Lumped parameter reactor

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. 

In this work, the characteristic "living" polymer phenomenon was utilized by preparing a seed polymer in a batch reactor. The seed polymer and styrene were then fed to a constant flow stirred tank reactor. This procedure allowed use of the lumped parameter rate expression given by Equations (5) through (8) to describe the polymerization reaction, and eliminated complications involved in describing simultaneous initiation and propagation reactions. 

In some cases, where the wall of the reactor has an appreciable thermal capacity, the dynamics of the wall can be of importance (Luyben, 1973). The simplest approach is to assume the whole wall material has a uniform temperature and therefore can be treated as a single lumped parameter system or, in effect, as a single, well-stirred tank. 

The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999 Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen... 

Flow patterns in a stirred tank (lumped parameter system) and a tubular reactor (distributed parameter system). 

The PSR is a lumped parameter system, where the temperature is uniform over the entire reactor. As a result, the fuel and NOj, emissions are strongly temperature dependent. As extinction is approached in the PSR, the radical mole fractions decrease sharply, and so do the NO, species. Thus, turbulent mixing in a PSR is responsible for the high sensitivity of NOj, to temperature. 

Bailey, J. E., 1977, Periodic phenomena in chemical reactor theory. In Chemical Reactor Theory (Edited by Lapidus, L. and Amundson, N. R.). Prentice-Hall, Englewood Cliffs, NJ. Balakotaiah, V. and Luss, D., 1982, Structure of the steady state solutions of lumped parameter chemical reacting systems. Chem. Engng Sci. 37, 1611-1623. 

Tronconni E, Forzatti P. Adequacy of lumped parameter models for SCR reactor with monolith structure. AIChE J 1992 38 201-210. 

Mathematical modeling of systems for which characteristic variables are time-dependent only and not space-dependent is done by ordinary differential equations (ODEs). The situation is found in a nearly well-mixed batch reactor. There one may find differences in temperature or concentrations from one site to another due to imperfect mixing. When space changes are not important to the model, the process variables can be approximated by means of lumped parameter models (LPMs). When the... 

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). 

Stability, on the other hand, is strongly connected with the multiplicity phenomenon as discussed in chapter 4 for simple lumped parameter systems. In fixed bed catalytic reactors, the situation is much more complicated and may give rise to extremely complex behaviour. A relatively simple and practically oriented discussion of the problem is given by McGreavy (1984). 

Optimization requires that at/R have some reasonably high value so that the wall temperature has a significant influence on reactor performance. There is no requirement that 33 jR be large. Thus the method can be used for polymer systems that have thermal diffusivities typical of organic liquids but low molecular diffusivi-ties. The calculations needed to optimize distributed-parameter systems (i.e., sets of PDEs) are much longer than needed to optimize the lumped-parameter systems (i.e., sets of ODEs) studied in Chapter 6, but the numerical approach is the same and is still feasible using small computers. 

Illustrate temperature and molecular weight changes in a tubular reactor by constructing a lumped-parameter model of styrene polymerization in a tube. 

Analogous to the experimental approaches discussed in the previous section, mathematical models have been developed to describe mass transfer at all three levels—cellular, multi-cellular (spheroid), and tissue levels. For each level two approaches have been used—the lumped parameter and distributed parameter models. In the former approach, the region of interest is considered to be a perfectly mixed reactor or compartment. As a result, the concentration of each region has no spatial dependence. In the latter approach, a more detailed analysis of the mass transfer process leads to information on the spatial and/or temporal changes in concentrations. Models for single cells and spheroids were reviewed in Section III,A and are part of the tissue-level models (Jain, 1984) hence, we will focus here only on tissue-level models. 

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