Conditionally stable case

The system (2.30), and a more complicated one, can be found in [5]. HI. BOUNDARY VALUE PROBLEM A. Conditionally Stable Case... 

Then we have the critical conditionally stable case. In this case, we must impose the following boundary conditions p + q q k) conditions at t = 0, and m-p-q conditions at t = 1. 

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. 

Pulsed detonation devices based on the DDT principles should contain wider chambers in the ignition section (fore-chambers) to guarantee a stable periodical operating mode insensitive to initiating conditions. In case of direct initiation of detonation, such chambers prevent detonation formation and propagation. 

Actually, as has been pointed out by A. Turkevich, 1 Mev has to be subtracted from this. The reason is that the final nucleus is not the most stable nucleus but has, in most cases, a smaller charge. In the case of odd mass, the Z of the final nucleus can differ from Zo by as much as db. Hence, the excess energy is, in this case, the average of aZ for Z between — and J, i.e., a/12 = 0.08a. The condition, in case of even mass... 

A system is bistable if, for a given value of perturbation in a variable x, its response to another variable y may take two stable values under steady-state conditions. In case of electro-kinetic phenomenon, current provides the perturbation, while the corresponding potential difference yields the response (Chapter 8). 

Note that the scheme is conditionally stable if it is applied to an exponentially damped problem (i.e., cd = i v), where v is real and positive. In this case, the stability eon-dition is v Ar < 2. 

Remark 1 We call this case conditionally stable since the rest point HqZ = 0 is now not asymptotically stable, and attracts only those solutions whose initial points lie on the separatrix 1. 

Since sin pzk/2 > 0, the inequality always holds if At > 0, thus guaranteeing stability. (Of course, mesh sizes must be kept small in order to reduce truncation errors and to ensure convergence to solutions of the PDE.) Unlike the conditionally stable explicit scheme studied earlier, this implicit scheme, which requires only tridiagonal matrix inversion, is unconditionally stable. We have tacitly assumed a positive time step At > 0 in arriving at this stability, which is the usual case. But in Chapter 21, we will introduce reverse time integration where we have At < 0. For such applications, the stability requirements are altered, and the nature of the numerical truncation errors changes. 

It is quite clear, first of all, that since emulsions present a large interfacial area, any reduction in interfacial tension must reduce the driving force toward coalescence and should promote stability. We have here, then, a simple thermodynamic basis for the role of emulsifying agents. Harkins [17] mentions, as an example, the case of the system paraffin oil-water. With pure liquids, the inter-facial tension was 41 dyn/cm, and this was reduced to 31 dyn/cm on making the aqueous phase 0.00 IM in oleic acid, under which conditions a reasonably stable emulsion could be formed. On neutralization by 0.001 M sodium hydroxide, the interfacial tension fell to 7.2 dyn/cm, and if also made O.OOIM in sodium chloride, it became less than 0.01 dyn/cm. With olive oil in place of the paraffin oil, the final interfacial tension was 0.002 dyn/cm. These last systems emulsified spontaneously—that is, on combining the oil and water phases, no agitation was needed for emulsification to occur. 

Many optical studies have employed a quasi-static cell, through which the photolytic precursor of one of the reagents and the stable molecular reagent are slowly flowed. The reaction is then initiated by laser photolysis of the precursor, and the products are detected a short time after the photolysis event. To avoid collisional relaxation of the internal degrees of freedom of the product, the products must be detected in a shorter time when compared to the time between gas-kinetic collisions, that depends inversely upon the total pressure in the cell. In some cases, for example in case of the stable NO product from the H + NO2 reaction discussed in section B2.3.3.2. the products are not removed by collisions with the walls and may have long residence times in the apparatus. Study of such reactions are better carried out with pulsed introduction of the reagents into the cell or under crossed-beam conditions. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

The condensation conditions must be as mild as possible, because we want to get only the most stable of the three possible enols (from the aldehyde). Though you could not haye predicted the exact conditions either for the double bond. cleayage or for the condensation, you should haye seen that control was possible as in each case the two functional groups are different enough. ( J. Amer. Chem. Soc.. 1960, 636 J. Org. Chem.. 1964, 29, 3740 ... 

Synthetic Fiber and Plastics Industries. In the synthetic fibers and plastics industries, the substrate itself serves as the solvent, and the whitener is not appHed from solutions as in textiles. Table 6 Hsts the types of FWAs used in the synthetic fibers and plastic industries. In the case of synthetic fibers, such as polyamide and polyester produced by the melt-spinning process, FWAs can be added at the start or during the course of polymerization or polycondensation. However, FWAs can also be powdered onto the polymer chips prior to spinning. The above types of appHcation place severe thermal and chemical demands on FWAs. They must not interfere with the polymerization reaction and must remain stable under spinning conditions. 

If conditions aie such that the film does not form, such as in the case of acids, then the reaction proceeds until all the metal is consumed. The reaction of magnesium with hydrofluoric acid [7664-39-3J is an exception to this rule, because a stable fluoride film forms. 

The rate of ion propagation, is independent of the counterion and has been found to be about 46 X 10 in all cases for CF SO", AsF, SbF, SbCFg, PF g, and BF/ counterions. Conditions were the same for all counterions, ie, 8.0 M of monomer in CCI4 solvent and 25°C polymerization temperature. With less stable counterions such as SbCF and BF at most temperatures, the influence of transfer and termination reactions must be taken into account (71). 

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