The Effective Transport Concept

The one-dimensional models discussed so far neglect the resistance to heat and mass transfer in the radial direction and therefore predict uniform temperatures and conversions in a cross section. This is obviously a serious simplification when reactions with a pronounced heat effect arc involved. For such cases there is a need for a model that predicts the detailed temperature and conversion pattern in the reactor, so that the design would be directed towards avoiding eventual detrimental overtemperatures in the axis. This then leads to two-dimensional models. 

The model discussed here uses the effective transport concept, this time to formulate the fiux of heat or mass in the radial direction. This flux is superposed on the transport by overall convection, which is of the plug flow type. Since the effective diffusivity is mainly determined by the flow characteristics, packed beds are not isotropic for effective diffusion, so that the radial component is different from the axial mentioned in Sec. 11.6.b. Experimental results concerning D are shown in Fig. 11.7.a-l [61, 62,63]. For practical purposes Pe may be considered to lie between 8 and 10. When the effective conductivity, X , is determined from heat transfer experiments in packed beds, it is observed that X decreases strongly in the vicinity of the wall. It is as if a supplementary resistance is experienced near the wall, which is probably due to variations in the packing density and flow velocity. Two alternatives are possible either use a mean X or consider X to be constant in the central core and introduce a new coefficient accounting for the heat transfer near the wall, a , defined by  

The data for are very scattered. Recently De Wasch and Froment [19] published data that are believed to have the high degree of precision required for the accurate prediction of severe situations in reactors. The correlations for air are of the form  

In the absence of flow the following mechanisms contribute to the effective conduction, according to Kunii and Smith [68]. 

Zehner and SchlUnder [74, 75] arrived at the following formula for the static contribution  

7 TWO-DIMENSIONAL PSEUDOHOMOGENEOUS MODELS 11.7.1 The Effective Transport Concept 

Peclet number for radial effective diffusion, based on particle diameter, versus Reynolds number. 1, Fahien and Smith [1955] 2 Bernard and Wilhelm [1950] 3, Dorweiler and Fahien [1959] 4, PLautz and Johnstone [1955] 5, Hiby [1962]. 

Strongly in the vicinity of the wall. It is as if a supplementary resistance is experienced near the wall, which is probably due to variations in the packing density and flow velocity. Two alternatives are possible either use a mean or consider to be constant in the central core and introduce a new coefficient accounting for the heat transfer near the wall, a, defined by 

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Partial differential equations are involved, and a simple analytical solution is usually impossible. One has to use advanced numerical techniques and computing aids to solve such models. The two-dimensional models use the effective transport concept to formulate the flux of heat and mass in the radial direction. This flux is superimposed upon the transport by the overall continuity equation for the key reacting component A and the energy equation. For a single reaction and at steady state, the equations can be written for the pseudo-homogeneous model as follows ... 

For liquids, there is no complete theory of multicomponent diffusion yet available. For this reason only rough theoretical approaches, as used for the description of mass transport in the porous particles filled with a liquid are discussed. The effective diffusivity concept just described is the only known approach and... 

More complex cases, such as cases with S- and P-inhibition kinetics, have been solved numerically by Moo-Young and Kobayashi (1972). Recently criteria have been developed specifically for Monod-, S-inhibited-, Teissier-, and maintenance-type kinetics to quantify and predict diffusional control within whole cells and cell floes (Webster, 1981). Further details concerning the use of the effectiveness factor concept for the quantification of biological processes will be given in Sect. 5.8, presenting simple formal kinetics in the case of internal transport limitation. 

A detailed study on velocity profiles, pressure drop and mass transport effects is given in [3]. This, in quantitative terms, precisely underlines the advantages (and limits) of the micro reactor concept. 

The major design concept of polymer monoliths for separation media is the realization of the hierarchical porous structure of mesopores (2-50 nm in diameter) and macropores (larger than 50 nm in diameter). The mesopores provide retentive sites and macropores flow-through channels for effective mobile-phase transport and solute transfer between the mobile phase and the stationary phase. Preparation methods of such monolithic polymers with bimodal pore sizes were disclosed in a US patent (Frechet and Svec, 1994). The two modes of pore-size distribution were characterized with the smaller sized pores ranging less than 200 nm and the larger sized pores greater than 600 nm. In the case of silica monoliths, the concept of hierarchy of pore structures is more clearly realized in the preparation by sol-gel processes followed by mesopore formation (Minakuchi et al., 1996). 

In the case of nonequimolal cpunterdiffusion, equation 12.2.6 suffers from the serious disadvantage that the combined diffusivity is a function of the gas composition in the pore. This functional dependence carries over to the effective diffusivity in porous catalysts (see below), and makes it difficult to integrate the combined diffusion and transport equations. As Smith (12) points out, the variation of 2C with composition (YA) is not usually strong, and it has been an almost universal practice to use a composition independent form of Q)c (12.2.8) in assessing the importance of intrapellet diffusion. In fact, the concept of a single effective diffusivity loses its engineering utility if the dependence on composition must be retained. 

The (isotropic) eddy viscosity concept and the use of a k i model are known to be inappropriate in rotating and/or strongly 3-D flows (see, e.g., Wilcox, 1993). This issue will be addressed in more detail in Section IV. Some researchers prefer different models for the eddy viscosity, such as the k o> model (where o> denotes vorticity) that performs better in regions closer to walls. For this latter reason, the k-e model and the k-co model are often blended into the so-called Shear-Stress-Transport (SST) model (Menter, 1994) with the view of using these two models in those regions of the flow domain where they perform best. In spite of these objections, however, RANS simulations mostly exploit the eddy viscosity concept rather than the more delicate and time-consuming RSM turbulence model. They deliver simulation results of in many cases reasonable or sufficient accuracy in a cost-effective way. 

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