Dispersion, laminar flows

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

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. 

Daskopoulos, P., Lenhoff, A. M., Dispersion coefficient for laminar flow in curved tubes, AIChE J. 34, 12 (1988) 2052-2058. 

The gas flow direction was from the top to bottom of the figure. No divergence is observed in the dispersion curve of the capillary, indicating that under the given conditions the dispersion of flow is small, and that this scheme is thus adequate to study the dispersion within a device of interest. This may appear unexpected, as microfluidic devices are usually assumed to exhibit laminar flow, however it can be explained by the fast lateral diffusion of individual gas molecules, which uniformly sample the whole cross section of the tube in a very short time compared with the travel time. Below each image, its projection is shown together with an independ-... 

Powell, R. L., and Mason, S. G., Dispersion by laminar flow. AlChE J. 28, 286-293 (1982). Rallison, J. M A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 109, 465-482 (1981). 

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. 

Eq. 7.2, where is then the eddy diffusion coefficient (Taylor and Spencer 1990). The height of the turbulent zone, within the atmospheric boundary layer, is orders of magnitude greater than that of the laminar flow layer, and dispersion of contaminant vapors in the turbulent zone is relatively rapid. 

Under some laminar flow circumstances, the dispersion model (see Sect. 5.4) may be a better description of reality than the laminar-flow model discussed above. Ananthakrishnan et al. [48] discuss conditions under which one should opt for a particular model and their data are presented in modified form by Levenspiel [26]. If the tubular reactor aspect ratio, i.e. the length-to-diameter ratio, is less than 100 and the product of the reactor Reynolds, dup/p, and Schmidt, p/p S numbers is greater than approximately 10", then the laminar-flow model should give good predictions of reactor performance. 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

Figures 13.15 and 13.16 show the findings for flow in pipes. This model represents turbulent flow, but only represents streamline flow in pipes when the pipe is long enough to achieve radial uniformity of a pulse of tracer. For liquids this may require a rather long pipe, and Fig. 13.16 shows these results. Note that molecular diffusion strongly affects the rate of dispersion in laminar flow. At low flow rate it promotes dispersion at higher flow rate it has the opposite effect.

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

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

Laminar Flow in Empty Tubes. As will be discussed in Section II, Dij, the radial coefiicient for the dispersed-plug flow model for laminar flow is merely the molecular diffusivity. 

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... 

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