Rate Fields

f is the normalized rate vector (normalized to unit magnitude). Note that when scaling is performed, information regarding the magnitude of r is lost, which may be important in understanding how large (how fast) reaction rates are relative to each other in space. 

For brevity, let x = Cx, y = Cy, and z = c. The kinetics associated with this system is assumed to be artificial, given by the following rate expressions for components X and Y  

Answer a Vector field in x-y space We are interested in plotting the rate field in x-y space. The concentration vector is hence defined by C = [x, y]. Since the kinetics depends on x and y only, a grid of points in x-y space may be generated over the desired range the 

The vectors are scaled to equal magnitude in order to only indicate the direction of the vectors (the direction of concentration movement). 

Answer b Vector field in x-z space Plotting the rate field in x-z space is slightly more complicated than that in x-y space. Since the kinetics is not infiuenced by the concentration of component Z directly, plotting the rate field in x-z space requires that the corresponding concentration is first computed for the x-y pair, and then back calculated by mass balance to find the x-z pair. We can achieve this transformation by writing a steady-state component molar balance in terms of the extent of reaction and the feed point as follows  

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Equipment Feed rate. Field Dia Width Apphcation... 

Based on the flame-hole dynamics [59], dynamic evolutions of flame holes were simulated to yield the statistical chance to determine the reacting or quenched flame surface under the randomly fluctuating 2D strain-rate field. The flame-hole d5mamics have also been applied to turbulent flame stabilization by considering the realistic turbulence effects by introducing fluctuating 2D strain-rate field [22] and adopting the level-set method [60]. 

Baker A, Smart PL (1995) Recent flowstone growth rates Field measurements in comparison to theoretical predictions. Chem Geol 122 121-128... 

Models conditioned on the turbulent dissipation rate attempt to describe non-stationary effects due to the fluctuating strain-rate field, and thus should be adequate for flamelet applications which require a model for the mixture-fraction dissipation rate at the stoichiometric surface. 

Figure 1.2 Schematic of a compact combustor using countercurrent shear layer (a) and comparison of measured strain rate fields in a single-stream (6) and countercurrent (c) shear layers [9]...

The results from Fig. 14.1 show the developing turbulent flame zone. The nonsymmetries of the reaction rate field are due to inhomogeneity of the polydis-persed mixture, i.e., nonsymmetrical distribution of model particles and their velocities. The reaction front is under formation oxygen and partially the volatiles in the center are burnt out, but the reaction front is not sphere-shaped yet. The nonuniformity of the model particles distribution was induced initially due to the stochastic modeling of the particulate phase. 

The shear rate field that results from such cyclic deformation is... 

This dependence is certainly different from the amplitude of the RR stress and strain-rate fields which is Kjlt. This is an illustration of why the amplitude of the RR-field, C(t), is not necessarily the crack driving force parameter. This is in contrast to the ambient temperature situation wherein the strain energy release rate correlates exactly with either G (= K2/E) or /, both of which also govern the amplitude of the appropriate elastic or elastic-plastic stress fields. 

Because of the presence of a creep zone at the tip of a growing creep crack, the crack tip fields can be expressed in terms of stress-strain rate fields as a function of the distance from the crack tip within the creep zone. These... 

For different polymers the results can be more readily appredated by examining the change in pressure with extrusion ratio R for a constant extrudate velocity. Results for different polyethylenes are shown in Fig. 17, where the rapid upturn occurs at comparatively low extrusion ratios. For different polymers results are shown in Fig. 18, together with the best analytical Gts based on modifled Hoffman-Sachs analysis, which incorporates the strain, strain rate and pressure dependent flow stress according to Eq. (4) and the Avitzur strain rate field of Eq. (5). Figure 17... 

Sluckin adopted a quasi-equilibrium thermodynamic approach to understanding the effect of a strain rate field on the isotropic-anisotropic transition in polymer solutions. He derived a Clausius-Clapeyron-like equation which connects the shift in the critical polymer mole fraction C, and Cj, which are concentrations of isotropic and nematic phases, respectively, to the applied strain rate. 

If the ratio f lf2 I is greater than unity the torques induced by the symmetric and antisymmetric strain rates respectively will never cancel out and the antisymmetric pressure will never vanish. This means that the director continues rotating for ever. The liquid crystal is said to be flow unstable and complicated flow patterns arise. TTiey have been studied comprehensively both experimentally and theoretically [30]. Some nematic liquid crystals are flow stable whereas others are not. For example, 4-n-pentyl-4 -cyanobiphenyl (5CB) is flow stable whereas 4-n-octyl-4 -cyanobiphenyl (8CB) is flow unstable. The only difference between this two substances is the length of the hydrocarbon chain attached to the cyanobiphenyl skeleton. Nematic liquid crystals that are flow stable usually become flow unstable close to the nematic-smectic A transition. The reason for this is that there is an emergent layer structure in the fluid that is incommensurate with the strain rate field. 

The nematic phases of both prolate and oblate Gay-Beme ellipsoids seem to be flow stable. The prolate system becomes flow unstable near the nematic-Smectic A transition because the smectic layer structure is incommensurate with the Couette strain rate field. The effective viscosity of nematic phases of... 

In the general case, the TE and WP surfaces differ substantially. The determination of the ECM rate field requires consideration of the transfer processes in the IEG, which is a complex-shaped, three-dimensional channel. In this case, the quasi-steady-state approximation method is employed. 

The method of quasi-steady-state approximation (the step method) implies the use of the explicit difference scheme for solving the problem with moving boundaries. In doing so, the ECM rate field at the instant time t is calculated from the transfer equations (8) in the region (Fig. 7) with... 

The quasi-steady-state approximation may be used, because the rates of the transfer processes in the I EG (meters per second) are considerably higher than the rate of the variation of the WP surface (millimeters per minute). Within the framework of the quasi-steady-state approximation, it is possible to divide the initial problem into two subproblems (1) Calculation of the transfer processes in the I EG and the determination of the ECM rate field Va. (2) Calculation of the WP surface evolution for the direct problem or correction of the TE surface for the inverse problem. However, even under this simplification, solving the direct and inverse ECM problems, especially for sculptured WPs, involves great difficulties. 

Non-Newtonian mixtures are homogenized considerably more slowly than Newtonian liquids in laminar and transition ranges. This is also due to a shear rate field existing in the tank, which causes strong viscosity differences. 

Tennekes and Lumley ([167], p. 258) presented a different mechanism. They imagined that the smaller eddies are exposed to the strain-rate field of the larger eddies. Because of the straining, the vorticity of the smaller eddies increases, with a consequent increase in their energy at the expense of the energy of the larger eddies. 

Considering the overall flux of material through the region affected by the inclusion, this is greater when diffusion is taken into account. On the other hand, the rate at which the inclusion changes ellipticity is slightly less. In terms of velocity fields, the strain-rate field by itself is smaller when diffusion operates than when it does not, but the sum of strain-rate field plus the velocity field for the self-diffusive movements is greater than the strain-rate field in the no-diffusion case. 

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