Radial profile

In Fig. 19-5, the conveyor belt is radially profiled at the point of sample extraction with contouring idlers set to match the path of the cutter moving from its driveshaft rotation axis. Cutter edges are posi-... 

The bubble size distribution is closely related to the hydrodynamics and mass transfer behavior. Therefore, the gas distributor should be properly designed to give a good performance of distributing gas bubbles. Lin et al. [21] studied the influence of different gas distributor, i.e., porous sinter-plate (case 1) and perforated plate (case 2) in an external-loop ALR. Figure 3 compares the bubble sizes in the two cases. The bubble sizes are much smaller in case 1 than in case 2, indicating a better distribution performance of the porous sinter-plate. Their results also show the radial profile of the gas holdup in case 1 is much flatter than that in case 2 at the superficial gas velocities in their work. 

Fig. 3. Radial profiles of the bubble size with Fig. 4. Influence of internals on the gas holdup different gas distributors (air-water system) [22]. (air-water-solid slurry system) [23].

Fig. 3. Radial profile of the gas holdup at different solid holdups...

Figure 3 shows the radial profile of the gas holdup in the riser with increasing superficial gas velocity under different solid holdups. The gas holdup increases with increasing superficial gas velocity at the different solid holdups. At a low superficial gas velocity, the liquid velocity... 

However, bubble nonhomogeneous distribution exists in two-phase shear flow. As yet, the following general trends in void fraction radial profiles are being identified for bubbly upward flow (Zun, 1990) concave profiles (Serizawa et al., 1975) convex profiles (Sekoguchi et al., 1981), and intermediate profiles (Sekoguchi et al., 1981 Zun, 1988). Two theories are currently dominant ... 

Radial profile Uniform and nonuniform (radial and corner peaking)... 

Figure 13. Radial profiles of solid volume fraction in fast fluidized bed. Fr = Us/Jgd P,M = Gs/ pgUg (Fro m Beaude and Louge, 1995.)...

Figure 14. Radial profiles of solid mass flux in fast fluidized bed. (From Herb, Dou, Tuzla and Chen, 1992.)...

Beaude, F., and Louge, M., Similarity of Radial Profiles of Solid Volume Fraction in a Circulating Fluidized Bed, Fluidization VIII, pp. 245-253 (1995)... 

Figure 1.2 (a) Lowest s- and p-type valence atomic orbitals of rare-gas atoms, showing radial profiles (left) and contour plots (right). (Each plot is 3 A wide, and only the four outermost contours are plotted see note 26.) (b) Similar to Fig. 1.2(a), for valence 4s, 4p, and 3d atomic orbitals of Kr, corresponding directly to the surface plots of Fig. 1.1. 

Figure 2.12 Radial profiles of filled (ls)u (left) and (2s)f (right) NAOs, showing the increasing nodal feature at the opposite nucleus in each orbital as R decreases.

Figure 4.96 Contour plots (left) and radial-profile plots (right) of overlapping hM and hH NHOs of ctMh bonds for group 10 dihydrides (NiH2, PdH2, and PtH2). Numerical (hM F, hH> Fock-matrix interaction elements are shown for each case.

Fig. 3 Doubling the bead radius causes oxygen deficiency inside the bead (lower curve) as these radial profiles show.

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

FIGURE 8.21 Comparison of the radial profiles for scattering cross-section gvv and temperature (uncorrected) as a function of radial position for a coannular ethane diffusion flame (from Santoro et al. [85]). 

The lack of software for the decomposition method in two dimensions in application to electron diffraction texture patterns was avoided by Zvyagin and Zhukhlistov by using one-dimensional methods of intensity extraction. Later the decomposition method for one-dimensional radial profiles (starting at the centre of the diffraction pattern and passing through... 

In the texture pattern described in this chapter, the background was approximated by a Gaussian curve from three different radial profiles, since there can be some deviations from the average radial profile in different directions which can lead to over- or underestimations of the background. Each profile was averaged over 3° in the azimuthal direction the directions... 

Swirl Since swirl is present in many propulsion burners, studies were undertaken to test the robustness of the control system by imposing swirl in the central air flow. Another objective of these studies was to see if added swirl could improve the operation of the controller. The swirl was introduced by letting the airflow into the central tube from two off-axis 90-degree inlets. The percent swirl quoted here is the percent of the total central air flow that entered from these swirl inlets as opposed to the nonswirling inlets. The average axial velocity radial profile showed a strong minimum on the centerline, and at higher levels of swirl the flame could actually be sucked back into the airflow. 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Simulations can be performed using additional radial collocation points for the concentration profiles and compared to those using the earlier model. In the simulations, the bulk concentration can then be obtained by integrating the radial profiles ... 

Figs. 25 to 28 show that the metal deposition in CoMo/A1203 hydrotreating catalysts is a function of the radial position within the catalyst and the axial location of the catalyst sample within the fixed-bed reactor. Nickel and vanadium both exhibit radial profiles with internal maxima, termed M-shaped profiles, at the reactor entrance. These maxima shift to the pellets edge at the reactor outlet, generating the classic U-shaped profile. 

In addition to catalyst pore structure, catalytic metals content can also influence the distribution of deposited metals. Vanadium radial profile comparisons of aged catalysts demonstrated that a high concentration of Co + Mo increases the reaction rate relative to diffusion, lowering the effectiveness factor and the distribution parameter (Pazos et al., 1983). While minimizing the content of Co and Mo on the catalyst is effective for increasing the effectiveness factor for HDM, it may also reduce the reaction rate for the HDS reactions. Lower space velocity or larger reactors would then be needed to attain the same desulfurization severity. 

Fig. 20 Radial profile of (°) and n close to (but not exactly at) = n/2. Note the regions of dilatation and compression of the layers caused by the flow alignment...

Fig. 8. Effect of superficial gas and liquid velocities on the liquid saturation Fig. 9 Effects of gas and liquid superficial velocities on the cross-radial profile at axial position of 2.5D. sectionally averaged liquid saturation.

Small bubbles and flow uniformity are important for gas-liquid and gas-liquid-solid multiphase reactors. A reactor internal was designed and installed in an external-loop airlift reactor (EL-ALR) to enhance bubble breakup and flow redistribution and improve reactor performance. Hydrodynamic parameters, including local gas holdup, bubble rise velocity, bubble Sauter diameter and liquid velocity were measured. A radial maldistribution index was introduced to describe radial non-uniformity in the hydrodynamic parameters. The influence of the internal on this index was studied. Experimental results show that The effect of the internal is to make the radial profiles of the gas holdup, bubble rise velocity and liquid velocity radially uniform. The bubble Sauter diameter decreases and the bubble size distribution is narrower. With increasing distance away from the internal, the radial profiles change back to be similar to those before contact with it. The internal improves the flow behavior up to a distance of 1.4 m. 

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