Stability limit

Table 1. Stability Limits and Resonant Tiraesteps for the Ver-let and Implicit Midpoint Schemes...

E. Barth and T. Schlick. Overcoming stability limitations in biomolecular dynamics I. combining force splitting via extrapolation with Langevin dynamics in LN. J. Chem. Phys., 109 1617-1632, 1998. 

This gives the well-known stability limit... 

The relay has voltage as well as current settings. The former defines the stability limit against through-faults, as discussed above, while the latter determines the sensitivity of the protected zone. 

The element Pg/sin B can be considered as the steady-state stability limit of the line, say P ax- I tie length compensation can improve the voltage profile and hence the power transfer capability of the line as follows. 

Retaining the stability limit of the system thus modified. 

In Figure. 2(b) we plot the stability limit temperatures for Pd-concentration x = 0.1,... 

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

Tahle 7. Electrochemical stability ranges or anodic stability limits of several nonaqueous electrolytes... 

Assigned as oxidation voltage, the anodic stability limit is slightly lower. 

Table 7 lists the electrochemical windows or the anodic stability limits of several nonaqueous electrolytes, or their anodic stabilty, the reference electrodes Rref used, the working electrode material Ew, the experimental conditions, and the references. It shows the following features ... 

Comparison of stability limits of low-temperature molten salts and lithium salts with a common anion shows the influence of the solvent, which limits the anodic stability range of solutions based on LiMe or Lilm. The 1,2-dimethyl-3-propylimidazolium methide... 

Figure 7. Anodic stability limits of lithium ben-zenediolatoborates, Li[B(C6H4, in PC.

Figure 10. Stability limits of lithium chelatoborates for details see the text.

Turbulence is known to occur in pipe flow at a Reynolds number (Re) above 2300. Beyond this stability limit, any disturbance will grow exponentially in time and the flow becomes fully chaotic at Re 4000, where Re is defined by ... 

C.G. Vayenas, and J. Michaels, On the Stability Limit of Surface Platinum Oxide and its role in the oscillatory behavior of Platinum Catalyzed Oxidations, Surf. Sci. 120, L405-L408 (1982). 

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

Flow in a Slit. Turning to a slit geometry, a flat velocity profile gives the simplest possible solution using Euler s method. The stability limit is independent of y ... 

J. Quinard, G. Searby, and L. Boyer. Stability limits and critical size of structures in premixed flames. Progress in Astronautics and Aeronautics, 95 129-141, 1985. 

R. Relce and R. Glavin. Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames. /. Fluid Mech., 124 219-237, 1982. 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

A point meriting attention is the voltage difference above. Doped polymers are rather electropositive (up to more than 4 V vs. a lithium electrode in the same solution), so much so that charging may have to be limited in order not to exceed the stability limits of the electrolyte (typically, propylene carbonate or acetonitrile as aprotic nonaqueous solvents). 

Two lines on the Pourbaix diagrams (Figure 5.2) deserve reference. These are the sloping dashed lines, and they represent the stability limits for water they are essentially the domain of thermodynamic stability of water superimposed on potential-pH diagrams. At the upper line, oxidation of water goes according to ... 

The closed-loop poles may lie on the imaginary axis at the moment a system becomes unstable. We can substitute s = jco in the closed-loop characteristic equation to find the proportional gain that corresponds to this stability limit (which may be called marginal unstable). The value of this specific proportional gain is called the critical or ultimate gain. The corresponding frequency is called the crossover or ultimate frequency. 

Kozlov, B. K., 1954, Forms of Flow of Gas-Liquid Mixtures and Their Stability Limits in Vertical Tubes, Zh Tekh. Fiz. 24( 12) 2285—2288. Transl. RJ-418, Assn. Tech. Service, East Orange, NJ. (3) Kramer, A. M., 1958, Boiling Water Reactors, Addison-Wesley Pub. Co., New York, (4)... 

Careful sizing of the treatment and choice of injection rates is required to prevent inadvertent overtreatment i.e., excessive treatment of oil-containing rock. The post-treatment fluid injection rate is usually significantly less than that prior to treatment. While successful applications of this technology in waterfloods and in surfactant polymer floods have been reported, temperature and pH stability limitations of the polymer and the crosslinking chemistry result in few if any applications in steam and CO2 injection wells. 

Ryles, R.G. "Chemical Stability Limits of Water-Soluble Polymers Used in Oil Recovery Processes," SPE Paper 13585, 1985 International Symposium on Oilfield and Geothermal Chemistry, Phoenix, April 9-11. 

The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations... 

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