Multi-zone models

To account to a certain extent for the effects of macro-mixing in the species balance equations, so-called multi-zone models have been developed. The reactor is divided in multiple, not necessarily geometrical, zones and, within each zone, an idealized mixing or flow pattern is assumed in the derivation of the species... 

Similar to RTD-based models (Section 12.6), multi-zone models do not explicitly account for the effects of micro-mixing. The latter may affect macromixing, that is, the transport rates between zones. Micro-mixing may also affect the averaged reaction rates at the scale of the zones or the macro-scale, as a result of non-uniformities within the individual zones. Micro-mixing effects can eventually be accounted for by the methods discussed in Sections 12.5 and 12.4. 

A multi-zone model similar to the finite volume methods used in Computational Fluid Dynamics (CFD, see Section 12.5) is based on geometrical zones and assumes uniform conditions within each zone, that is, completely mixed (CSTR) zones, while essentially allowing conditions to differ between zones. For statistically stationary flow, the species balance equations for species A in zone i of iVcan then be written as ... 

The transport rates Fin (12.7.1-3) can be measured experimentally or calculated from CFD simulations (see Section 12.5). With respect to the latter, multi-zone models may offer some advantages compared to a fully CFD-based approach in case the scale of the non-uniformities in the species concentration field is larger than the scale of the non-uniformities in the velocity field. This allows a coarser grid for the species concentration field calculation than for the velocity field calculation, that is, less zones than CFD finite volumes, and significantly reduces the computational effort. 

Pelletret R. COMES V3.0 A new simulation environment for multi-zone air flow modelling. RoomVent 96 conference, Tokyo, May 1996. 

Modification of catalyst introduction into reaction zone in particular realization of multi-step catalyst introduction into tubular turbulent apparatus ("zone" model) is the convenient way of polymer MM and MMD control [62]. For "zone" model realization in turbulent regime the following conditions should be fulfilled [1, 27, 66] ... 

In our present work, we want to find out the suitability of CFAST in fire simulation for fire safety onboard ship, which is a multi compartment zone model, comparing the results with CFX- a general purpose CFD... 

B. Agostini, J. R. Thome, Comparison of an extended database of flow boiling heat transfer coefficient in multi-microchannel elements with the three-zone model, in ECI International Conference on Heat Transfer and Fluid Flow in Microscale, 2005, Castelvecchio Pascoli, Italy. 

Clarke,B.L., 1974. Stability analysis of a model reaction network using graph theory. J.Chem.Phys. 60,1493-1501. Constales, D., Yablonsky, G.S., Marin, G.B., Gleaves, J.T., 2001. Multi-zone TAP-reactors theory and application I. 

MULTI-ZONE, TANKS-IN-SERIES AND AXIAL DISPERSION MODELS... 

Multi-zone, Tanks-in-Series, and Axial dispersion models (Fig. 12.3-1 F) Other, less fundamental approaches accounting for mixing limitations in reactors are described in Section 12.7. They are based on simplified descriptions of the mixing pattern, e.g., a ID axial dispersion approach, or on the decomposition of the complex flow reactor into multiple interconnected regions or zones, each of these being described by a different idealized mixing pattern. Such semi-empirical models contain model parameters which have to be determined, experimentally or a posteriori from PDF, CFD, or RTD data. 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

Figure 4 shows the isotopic anomalies of the iron peak elements predicted by the multi-zone mixing model as compared with the average excesses as observed in Ca-Al-rich inclusions. The match between the two data sets is impressive, except for Fe and Zn. In the case of Fe no significant anomalies have been measured, but the multi-zone mixing model only predicts a Fe excess of approximately 1 part in 10", which is at the limit of present mass spectrometric capability. In the case of Zn, the excess in Zn is approximately an order of magnitude less than that expected. This can be explained in terms of the volatility of Zn with respect to the other iron peak elements, as it would be the last of these elements to condense from the expelled stellar material. The correlation between anomalies in neutron-rich isotopes in the iron peak elements can therefore be explained in terms of the nuclear statistical equilibrium processes, which took place at a late stage in the evolution of massive stars. 

Figure 4 (A) Isotopic excesses of several iron peak elements as predicted by the multi-zone mixing model. They are given as e units (parts in 10 ) relative to normal terrestrial isotopic composition. (B) Average excesses observed in normal Allende inclusions are displayed.

Zone models These are applied to fire situations in partially enclosed volumes i.e. allowance is made for the presence of doors, windows, and vents in the problem definition. These models can be equally well applied to both single and multi-room configurations. A particular distinction is made between a lower fire plume and an upper smoke layer within the room (s). 

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