Radial gradient

In either laminar or turbulent flow, rotational eireulation of a proeessed material around its own hydraulie eenter in eaeh ehannel of the mixer eauses radial mixing of the material. All proeessed material is eontinuously and eompletely intermixed, virmally eliminating radial gradients in temperature, veloeity, and material eomposition. 

The Kenics mixer has been shown to provide thorough radial mixing. This results in a reduction in radial gradients in velocity, composition, and temperature. Because the unit possesses nearly plug flow characteristics, both temperature and product quality controls are achieved. 

The scheme of commercial methane synthesis includes a multistage reaction system and recycle of product gas. Adiabatic reactors connected with waste heat boilers are used to remove the heat in the form of high pressure steam. In designing the pilot plants, major emphasis was placed on the design of the catalytic reactor system. Thermodynamic parameters (composition of feed gas, temperature, temperature rise, pressure, etc.) as well as hydrodynamic parameters (bed depth, linear velocity, catalyst pellet size, etc.) are identical to those in a commercial methana-tion plant. This permits direct upscaling of test results to commercial size reactors because radial gradients are not present in an adiabatic shift reactor. 

For optical fibers, improved control over the stracture of the thin films in the preform will lead to fibers with improved radial gradients of refractive index. A particnlar challenge is to achieve this sort of control in preforms created by sol-gel or related processes. 

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... 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

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. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

Compare Equation (11.42) with Equation (9.1). The standard model for a two-phase, packed-bed reactor is a PDE that allows for radial dispersion. Most trickle-bed reactors have large diameters and operate adiabaticaUy so that radial gradients do not arise. They are thus governed by ODEs. If a mixing term is required, the axial dispersion model can be used for one or both of the phases. See Equations (11.33) and (11.34). 

Model of ID dissipation spectrum from Pope [19] (line) and measured, noise-corrected spectrum of the square of the radial gradient of fluctuating temperature in a CH4/I-I2/N2 jet flame (Re = 15,200) (symbols). Each spectrum is normalized by its maximum value. The arrow indicates the 2% level, which corresponds to the normalized wavenumber k = 1 according to the model spectrum. (From Barlow, R.S., Proc. Combust. Inst., 31, 49,2007. With permission.)... 

Extraction of rhizosphere soil (22,34,51,52) is an approach that can provide information about long-term accumulation of rhizosphere products (root exudates and microbial metabolites) in the soil. Culture systems, which separate root compartments from adjacent bulk soil compartments by steel or nylon nets (52-54) have been employed to study radial gradients of rhizosphere products in the root environment. The use of different extraction media can account for different adsorption characteristics of rhizosphere products to the soil matrix (22,34). However, even extraction with distilled water for extended periods (>10 min) may... 

Dapremont, O., Cox, G.B., Martin, M., Hilaireau, P., and Colin, H., Effect of radial gradient of temperature on the performance of large-diameter high-performance liquid chromatography columns I. Analytical conditions,. Chromatogr. A, 796, 81, 1998. 

Another typical example of inhomogeneity in rheometry is the oxidation of a polymer in a rotational rheometer in which a disk-shaped sample is held between metal fixtures. The oxygen enters the sample through the free surface (at the outer diameter) and diffuses radially inwards. The result is a radial gradient in properties which changes with time. If the reaction with oxygen results in... 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

Radial gradients in both temperature and concentration may develop for extremely exothermic reactions. This results from the positive interaction between reaction rate and temperature. The temperature rises more rapidly at the center (r = 0) of the bed, if... 

Voids often look similar to air bubbles. The appearance of voids in filaments or films, however, results for different reasons. Voids can be produced during stretching in the area of necking by a kind of folding mechanism. The formation of voids may also depend on the generation of a radial gradient structure during solidification of the threads. 

Example 2.2. Fluid is flowing through a constant-diameter cylindrical pipe sketched in Fig. 2.2. The flow is turbulent and therefore we can assume plug-flow conditions, i.e., each slice of liquid flows down the pipe as a unit. There are no radial gradients in velocity or any other properties. However, axial gradients can exist. 

Example 2.5. Instead of fluid flowing down a pipe as in Example 2.2, suppose the pipe is a tubular reactor in which the same reaction A A B of Example 2.3 takes place. As a slice of material moves down the length of the reactor the concentration of reactant decreases as A is consumed. Density p, velocity v, and concentration can aU vary with time and axial position z. We stiU assume plug-flow conditions so that there are no radial gradients in velocity, density, or concentration. 

Judice TN, Nelson NC, Beisel CL, Delimont DC, Fritzsch B, et al. 2002. Cochlear whole mount in situ hybridization identification of longitudinal and radial gradients. Brain Res Prot 9 65-76. 

Consider a packed bed with heat exchange (Figs. 19.1a and 19.16). For an exothermic reaction Fig. 19.2 shows the types of heat and mass movement that will occur when the packed bed is cooled at the walls. The centerline will be hotter than the walls, reaction will be faster, and reactants will be more rapidly consumed there hence, radial gradients of all sorts will be set up. 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

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