Velocity tubes model

Plug flow A simple convective flow pattern in pipes and tubes that is characterized by a fluid velocity independent of radial position, complete mixing in the radial direction, and no mixing in the axial direction. Also called the parallel tube model or tubular flow. See Eqs. (7) and (8) and Figure 3. 

Figure 1. The BPT (bundle of parallel tubes) model. The flow-rate in a tube with residence time between t and t +dt is dQ = E(t )dt. The fluid flows in the tubes at constant velocity. All particles on a vertical have the same age. The height of the pile is Q. The locus of tube extremities is a picture of F-curve...

It has already been noted that transverse variation of velocity is a major factor in the mixing process. Therefore, models which neglect this variation will require an inflated coefficient such as Di,. This article will therefore focus on the so-called stream-tube models, which do include depth and velocity variations across the channel. However, solutions which do not consider such variations will also be presented, since sometimes data may not exist to enable use of the more complicated model. 

As discussed in Section II,G, stream-tube models generally allow one to incorporate lateral variation of depth and velocity into the model. Yotsukura and Cobb W6) showed that the solutions of Eq. (20) for = 1 are not too sensitive to variations in the specific distribution of h Uj Dy with respect to as long as its averaged value over the total river flow Q remains the same. This finding, coupled with their further experience, indicates that one can, for many natural channels, obtain a reasonable approximation by defining a constant diffusion factor as ... 

Figure 12.6. PDMS slip velocity data of Leger and co-workers and predictions of the stochastic tube model. Reprinted from Xu et al., /. Rheol, 51,451 (2007).

From the point of view of tube models, the two key elements of nonlinear behavior are tube orientation and tube or chain stretch. The former nonlinearity can be probed using shear flow, but shear flows are not effective in generating significant chain stretch. As we have seen, chain stretch in shear is strongly suppressed by the mechanism of convective constraint release (CCR) up to extremely high shear rates. The CCR mechanism of relaxation is qualitatively much less important in extensional flows than in shear flows, because in the former molecules on neighboring streamlines move at the same velocity. Thus, extensional flows are of particular importance in the study of nonlinear viscoelasticity. 

So far we have considered the free-falling velocity in air which is at rest. But the falling phenomenon in a tube differs from this because the falling of solid particles itself causes an airflow upwards (see Fig. 14.7). The modeling idea of Fig. 14.7 is from Weber. The air volume replaced by the falling parti cle flows upward and this airflow rate is... 

One of the possible ways to account for the effect of roughness on the pressure drop in a micro-tube is to apply a modified-viscosity model to calculate the velocity distribution. Qu et al. (2000) performed an experimental study of the pressure drop in trapezoidal silicon micro-channels with the relative roughness and hydraulic diameter ranging from 3.5 to 5.7% and 51 to 169 pm, respectively. These experiments showed significant difference between experimental and theoretical pressure gradient. 

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 ... 

Heat Stability of 2,3,7,8-Tetrachlorodibenzo- -dioxin. A sample boat containing 1-3 mg of 2,3,7,8-tetrachlorodibenzo-p-dioxin was placed in a Sargent Tube Heater (Model J-807) which contained a 1/2 inch i.d. quartz tube. Air was aspirated through the tube at a velocity of 0.02 ft/sec. This velocity gave a residence time of 21 sec. (Residence time was extended by connecting a second heater in series with the first.)... 

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