x [Fig. 2.38(a)] or is subjected to a peripheral heat flux q" [Fig. 2.38(b)]. The wall thickness of the tube is negligible. Based on a radially lumped analysis, we wish to know the axial temperature distribution in the fluid. 

Neglecting the effect of axial conduction, considering a radially lumped analysis, and assuming a peripherally uniform q", this formulation [obtained by an appropri-... 

In the case of C02, system was broken down into muscle and nonmuscle tissue compartments with unsteady lumped analysis being applicable in each. For 02, only one tissue compartment was employed. 

---