Velocity profile mixed convection

With the carrier stream unsegmented by air bubbles, dispersion results from two processes, convective transport and diffusional transport. The former leads to the formation of a parabolic velocity profile in the direction of the flow. In the latter, radial diffusion is most significant which provides for mixing in directions perpendicular to the flow. The extent of dispersion is characterized by the dispersion coefficient/). 

The use of the Coanda effect is based on the desire to have a second passive momentum to speed up mixing in addition to diffusion [55, 163], The second momentum is based on so-called transverse dispersion produced by passive structures, which is in analogy with the Taylor convective radial dispersion ( Taylor dispersion ) (see Figure 1.180 and [163] for further details). It was further desired to have a flat ( in-plane ) structure and not a 3-D structure, since only the first type can be easily integrated into a pTAS system, typically also being flat A further design criterion was to have a micro mixer with improved dispersion and velocity profiles. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

Velocity profiles in fully developed mixed convective flow in a vertical plane channel. Results are for assisting flow. 

Here, D is diffusivity, I is the characteristic length scale (typically chaimel height), and U is the flow velocity. The Peclet number can thus be viewed as the ratio of characteristic time for diffusion (tdiff) to the characteristic convection time (tconv)- When mixing occurs only via diffusion in a microchannel with a flat velocity profile, t ix = tdiff = "tconYPe. 

This general analysis of the chlorine-toluene system based upon first order reacticxi was developed in parallel with a series of experimental measurements (9), in which chlorine was absorbed in toluene in a laminar jet. This absorption device provides remarkable control of surface area and with a flat velocity profile the penetration time is reasonably well defined so that the penetration theory can be directly applied without any uncertainty concerning the complications of convective transport. Experimental mixing cup temperatures for Cl -toluene ranged from i C to 6°C. These can be interpreted as the amount of heat accumulated per unit of Jet surface as the jet plunges into the receiver via the equation... 

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective mixing have been lumped into the dispersion coefficient. With these model simplifications the semi-empirical correlations for the dispersion coefficients will be system- and scale specific and far from general. 

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. 

Unstable Conditions In unstable conditions there is usually an inversion base height at z = Zi that defines the extent of the mixed layer. The two parameters that are key in determining Kzz are the convective velocity scale wr and Zi- We expect that a dimensionless profile Kzz = Kzz/w,z., which is a function only of z/z., should be applicable. This form should be valid as long as Kzz is independent of the nature of the source distribution. Lamb and Duran (1977) determined that Kzz does depend on the source height. With the proviso that the result be applied when emissions are at or near ground level, Lamb et al. (1975) and Lamb and Duran (1977) derived an empirical expression for Kzz under unstable conditions, using the numerical turbulence model of Deardorff (1970) ... 

Usually, the radial diffusive transport term is taken into account for mass and energy balances, while for momentum balance it could be neglected since the presence of catalyst pellets produces a mixing effect leading to a uniform gas velocity radial profile. Moreover, the radial diffusive terms are always much greater than the convective radial terms, which can be neglected. 

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