Combined flow models

When we have a combination of recycled flow, by-pass connections, the presence of dead regions and a complex series and/or parallel coupling of the basic PM and PF models in a system, we have an important class of flow models recognized as combined flow models (CFM). 

The next section will first show the importance of flow in a concrete modelling problem such as the slip flow effect on the efficiency of a permanent mechanically mixed reactor. Then the characterization of the combined flow models where the slip flow occurs will be presented. 

Beven, K.( and P. Germann. 1981. Water flow in soil macropores. II A combined flow model. J. Soil Sci. 32 15-29. 

Table 5.1 Some Typical Combined Flow Models...

Fig. 8. Combined flow reactor models (a) parallel flow reactors with longitudinal diffusion (diffusivities can differ), (b) internal recycle—cross-flow reactor (the recycle can be in either direction), comprising two countercurrent plug-flow reactors with intercormecting distributed flows, (c) plug-flow and weU-mixed reactors in series, and (d) 2ero-interniixing model, in which plug-flow reactors are parallel and a distribution of residence times dupHcates that...

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Methods for determining the drop in pressure start with a physical model of the two-phase system, and the analysis is developed as an extension of that used for single-phase flow. In the separated flow model the phases are first considered to flow separately and their combined effect is then examined. 

The statistic models consider surface roughness as a stochastic process, and concern the averaged or statistic behavior of lubrication and contact. For instance, the average flow model, proposed by Patir and Cheng [2], combined with the Greenwood and Williamsons statistic model of asperity contact [3] has been one of widely accepted models for mixed lubrication in early times. 

Compared with the use of arbitrary grid interfaces in combination with reduced-order flow models, the porous medium approach allows one to deal with an even larger multitude of micro channels. Furthermore, for comparatively simple geometries with only a limited number of channels, it represents a simple way to provide qualitative estimates of the flow distribution. However, as a coarse-grained description it does not reach the level of accuracy as reduced-order models. Compared with the macromodel approach as propagated by Commenge et al, the porous medium approach has a broader scope of applicability and can also be applied when recirculation zones appear in the flow distribution chamber. However, the macromodel approach is computationally less expensive and can ideally be used for optimization studies. 

Cold flow studies have several advantages. Operation at ambient temperature allows construction of the experimental units with transparent plastic material that provides full visibility of the unit during operation. In addition, the experimental unit is much easier to instrument because of operating conditions less severe than those of a hot model. The cold model can also be constructed at a lower cost in a shorter time and requires less manpower to operate. Larger experimental units, closer to commercial size, can thus be constructed at a reasonable cost and within an affordable time frame. If the simulation criteria are known, the results of cold flow model studies can then be combined with the kinetic models and the intrinsic rate equations generated from the bench-scale hot models to construct a realistic mathematical model for scale-up. 

Some early spray models were based on the combination of a discrete droplet model with a multidimensional gas flow model for the prediction of turbulent combustion of liquid fuels in steady flow combustors and in direct injection engines. In an improved spray model,[438] the full Reynolds-averaged Navier-Stokes equations were... 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

In the spatially ID model of the monolith channel, no transverse concentration gradients inside the catalytic washcoat layer are considered, i.e. the influence of internal diffusion is neglected or included in the employed reaction-kinetic parameters. It may lead to the over-prediction of the achieved conversions, particularly with the increasing thickness of the washcoat layer (cfi, e.g., Aris, 1975 Kryl et al., 2005 Tronconi and Beretta, 1999 Zygourakis and Aris, 1983). To overcome this limitation, the effectiveness-factor concept can be used in a limited extent (cf. Section III.D). Despite the drawbacks coming from the fact that internal diffusion effects are implicitly included in the reaction kinetics, the ID plug-flow model is extensively used in automotive industry, thanks to the reasonable combination of physical reliability and short computation times. 

Calculation of Solute Separation and Product Rate. Once the pore size distribution parameters R, ou R >,2, 02, and h2 are known for a membrane and the interfacial interaction force parameters B and D are known for a given system of membrane material-solute, solute separation f can be calculated by eq 6 for any combination of these parameters. Furthermore, because the PR-to-PWP ratio (PR/PWP) can also be calculated by the surface force-pore flow model (9), PR is obtained by multiplying experimental PWP data by this ratio. 

The following discussion assumes any reaction present is in the slow kinetic regime. It also combines the batch and continuous-flow models, since there is so little difference between them. 

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