Local steady-state values

Figure 16. The stability of a steady state, as determined by a rate balance plot [287]. Left panel The rate of synthesis and consumption vcon of a substrate S. Right panel The (net) flux difference vnet vco . The steady state value S° is locally stable After transient perturbation...

This equation says that if the advance corresponds to 1 mole of oxygen, (2/4) moles of vacancies arrived at the reaction front (4 - (A, B) O - (A, B)304). As long as k attains a constant (= steady state) value, the balance of B cations yields (with respect to the local reaction A1 NoB O = (1 —A)-A, BnO+(A/4)-AB204 + A2+ions)... 

If the values of local mean bubble diameter and local gas flux are available, a fluid dynamic model can estimate the required influence of mass transfer and reactions on the fluid dynamics of bubble columns. Fortunately, for most reactions, conversion and selectivity do not depend on details of the inherently unsteady fluid dynamics of bubble column reactors. Despite the complex, unsteady fluid dynamics, conversion and selectivity attain sufficiently constant steady state values in most industrial operations of bubble column reactors. Accurate knowledge of fluid dynamics, which controls the local as well as global mixing, is however, essential to predict reactor performance with a sufficient degree of accuracy. Based on this, Bauer and Eigenberger (1999) proposed a multiscale approach, which is shown schematically in Fig. 9.13. 

Figure 8.1 sketches the basic mechanism underlying this type of stationary pattern formation [5] The concentration c of the activator (red line) is locally slightly perturbed from its steady-state value Css (s) This initiates the self-enhanced production of the activator that now slowly diffuses into neighboring regions (b). As a consequence, the inhibitor, that is, the double-layer potential grows (blue dashed line) that propagates more rapidly out of the activator spot (c). Since the inhibitor consumes the activator, the concentration of the latter will always be suppressed outside the spot (d). In this way, a stationary concentration profile may develop surrounded by a halo of increased inhibitor < dl-... 

Recently, the theory of dielectrophoresis was applied to explain the microscopic physics of the movement of pigments in electrophoretic image displays and to prove the discrepancies between theory and measurement [9], Dielectrophoresis is induced by the interaction of the electric field and the induced dipole and is used to describe the behavior of polarizable particles in a locally nonuniform electric field. For example, the phenomenon of the delay time can be explained by the principle of dielectrophoresis. In electrophoresis, when the backplane voltage is switched, the particles on the electrode have to move instantaneously under a given electric field. However, the particles need a removal time which results in a delay time in the switching process. The time constant to obtain an induced dipole from a particle at rest is derived by Schwarz s formula [10] and used to compute the dielectrophoretic force at its steady-state value. The force and the velocity fields under a nonuniform electric field due to the presence of pigments also help to estimate realistic values for physical properties. 

Here, the second subscript denotes the steady state value. The roots of the quadratic characteristic equation (eigenvalues) of the matrix A determine the stability of the equations the system will converge exponentially to the steady state if all roots have a negative real part and, therefore, is asymptotically stable. It will show a limit cycle if the roots are imaginary with zero real parts. It is unstable if any of the roots has a positive real part. Since the perturbations will decay asymptotically if and only if all the eigenvalues of the matrix A have a negative real part, it follows that the necessary and sufficient conditions for local stability are ... 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

In Fig. 3b, L2 and L4 have substantially different STD values, with L4 showing significantly smaller effect, even though these two protons are equidistant from the P3 proton. This is a simple consequence of the differences in the relaxation rates for these L2 and L4 protons due to differences in their local environments (e.g., in the asymmetric model, the L4-L5 distance is shorter than the L3-L2 distance, and thus L4-L5 protons experience a faster longitudinal relaxation rate than the L4-L5 protons). These observations suggest that caution is needed in quahtative attempts to relate the magnitudes of steady state STDs to spatial proximity of ligand protons to the protein protons. 

In the case of material with a significant concentration of localized states, it is possible to assnme that transport of a carrier over any macroscopic distance will involve motion in states confined to a single energy. Here it is necessary to note that a particn-larly important departnre from this limiting situation is (according to Rose [4]) a trap-limited band motion. In this case, transport of carrier via extended states is repeatedly interrnpted by trapping in localized states. The macroscopic drift mobility for such a carrier is reduced from the value for free carriers, by taking into acconnt the proportion of time spent in traps. Under steady-state conditions, we may write... 

Assume that the diffusion field is in a quasi-steady state and that local equilibrium is maintained at the surface and in the volume at a long distance from the surface, where yv = 0 and yA has the value characteristic of a flat surface. 

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