Wall velocity axial

Consider a fully developed steady-state laminar flow of a constant-property fluid through a circular duct with a constant heat flux condition imposed at the duct wall. Neglect axial conduction and assume that the velocity profile may be approximated by a uniform velocity across the entire flow area (i.e., slug flow). Obtain an expression for the Nusselt number. 

The above analysis can be applied for the case of bubble columns. From Figure I it can be seen that, near the wall the axial component of the liquid velocity is downwards, whereas the radial component of the liquid velocity is towards the wall in the top half of the circulation cell and away from the wall in the lower half of the circulation cell. As a result, for one circulation cell the enhancement factor is given by the following equation ... 

In trickle-bed microreactors (TBMR), it is frequently used a commercial catalyst sample and real feedstocks, however, the length of the catalyst bed and hence the reactor length to catalyst particle diameter ratio are low as compared to commercial reactors. In addition, low liquid velocities are used in order to match the liquid hourly space velocities (LHSV) of industrial units. These differences cause number of problems in testing catalyst having commercially applied size and shape, such as poor wetting of catalyst, wall effect, axial dispersion, maldistribution, and the data obtained in such TBMR may not be reliable [1,2],... 

In order to obtain the genuine wall velocity, i.e., the velocity joined to a mass point and including both radial and axial components, one may start from the analysis of the expanding path of the individual cylinder particle (Figure 5.17). 

Here the quantity/ (z) is the particle mass flux at the wall at axial coordinate z. For region 1, F , z) is the product of the particle mass concentration, Ppj (z), and the radial particle velocity at the wall, Up ... 

A positive value iadicates vertical movement. Thea, moving from the outer wall to the air core, the axial velocity iacreases to positive values. Thus, the fluid motioa is dowa the wall of the cycloae to the apex and up the air core through the vortex finder. In the cylindrical section, the axial velocity goes negative again, approaching the vortex-finder wall. The fluid flow is then down the inner cyclone wall and the outer vortex-finder wall. There is a locus of zero axial velocity. 

Two complementai y reviews of this subject are by Shah et al. AIChE Journal, 28, 353-379 [1982]) and Deckwer (in de Lasa, ed.. Chemical Reactor Design andTechnology, Martinus Nijhoff, 1985, pp. 411-461). Useful comments are made by Doraiswamy and Sharma (Heterogeneous Reactions, Wiley, 1984). Charpentier (in Gianetto and Silveston, eds.. Multiphase Chemical Reactors, Hemisphere, 1986, pp. 104—151) emphasizes parameters of trickle bed and stirred tank reactors. Recommendations based on the literature are made for several design parameters namely, bubble diameter and velocity of rise, gas holdup, interfacial area, mass-transfer coefficients k a and /cl but not /cg, axial liquid-phase dispersion coefficient, and heat-transfer coefficient to the wall. The effect of vessel diameter on these parameters is insignificant when D > 0.15 m (0.49 ft), except for the dispersion coefficient. Application of these correlations is to (1) chlorination of toluene in the presence of FeCl,3 catalyst, (2) absorption of SO9 in aqueous potassium carbonate with arsenite catalyst, and (3) reaction of butene with sulfuric acid to butanol. 

At the exit the absolute velocity has velocity component C2 on the large circumference parallel to the shaft of Example 3. Component is of no advantage if a duct is connected to the axial fan, since it disappears due to the friction between the walls of the duct and gas flow. 

There is also a standardized method based on the estimation of the flow rate on one measurement point only, In this method the velocity probe is placed in the duct so that the measured local velocity is equal to the mean axial velocity. In fully developed turbulent duct flow, this distance from the wall... 

Consider now the case where the axial heat transfer due to the temperature gradient is negligible compared to the heat transfer from the capillary wall and the friction caused by the velocity gradient in the x-direction is negligible compared to the momentum losses at the fluid-wall interface. 

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

Global velocity distribution behind flame front. Upward propagation in 5.15% methane/air mixture, (a) vector map, (b) and (c) scalar maps of axial and radial velocity components, respectively. Spots are caused by condensation of water vapor on the glass walls. 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

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