Laminar axial dispersion

The diameters of the first 11 branches fall off exponentially with branch number in both the right and left lungs [5], and the flow is complex and laminar. Axial dispersion [29] is appreciable here but less than that for parabolic flow, and it decreases steadily, ft is negligible for subsequent branches. There is no appreciable diffusion across these large ducts, but deposition of aerosols is quite significant, and it is currently an active area of research (Google). Size distribution is complex in the smaller airways, but shows very little species dependence [5, p. 57]. 

Kirelic/lranspon for Isothermal Laminar-Flow Reactor with no Axial Dispersion [See Shinohara and Christiansen (I974J for ilie non-isoihermul... 

Chapters 8 and Section 9.1 gave preferred models for laminar flow and packed-bed reactors. The axial dispersion model can also be used for these reactors but is generally less accurate. Proper roles for the axial dispersion model are the following. 

Laminar Pipeline Flows. The axial dispersion model can be used for laminar flow reactors if the reactor is so long that At/R > 0.125. With this high value for the initial radial position of a molecule becomes unimportant. 

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory ... 

Micromixing Models. Hydrodynamic models have intrinsic levels of micromixing. Examples include laminar flow with or without diffusion and the axial dispersion model. Predictions from such models are used directly without explicit concern for micromixing. The residence time distribution corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

Concerning the hydrodynamics and the dimensioning of the test reactor, some rules of thumb are a valuable aid for the experimentalist. It is important that the reactor is operated under plug-flow conditions in order to avoid axial dispersion and diffusion limitation phenomena. Again, it has to be made clear that in many cases testing of monolithic bodies such as metal gauzes, foam ceramics, or monoliths used for environmental catalysis, often needs to be performed in the laminar flow regime. 

Fig. 10. Axial dispersion in laminar flow in pipes, dispersed plug flow model. Adapted from (B13).

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

Taylor (T4, T6), in two other articles, used the dispersed plug-flow model for turbulent flow, and Aris s treatment also included this case. Taylor and Aris both conclude that an effective axial-dispersion coefficient Dzf can again be used and that this coefficient is now a function of the well known Fanning friction factor. Tichacek et al. (T8) also considered turbulent flow, and found that Dl was quite sensitive to variations in the velocity profile. Aris further used the method for dispersion in a two-phase system with transfer between phases (All), for dispersion in flow through a tube with stagnant pockets (AlO), and for flow with a pulsating velocity (A12). Hawthorn (H7) considered the temperature effect of viscosity on dispersion coefficients he found that they can be altered by a factor of two in laminar flow, but that there is little effect for fully developed turbulent flow. Elder (E4) has considered open-channel flow and diffusion of discrete particles. Bischoff and Levenspiel (B14) extended Aris s theory to include a linear rate process, and used the results to construct comprehensive correlations of dispersion coefficients. 

Checks on the relationships between the axial coefficients were provided in empty tubes with laminar flow by Taylor (T2), Blackwell (B15), Bournia et al. (B19), and van Deemter, Breeder and Lauwerier (V3), and for turbulent flow by Taylor (T4) and Tichacek et al. (T8). The agreement of experiment and theory in all of these cases was satisfactory, except for the data of Boumia et al. as discussed previously, their data indicated that the simple axial-dispersed plug-flow treatment may not be valid for laminar flow of gases. Tichacek et al. found that the theoretical calculations were extremely sensitive to the velocity profile. Converse (C20), and Bischoff and Levenspiel (B14) showed that rough agreement was also obtained in packed beds. Here, of course, the theoretical calculation was very approximate because of the scatter in packed-bed velocity-profile data. 

We will now find the RDT for several models of tubular reactors. We noted previously that the perfect PFTR cannot in fact exist because, if flow in a tube is sufficiently fast for turbulence (Rco > 2100), then turbulent eddies cause considerable axial dispersion, while if flow is slow enough for laminar flow, then the parabolic flow profile causes considerable deviation from plug flow. We stated previously that we would ignore this contradiction, but now we will see how these effects alter the conversion from the plug-flow approximation. 

Fig. 2.20. Dimensionless axial-dispersion coefficients for fluids flowing in circular pipes. In the turbulent region, graph shows upper and lower limits of a band of experimentally determined values. In the laminar region the lines are based on the theoretical equation 2.37...

Kinetic/transport for Isothermal Laminar-Flow Reactor with Axial Dispersion under Transient Open-Loop Operation... 

Figure 8-33. Correlation of axial dispersion coefficient for flow of fluids through pipes in laminar flow region (NRe < 2,000). (Source Wen, C. Y. and Fan, L. T, Models for Flow Systems and Chemical Reactors, Marcel Dekker Inc., 1975.)...

As our first application, we consider the classical Taylor-Aris problem (Aris, 1956 Taylor, 1953) that illustrates dispersion due to transverse velocity gradients and molecular diffusion in laminar flow tubular reactors. In the traditional reaction engineering literature, dispersion effects are described by the axial dispersion model with Danckwerts boundary conditions (Froment and Bischoff, 1990 Levenspiel, 1999 Wen and Fan, 1975). Here, we show that the inconsistencies associated with the traditional parabolic form of the dispersion model can be removed by expressing the averaged model in a hyperbolic form. We also analyze the hyperbolic model and show that it has a much larger range of validity than the standard parabolic model. 

The important result is that the two-mode models for a turbulent flow tubular reactor are the same as those for laminar flow tubular reactors. The two-mode axial dispersion model for turbulent flow tubular reactors is again given by Eqs. (130)—(134), while the two-mode convection model for the same is given by Eqs. (137)—(139), where the reaction rate term r((c)) is replaced by the Reynolds-averaged reaction rate term rav((c)). The local mixing time for turbulent flows is given by... 

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