Stability critical value

Dielectric Strength. Dielectric failure may be thermal or dismptive. In thermal breakdown, appHed voltage heats the sample and thus lowers its electrical resistance. The lower resistance causes still greater heating and a vicious circle, leading to dielectric failure, occurs. However, if appHed voltage is below a critical value, a stabilized condition may exist where heat iaput rate equals heat loss rate. In dismptive dielectric failure, the sample temperature does not iacrease. This type of failure is usually associated with voids and defects ia the materials. 

The tips of structures will be destroyed if the stability parameter critical value cr given by Eq. (93). Using the value G (Eq. (83)), one obtains from (93) a line of smooth transition from CD to FD structures in Fig. 6 ... 

The mechanical properties of ionomers are generally superior to those of the homopolymer or copolymer from which the ionomer has been synthesized. This is particularly so when the ion content is near to or above the critical value at which the ionic cluster phase becomes dominant over the multiplet-containing matrix phase. The greater strength and stability of such ionomers is a result of efficient ionic-type crosslinking and an enhanced entanglement strand density. 

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

In connection with the transition, Ryan and Johnson l0) have proposed a stability parameter Z. If the critical value Zc of that parameter is exceeded at any point on the cross-section of the pipe, then turbulence will ensue. Based on a concept of a balance between energy supply to a perturbation and energy dissipation, it was proposed that Z could be defined as ... 

The critical value of the Reynolds number (Remit) for the transition from laminar to turbulent flow may be calculated from the Ryan and Johnson001 stability parameter, defined earlier by equation 3.56. For a power-law fluid, this becomes ... 

Figure 6. Expanded view of neutral stability diagram for System I with k = 0.865. Critical values for bifurcation of families of cells in a 2Xc sample size are marked.

A review of the literature demonstrates some trends concerning the effect of the polymer backbone on the thermotropic behavior of side-chain liquid crystalline polymers. In comparison to low molar mass liquid crystals, the thermal stability of the mesophase increases upon polymerization (3,5,18). However, due to increasing viscosity as the degree of polymerization increases, structural rearrangements are slowed down. Perhaps this is why the isotropization temperature increases up to a critical value as the degree of polymerization increases (18). 

Saturated nucleate flow boiling of ordinary liquids. To maintain nucleate boiling on the surface, it is necessary that the wall temperature exceed a critical value for a specified heat flux. The stability of nucleate boiling in the presence of a temperature gradient, as discussed in Section 4.2.1.1, is also valid for the suppres-... 

For transporting foam, the critical capillary pressure is reduced as lamellae thin under the influence of both capillary suction and stretching by the pore walls. For a given gas superficial velocity, foam cannot exist if the capillary pressure and the pore-body to pore-throat radii ratio exceed a critical value. The dynamic foam stability theory introduced here proves to be in good agreement with direct measurements of the critical capillary pressure in high permeability sandpacks. 

Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems and stability problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed. 

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of The results of this analysis are shown in Figure 3.7. This relationship has been tested for shear thinning and for Bingham... 

There is some critical value of gain at which the G, B plot goes right through the (—1, 0) point. This is the limit of closedloop stability. See Fig. 13.3e. The value of K, at this limit should be the ultimate gain that we have dealt with before in making root locus plots of this system. We found in Chap. 10 that = 64 and Let us see if the frequency-domain Nyquist stability... 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

This crude analysis is based on the behavior postulated by the Born equation. However, ion-pair formation equilibrium constants have been observed to deviate ma edly from that behavior (22/ -222)1 Oakenful, and Fenwick (222) found a maximum in the ion-pair formation constants of several alkylamines with carboxylic acids when determined at various methanol-water solvent compositions as shown by their data in Fig. 54. The results demonstrate that in this system the stability constant decreases with increasing organic solvent concentration above a.critical value which yields maximum stability. The authors suggested that this was due to a weakening of hydrophobic interactions between the ion-pair forming species by increased alcohol concentrations. In practice the effect of added organic solvent has been either to decrease the retention factor or to have virtually no effect. 

The stability of a gas (i.e., N2, C02, air) bubble in a solution depends on its dimensions. A bubble with a radius greater than a critical magnitude will continue to expand indefinitely and degassing of the solution would take place. Bubbles with a radius equal to the critical value would be in equilibrium, while bubbles with a radius less than the critical value would be able to redissolve in the bulk liquid. The magnitude of the critical radius, Rcr, varies with the degree of saturation of the liquid (i.e., the higher the level of supersaturation, the smaller the Rcr). The work, W, required for the formation of the bubble of radius Rcr is given by La Mer ... 

When one fluid overlays a less dense fluid, perturbations at the interface tend to grow by Rayleigh-Taylor instability (LI, T4). Surface tension tends to stabilize the interface while viscous forces slow the rate of growth of unstable surface waves (B2). The leading surface of a drop or bubble may therefore become unstable if the wavelength of a disturbance at the surface exceeds a critical value... 

While precipitation in homogenous solutions is an exceedingly simple method, usually rather low concentrations of electrolytes must be used if well-dispersed uniform particles are to be achieved. This requirement is based on the need to keep the ionic strength of the system below a critical value in order to prevent the coagulation of the precipitates, which consist, almost without exception, of charged particles. In some instances, concentrations of the reactants can be increased, if stabilizers are added into the systems, although the latter may affect the particle uniformity and/ or shape. 

For 7 > 7C the system did not approach any steady state at all but rather oscillated with p and E amplitudes dependent on the increment of the current above the critical value. With the current increasing above 7C, the stabilized shape of the oscillations soon received a relaxation character, schematically depicted in Fig. 6.1.2. 

Mechanisms of Flame Stabilization. CRITICAL BOUNDARY VELOCITY GRADIENT. A flame stabilized at the port of a Bunsen burner does not actually touch the rim. There is a dark region, called the dead space, between the rim and the flame. Heat is removed and free radicals are destroyed by the solid surface the burning velocity is reduced to zero and the flame is quenched. Even beyond the dead space, where the flame is able to exist as a luminous reaction zone, the burning velocity only gradually rises to the value achieved at a distance from solid surfaces. 

If the critical value for R at the stability limit is denoted i , then... 

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