Conditionally stable

A difference scheme is said to be conditionally stable when it is stable only if r and h are related in some way, otherwise it becomes unconditionally stable. A scheme, stable for arbitrary values of r and h, is said to be absolutely stable. One can encounter schemes stable for sufficiently small h and r such that h [Pg.310]

The explicit scheme is stable only under condition (25) relating the grid steps h and r (a conditionally stable scheme). 

Scheme (3) is conditionally stable in the space C with respect to the initial data, the right-hand side and the boundary data. The maximum principle for the difference problem (3) may be of help in establishing the indicated properties with further reference to the canonical form... 

Another way to obtain, under suitable conditions, stable dispersions of sur-factant-stabihzed nanoparticles consists in the direct suspension of some materials in w/o microemulsions. The formation of stable dispersions of rutile (size 80-450 mn) and carbon black (200-500 nm) in AOT// -xylene and of rutile, lead chloride, aluminium, antimony in solutions of calcium soaps in benzene has been reported [219,220],... 

Stable in neutral, acidic and weakly alkaline conditions. Hydrolyzed in highly alkaline conditions. Stable to light and heat... 

Stable in an acidic to neutral aqueous solution. Unstable in strongly basic conditions Stable in most organic solvents... 

C <4.7x10 Pa at 20 °C Water 20 mg at 20 °C Readily soluble in polar organic solvents Stable in neutral and weakly basic conditions Unstable in strongly acidic or basic conditions Stable in most of organic solvents such as acetone, acetonitrile and carbon tetrachloride Undergoes hydrolysis to yield methomyl oxime in alkaline solutions... 

Stable in acidic and alkaline conditions Stable to heat and light Log K ow (4.3)... 

When one of the three draft tube velocities was increased to simulate upset conditions, stable operations were still possible. These upset conditions could also be detected by pressure drop differences among various draft tubes and downcomers when differences in draft tube velocities were large. For severe upset conditions, where some of the draft tubes become downcomers, pressure drop measurement alone could not distinguish the solids flow pattern inside the draft tubes. 

From equation 7.204 in Section 7.16.2, Volume 3, the conditionally stable conditions are given when ... 

Under the simulation conditions, the HMX was found to exist in a highly reactive dense fluid. Important differences exist between the dense fluid (supercritical) phase and the solid phase, which is stable at standard conditions. One difference is that the dense fluid phase cannot accommodate long-lived voids, bubbles, or other static defects, whereas voids, bubbles, and defects are known to be important in initiating the chemistry of solid explosives.107 On the contrary, numerous fluctuations in the local environment occur within a time scale of tens of femtoseconds (fs) in the dense fluid phase. The fast reactivity of the dense fluid phase and the short spatial coherence length make it well suited for molecular dynamics study with a finite system for a limited period of time chemical reactions occurred within 50 fs under the simulation conditions. Stable molecular species such as H20, N2, C02, and CO were formed in less than 1 ps. 

For values of gain greater than the system is closedloop unstable. Thus the system is conditionally stable the controller gain must lie between and. ... 

In both cases the open-loop system is unstable but the location of the poles makes the second one more difficult to control. Sketching the root locus for the transfer function in Eq.(33), it is easy to verify that the system is conditionally stable. There is only a range of controller gain, K G K nn, K ,ax), leading to a closed-loop stable reactor. 

Figure 4 shows the root locus for an example with a typical transfer function of a conditionally stable reactor. Note that from Eq.(35) and (36) different root locus can be deduced. The poles are the same but they have different zeros. 

Fig. 4. Root locus of a conditionally stable process. Kmin means the minimum gain from which the system is stable. For K > Ko the system starts to be underdamped. For K > Kmax the system is unstable.

Water-soluble l,3-bis(di(hydroxyalkyl)phosphino)propane derivatives were thoroughly studied as components of Pd-catalysts for CO/ethene (or other a-olefins) copolymerization and for the terpolymerization of CO and ethene with various a-olefins in aqueous solution (Scheme 7.17) [59], The ligands with long hydroxyalkyl chains consistently gave catalysts with higher activity than sulfonated DPPP and this was even more expressed in copolymerization of CO with a-olefins other than ethene (e.g. propene or 1-hexene). Addition of anionic surfactants, such as dodecyl sulfate (potassium salt) resulted in about doubling the productivity of the CO/ethene copolymerization in a water/methanol (30/2) solvent (1.7 kg vs. 0.9 kg copolymer (g Pd)" h" under conditions of [59]) probably due to the concentration of the cationic Pd-catalyst at the interphase region or around the micelles which solubilize the reactants and products. Unfortunately under such conditions stable emulsions are formed which prevent the re-use... 

Should any pole of the system lie on the imaginary axis (i.e. when its real part is zero) then the system is conditionally stable. Poles lying to the right of the imaginary axis (corresponding to roots of the characteristic equation with positive real parts) indicate unbounded behaviour, i.e. that the system is unstable. 

Hence with Kc= 1.8 and r, = 3.5, the polar plot of the open-loop transfer function passes through the point (-1, 0). This confirms the result obtained in Example 7.6 using the Routh-Hurwitz criterion, i.e. that with these controller parameters, the response of the controlled variable is conditionally stable. Figure 7.55 shows polar plots of the open-loop transfer function Gr(i) for different values of Kc and t>. 

Thus, from the Nyquist criterion (Section 7.10.5), the system will be conditionally stable if the polar plot of NG(s) passes through the point (-1,0) on the complex plane. Under these circumstances there will be one frequency o>x at which ... 

Hence, if Gr(icomplex plane, then a conditionally stable system is indicated by the conditions existing at the point of intersection of the two plots. This is illustrated in the following example. 

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