Diffusion streamline

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

Observation of operating conveyors indicates that the solids are rarely uniformly dispersed in the gas phase. With infrequent exceptions, the particles move in a laminar pattern, following a streamline along the duct wall where the flow velocity is at a minimum. Complete or even partial diffusion in the gas phase is rarely experienced even with low-specific-gravity particles. Air velocities may approach 20 to 30 m/s. It is doubtful, however, that even finer and lighter materials reach more than 80 percent of this speed, while heavier and larger... 

Smaller particles, particularly those below about 0.3//m in diameter, exhibit consideroble Brownian movement and do not move uniformly along the gas streamline. These particles diffuse from the gas to the surface of the collecting body and are collected. 

If 0 fluid streomline passes within one particle rodius of the collecting body, o particle traveling olong the streamline will touch the body and may be collected without the influence of inertia or brownian diffusion. 

Axial diffusion Mass transfer by diffusion along streamlines that occurs at... 

Another design method uses capture efficiency. There are fewer models for capture efficiency available and none that have been validated over a wide range of conditions. Conroy and Ellenbecker - developed a semi-empirical capture efficiency for flanged slot hoods and point and area sources of contaminant. The point source model uses potential flow theory to describe the flow field in front of a flanged elliptical opening and an empirical factor to describe the turbulent diffusion of contaminant around streamlines. 

The relation (103) will hold for any value of the scission yield, providing the macromolecules do not cross the boundary between the different streamlines, either by molecular diffusion or by flow turbulence. For samples having similar polydispersities and MWD, all the degradation data could be superimposed onto a single curve when plotted on a relative abscissa scale (e(0)/ef). 

Example 8.1 derived a specific example of a powerful result of residence time theory. The residence time associated with a streamline is t = LIVz. The outlet concentration for this streamline is ahatchit)- This is a general result applicable to diffusion-free laminar flow. Example 8.1 treated the case of a... 

To account for molecular diffusion, Equation (8.2), which governs reactant concentrations along the streamlines, must be modihed to allow diffusion between the streamlines i.e., in the radial direction. We ignore axial diffusion but add a radial diffusion term to obtain... 

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 ... 

Include the radial velocity term in the convective diffusion equation and plot streamlines in the reactor. 

Fig. 5.1.2 Non-ideal capillary flow reactor (a) propagators [13] and (b) corresponding RTDs calculated from the propagator data, (a) The propagators indicate the distribution of average velocities over each observation time (A) ranging from 50 ms to 1 s. As the observation time increases the spins exhibit a narrowing distribution of average velocities due to the motional narrowing effect of molecular diffusion across the streamlines. The dashed vertical line represents the maximum velocity that would be present in the absence of molecular...

FTIR instrumentation is mature. A typical routine mid-IR spectrometer has KBr optics, best resolution of around 1cm-1, and a room temperature DTGS detector. Noise levels below 0.1 % T peak-to-peak can be achieved in a few seconds. The sample compartment will accommodate a variety of sampling accessories such as those for ATR (attenuated total reflection) and diffuse reflection. At present, IR spectra can be obtained with fast and very fast FTIR interferometers with microscopes, in reflection and microreflection, in diffusion, at very low or very high temperatures, in dilute solutions, etc. Hyphenated IR techniques such as PyFTIR, TG-FTIR, GC-FTIR, HPLC-FTIR and SEC-FTIR (Chapter 7) can simplify many problems and streamline the selection process by doing multiple analyses with one sampling. Solvent absorbance limits flow-through IR spectroscopy cells so as to make them impractical for polymer analysis. Advanced FTIR... 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

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