First-order instabilities

Simple static instability. Flow excursion (Ledinegg instability) involves a sudden change in the flow rate to a lower value. It occurs when the slope of the channel demand pressure drop-versus-flow rate curve (internal characteristic of the channel) becomes algebraically smaller than the loop supply pressure drop-versus-flow rate curve (external characteristic of the channel). The criterion for this first-order instability is... 

Simulations of incommensurate surfaces showed a similar dependence on Vi, with first-order instabilities occurring if Vi < Vj, where Vj is some positive, critical value that depends on the degree of mismatch between the lattice constants of the top and bottom surfaces. This process leads to nonvanishing Fk as l o goes to zero. In the case where Vi < V, the atoms are dragged with the wall that exerts the maximum lateral force. It, in turn, leads to friction that scales linearly with the sliding velocity. As a result, the friction force will go to zero with vq. 

The effect of dimensionality was also considered in that study. It was found that systems with commensurate 2-D walls yield results that are similar to the 1-D case because the interference between Vt and V[, persists. This situation is no longer true for the incommensurate case, where the adsorbed atoms can circumnavigate the points of maximum lateral force, which permits first-order instabilities regardless of the nature of the higher order harmonics in the wall-atom potential. Thus, one would expect friction to remain finite as Vq... 

First-order instabilities may not only involve the translational motion of atoms confined within contacts, but they may also involve chemical reactions within the confined fluid itself. This has been demonstrated recently in first-principles studies of zinc phosphates, which are found in protective films formed in automobile engines.19,83 Here, we focus on simulations of systems containing phosphate molecules in which pressure-induced chemical reactions lead to hysteresis and energy dissipation. The reactions involving zinc phosphates are discussed below along with other tribochemical reactions. 

In the instability region, the radial increment of the moving path of the jet is constant, thus the fiber formation space is conical. The vertical speed of the emerging jet was estimated to be approximately 3 m/s [77] and the rotation direction of the jet can be both clockwise and counterclockwise depending on the first-order instability formation. 

Analyzing the Laplace pressure difference in a film sinusoidally perturbed (given by the two principal radii of curvature) it results that the pressure in a valley is larger than in a crest if the wavelength of the perturbation (A.) is larger than Ina, where a is the fiber radius. In this case, the liquid flows from the valley to the crest and the perturbation is increased. A first order instability analysis shows that the fastest growing perturbation has a wavelength... 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

Forced-Convection Flow. Heat transfer in pol3rmer processing is often dominated by the uVT flow advectlon terms the "Peclet Number" Pe - pcUL/k can be on the order of 10 -10 due to the polymer s low thermal conductivity. However, the inclusion of the first-order advective term tends to cause instabilities in numerical simulations, and the reader is directed to Reference (7) for a valuable treatment of this subject. Our flow code uses a method known as "streamline upwindlng" to avoid these Instabilities, and this example is intended to illustrate the performance of this feature. 

The reaction was followed by observing the appearance of the yellow colour of Ce(IV) at 400 m, with Pb(IV) present in excess concentration. Pb(IV) was varied in the region 8.6 x 10 M to 4.4 x 10 M while Ce(III) was kept at 4 x 10 M. A practical difficulty encountered was the photochemical instability of Ce(IV) acetate. Under the above conditions and in the temperature range 30-47 °C, the reaction is strictly first-order in each reactant. The observed rate coefficient at 30.0 °C is 1.48x10" l.mole sec and the apparent activation energy and... 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

Phase transitions, whether first-order or second-order, are potent sources of instability of solid drugs and can usually be detected and studied by thermal methods of analysis (e.g., DSC, TGA, TMA, ODSC, DMA, DEA). In crystalline solids, typical first-order transitions are polymorphic or desolvation transitions. In amorphous solids, second-order transitions, such as glass transitions, are common. 

Ethylene pressure studies have revealed a first-order dependence on ethylene for both the rate of chain propagation and the rate of chain transfer. This polymerization behavior together with X-ray analyses and DFT calculations has provided strong support for (1-11 transfer to an incoming monomer, which is responsible for the production of vinyl-terminated PEs. The calculations thus suggest that the catalysts disfavor (I-11 transfer to the Zr metal because of the extreme instability of the Zr hydride species that is produced in such a chain transfer process. 

Lovett, R., Ortoleva, P. Ross, J. (1978). Kinetic instabilities in first-order phase transitions. J. Chem. Phys., 69, 947-55. 

Owing to the great instability of the radicals formed from propane and larger molecules, reaction (3.99) is fast and effectively first-order thus, competitive reactions similar to (3.100) need not be considered. Thus, in reactions (3.94) and (3.95) the M has to be included only for ethane and, to a small degree, propane and in reaction (3.99) M is required only for ethane. Consequently, ethane is unique among all paraffin hydrocarbons in its combustion characteristics. For experimental purposes, then, ethane (like methane) should not be chosen as a typical hydrocarbon fuel. 

From the three eases above we can conclude that, for this third-order system with three equal first-order lags, the 4-2-dB specification is the most conservative, the 45° PM is next, and the 2 GM gives the controller gain that is closest to instability. 

The absolute instability of the "metastable" states in the framework of classical elasticity manifests itself in dynamics as well. The associated elastodynamical problem reduces to a solution of the nonlinear wave equation = o (uJu . It is convenient to rewrite it as a mixed type first order system... 

Recently, the assignment of the band at 980 cm to 28 has been doubted based on new calculations (this band is shifted to 976 cm if 28 is generated from 1,4-diiodobenzene (37), which is not unusual in the presence of iodine atoms. This shift may also be attributable to the change of the matrix host from argon to neon). ° On the other hand, ab initio calculations of the IR spectrum of 28 are complicated by the existence of orbital instabilities, the effect of which may (often) be negligible for first order properties (such as geometry and energy), but can result in severe deviations for second-order properties (vibrational frequencies, IR intensities). 

In the case of the nonisothermal first-order exothermic reaction heat is auto catalytic, for it raises the temperature and provokes an increase of reaction rate, yet is itself a product of the reaction. In the Gray-Scott scheme, B is plainly autocatalytic and its degeneration by the second reaction plays the role of the direct cooling in the non-isothermal case. This reaction appears in the chemical engineering literature in 1983,16 and is the keynote reaction in Gray and Scott s 1990 monograph on Chemical Oscillations and Instabilities.17 A justification of the autocatalytic mechanism in terms of successive bimolecular reactions is the subject of Chapter 12. 

Another type of stability problem arises in reactors containing reactive solid or catalyst particles. During chemical reaction the particles themselves pass through various states of thermal equilibrium, and regions of instability will exist along the reactor bed. Consider, for example, a first-order catalytic reaction in an adiabatic tubular reactor and further suppose that the reactor operates in a region where there is no diffusion limitation within the particles. The steady state condition for reaction in the particle may then be expressed by equating the rate of chemical reaction to the rate of mass transfer. The rate of chemical reaction per unit reactor volume will be (1 - e)kCAi since the effectiveness factor rj is considered to be unity. From equation 3.66 the rate of mass transfer per unit volume is (1 - e) (Sx/Vp)hD(CAG CAl) so the steady state condition is ... 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

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