Radial concentration gradients

Gas-phase reactors are often in laminar flow but have such high diffusivities that radial concentration gradients tend to be unimportant. Combustion reactions are fast enough to be exceptions. See... 

Assuming that there are only axial and no radial concentration gradients in the pore due to the negligible size of the pore diameter, the modeled concentration profiles of CO, H2, and H20 in a wax-filled cylindrical pore are given in Figure 12.2 (left, Presto Kinetics). For verification reasons of the underlying model and to obtain a better visual impression of the respective processes in the catalyst pore,... 

Figure 3 Schematic diagram of transportation of sample via the stream of carrier by (a) turbulent flow, (b) laminar flow, and (c) radial diffusion of the sample in response to radial concentration gradient, within the flowing stream.

Longitudinal dispersion coefficients can be readily obtained by injecting a pulse of tracer into the bed in such a way that radial concentration gradients are eliminated, and measuring the change in shape of the pulse as it passes through the bed. Since dC/dr is then zero, equation 4.34 becomes ... 

A comparison of the effects of axial and radial mixing is seen in Figure 4.7, which shows results obtained by Gunn and Pryce(28) for dispersion of argon into air. The values of Dl were obtained as indicated earlier, and DR was determined by injecting a steady stream of tracer at the axis and measuring the radial concentration gradient across the... 

The lumped pore model (often referred to as the POR model) was derived from the general rate model by ignoring two details of this model [5]. The first assumption made is that the adsorption-desorption process is very fast. The second assumption is that diffusion in the stagnant mobile phase is also very fast. This latter assumption leads to the consequence that there is no radial concentration gradient within a particle. Instead of the actual radial concentration profile across the porous particle, the model considers simply its average value. 

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 ... 

In his analysis of the open tube distillation column Westhaver (1942) goes into a detailed consideration of radial concentration gradients which is very similar to Taylor s approach. His final formula, however, is the same as if he had assumed a constant velocity profile and an effective diffusion coefficient (Dt + llU2r2l48Dt). This is just the diffusion coefficient that we have found for viscous flow in the presence of a film on the tube wall in which the solute concentration is infinitely greater than in the fluid. This is clearly the case for... 

The goal of a SV experiment is to monitor the rate of movement of the solvent—solution boundary (or, the evolution of the radial concentration gradient dc/dx as a function of time), described by the Lamm equation which takes into account the combined effects of sedimentation and diffusion, as follows ... 

Plug flow reactor Sometimes called a piston flow or a perfect flow reactor. The plug flow reactor has continuous input and output of material. The plug flow assumption generally requires turbulent flow. No radial concentration gradients are assumed. 

This conclusion can be drawn because the small amount of vanadium deposited is almost uniformly distributed in the grain volume. A vanadium radial concentration gradient induced by a diffusion limitation may be one reason for the discrepancies found in the literature. In such cases, the active phase inside the grain is less affected by the vanadium, but can be affected by the carbon deposit. 

For a particle with radial concentration gradient C, and radius r, the emission intensity /, should be the sum of the contributions from all the dopants within the particle ... 

When a fixed bed of catalyst is operated isothermally as an id al piston-flow reactor without radial concentration gradients and without change in the gas density, the differential mass balance for reactant along the bed is given by... 

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