Normalized radial concentration

Figure 6. Normalized radial concentration distribution for NE. Solid line theoretical for central injection with Oh - 0.188 mm. Circles experimental. The dashed lines are the estimated position of the column walls. (Reproduced from ref. 13. Copyright 1988 American Chemical Society.)...

Watson (1979a) carried out tracer diffusion experiments by loading a small amount of " Ca tracer onto one surface of a cylinder. The cylinder was heated up and " Ca diffuses into the cylinder. Assume that diffusion is along the axis of the cylinder (i.e., there is no radial concentration gradient). Assume that D=10 m /s. The cylinder is 3 mm long. Calculate the diffusion profile (concentration normalized to the surface concentration) at t = 2h and t=8h. How does the concentration profile at 8 h look like when compared to that at 2 h ... 

Figure 6 shows the normalized concentration observed for one peak as a function of radial position. Superimposed on the curve is the dispersion expected for a centrally Injected substance calculated for the column conditions and flow rate (1.0 mL mln ) employed. It is clear that the dispersion is much greater for the experimental data. It is likely that the observed radial concentration profile is a consequence of the sasiple introduction processes as previously proposed by Kirkland (14). 

Rachedi et al. [22] continued Zeaton et al. s work by examining the behavior of a real jet fuel (JP-10), and compared it to Zeaton et al. s carbon dioxide data. Their results showed that carbon dioxide could be used as a surrogate fluid for JP-10, since their behaviors were very similar. Both Rachedi et al. and Zeaton et al. found that the injected fluid radial concentration profile was well described by a Gaussian profile when the radius was normalized by the jethalf-radius. The jethalf-radius was defined as the radius where the concentration is half of the maximum (centerline) value. 

Gas exchange between the jet and the outside emulsion phase was studied by tracer gas injection and by integration of gas velocity profiles in the jet at various heights above the jet nozzle in a 28.6 cm diameter bed with a 3.5 cm jet using polyethylene beads as bed material (Yang et al., 1984a). The concentration profiles obtained at different elevations were found to be approximately similar if the local tracer concentration is normalized with the maximum tracer concentration at the axis, C/Cm, and plotted against a normalized radial distance, r/(ri/2)c, where (ri/2)( is the radial position where the tracer concentration is just half the maximum tracer concentration at the axis. Thus in a permanent flamelike jet in a fluidized bed, not only the velocity profiles in the jet but also the gas concentration profiles are similar. 

FIGURE 3.25 Solution of the single pore model in the ID Poisson-Boltzmann limit (a) the electrostatic effectiveness factor, as a function of the metal surface charge density, ctm, for various values of Rp (b) radial variation of the normalized proton concentration in the pore for various values of Rp at gm = —0.05 C (Reprinted from Chan, K. and Eikerling, M. 2011. /. Electrochem. Soc., 158(1), B18-B28, Figures 1,2,3,4,5,6. Copyright (2011), the Electrochemical Society. With permission.)... 

A considerable reduction in stress concentration could be achieved by using a cross-bore which is eUiptical in cross-section, provided the major axis of the eUipse is normal to the axis of the main cylinder. A more practical method of achieving the same effect is to have an offset radial hole whose axis is parallel to a radius but not coincident with it (97,98). Whenever possible the sharp edges at the intersection of the main bore with the cross bore are removed and smooth rounded corners produced so as to reduce the stress raising effects. 

Air is compressed to modest pressures, typically 100 to 200 kPa ( 15-30 psig) with either a centrifugal or radial compressor, and mixed with superheated vaporized butane. Static mixers are normally employed to ensure good mixing. Butane concentrations are often limited to less than 1.7 mol 1 to stay below the lower flammable limit of butane (144). Operation of the reactor at butane concentrations below the flammable limit does not eliminate the requirement for combustion venting, and consequendy most processes use mpture disks on both the inlet and exit reactor heads. A dow diagram of the Huntsman fixed-bed maleic anhydride process is shown in Figure 1. 

Pattemators may comprise an array of tubes or concentric circular vessels to coUect Hquid droplets at specified axial and radial distances. Depending on the pattemator, various uniformity indexes can be defined using the accumulated relative values between the normalized flow rate over a certain sector or circular region and a reference value that represents a perfectly uniform distribution. For example, using an eight-sector pie-shaped coUector, the reference value for a perfectly uniform spray would be 12.5%. The uniformity index (28) could then be expressed as foUows, where is the normalized volume or mass flow rate percentage in each 45-degree sector. 

When the discs are of the shrunk-on design, they are made up individually and stacked onto the shaft by first heating the disc to dilate the bore. They are then allowed to cool and thus attach themselves to the shaft. Keys are normally not used. When the discs are of the stacked design, the discs are equipped with rabbet fit to radially lock the discs to maintain concentricity of assembly. The through-bolts are usually tensioned by stretching hydraulically to a precise value to ensure the mechanical integrity of the assembly. 

Laminar flow reactors have concentration and temperature gradients in both the radial and axial directions. The radial gradient normally has a much greater effect on reactor performance. The diffusive flux is a vector that depends on concentration gradients. The flux in the axial direction is... 

Quantitation of Human IgG. Radial immunodiffusion technique (Tri-Partigen Calbiochem-Behring Corp.) was used to measure normal serum and serum from byssinotic persons. After filling the wells, diffusion was allowed to proceed for 50 hr at 4°C. Precipitin rings were measured and the concentration of IgG in mg/dl was obtained from the Table of References supplied with the plates. 

More recent versions of this design feature an added focusing system, called a radial drift chamber, in front of the detector. Gas ionization occurs within this drift chamber, which is composed of a set of concentric charged rings sandwiched between two lens-shaped metal grids. The cluster of electrons is then focused by the drift chamber, making its direction normal to the surface of the multiwire detector. 

To obtain an expression for the normal stress 0 in the tangential direction, consider the equilibrium of an element cut from a sphere by two concentric spherical surfaces of radii R and R + dR and by a circular cone with a small angle dtp, as shown in Fig. 2.6. The force balance along the radial direction gives... 

The effective radial diffusivity D is normally different from the axial dilfusivity. It is often safe to neglect the radial variation of species concentration due to the relatively fast radial mixing. The effective conductivity k has to be determined from heat-transfer experiments preferably with the actual bed and fluids. This coefficient can be... 

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