Concentration field

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Recalling that and are smoothed concentration fields for the macro-... 

It arises solely because we continue Co describe micropore diffusion in terms of smooch macropore concentration fields and their gradients, even under reactive conditions where these no longer adequately describe Che actual concentration gradients in the micropores. 

The mmen is not functional at birth and milk is shunted to the abomasum. One to two weeks after birth, the neonate consumes soHd food if offered. A calf or lamb that is nursing tends to nibble the mother s feed. An alternative method of raising the neonate is to remove it from its mother at a very young age, <1 week. A common example of an early weaning situation is the dairy calf that is removed from the cow soon after birth so that the cow s milk supply might be devoted entirely to production. In this instance, the neonate requires complete dietary supplementation with milk replacer. Sources of milk replacer protein have traditionally included milk protein but may also include soybean proteins, fish protein concentrates, field bean proteins, pea protein concentrates, and yeast protein (4). Information on the digestibiUty of some of these protein sources is available (4). 

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

Figure 10-11c. The concentration field in the 3-D blending simulation at Time = 10 s (see full-color version on CD).

Actually, this is not really diffusion-XimiiQd, but rather Laplacian growth, since the macroscopic equation describing the process, apart from fluctuations, is not a diffusion equation but a Laplacian equation. There are some crucial differences, which will become clearer below. In some sense DLA is diffusion-limited aggregation in the limit of zero concentration of the concentration field at infinity. 

Kamenskaya [221] and Konstantinov [37] investigated the concentration field adjacent to the KF - NbF5 side of the ternary interconnected system K+, Nb5+//Q2, F. Fig. 55 shows the projection of the crystallization fields. 

The spatio-temporal variations of the concentration field in turbulent mixing processes are associated wdth very different conditions for chemical reactions in different parts of a reactor. This scenario usually has a detrimental effect on the selectivity of reactions when the reaction time-scale is small compared with the mixing time-scale. Under the same conditions (slow mixing), the process times are increased considerably. Due to mass transfer inhibitions, the true kinetics of a reaction does not show up instead, the mixing determines the time-scale of a process. This effect is known as mixing masking of reactions [126]. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

Figure 2.40 Zigzag micro mixer with concentration field (left) and flow stream lines (right) obtained from a CFD simulation for a Reynolds number of 38. In [135] a sawtooth geometry of larger amplitude was considered and distinctive recirculation zones were found only at Reynolds numbers larger than 80.

Concentration averaged over the cross section of a tube cf (co) Laplace transform of the effective concentration field... 

A. Frank, G. G. Lipscomb, M. Dennis 2000, (Visualization of concentration fields in hemodialyzers by computed tomography),/. Membrane Sci. 175, 239-251. 

The spatial and temporal evolution of the concentration field is dependent on the velocity field vector v(r,t), the diffusion tensor D(r,t) and any reactions occurring in the system R(r,t). Non-dimensionalization of Eqn. (5.1.4) generates the Pedet... 

If kinetic processes on catalytic surfaces in S02 oxidation are assumed to be at steady state, temperature and concentration fields in a radially symmetrical, adiabatic catalyst bed are described by the equations collected in Table IX for the reactor space 0 and time I > 0 (Matros, 1989 ... 

A pharmacotectonics concept was illustrated by researchers, in which drug-delivery systems were arranged spatially in tissues to shape concentration fields for potent agents. NGF-releasing implants placed within 1-2 mm of the treatment site enhanced the biological function of cellular targets, whereas identical implants placed mm from the target site of treatment produced no beneficial effect (Mahoney and Saltzman, 1999). Because of some limitations with controlled delivery systems, alternatives such as encapsulation of cells that secrete these factors are discussed in the next section. 

The destabilizing influence of the electrostatic and concentration fields are expressed by the dimensionless groups Fe and Fc, while the stabilizing effect of surface-excess... 

Fig. 6. A sample of Hartmann s results (2005). The tracer is injected (with zero speed) at the top of the tank in a plane midway between two baffles (black dot in the plots). The concentration fields shown are also in the midway baffle plane. T/D = 3, c, denotes the final uniform concentration.

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