Stability conditions first order

The governing equations (12-307) for a 2D disturbance (ay = v = 0) can be combined to obtain a single higher-order equation that can be used to study the stability conditions. First, we note that the continuity equation (12-307d) can be satisfied by introducing a function [Pg.875]

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

The Cauchy problem for a system of differential equations of first order. Stability condition for Euler s scheme. We illustrate those ideas with concern of the Cauchy problem for the system of differential equations of first order... 

Polymers designed with this technique have a number of important aspects in common with proteins. First of all, the transition from a liquid-like globule into a frozen state occurs as a first order phase transition. Further, the frozen state itself has an essential stability margin, which is determined by the design parameters. As in real proteins, neither a large variation of temperature or other environmental conditions, nor a mutational substitution of several monomers leads to any change in basic state conformation. In this respect the ability of sequence design to capture certain essential characteristics of proteins seems quite plausible. 

An initial use of reflectance spectroscopy in the characterization of pharmaceutical solids concerned studies of the stability of coloring agents in tablet formulations. With the description of a device that enabled the surface of intact tablets to be studied [14], the photostability of various dyes and lakes in tablets was followed [15,16]. Exposure of formulations to both normal and exaggerated light conditions was investigated, and the kinetics of the photodegradation evaluated. In most cases, the photoreactions appeared to follow first-order kinetics. 

The reactant R2 can also be considered to be a solvent molecule. The global kinetics become pseudo first order in Rl. For a SNl mechanism, the bond breaking in R1 can be solvent assisted in the sense that the ionic fluctuation state is stabilized by solvent polarization effects and the probability of having an interconversion via heterolytic decomposition is facilitated by the solvent. This is actually found when external and/or reaction field effects are introduced in the quantum chemical calculation of the energy of such species [2]. The kinetics, however, may depend on the process moving the system from the contact ionic-pair to a solvent-separated ionic pair, but the interconversion step takes place inside the contact ion-pair following the quantum mechanical mechanism described in section 4.1. Solvation then should ensure quantum resonance conditions. 

The reaction is acid-catalyzed and yields isocyclosporin A (iso-CsA, 6.58, Fig. 6.23) as the major product. At 50°, the kinetics of the first-order reaction were k=l x 10 6 s 1 (tV2 ca. 1.1 d) at pH 1, and k= 1. 7 x 10 8 s 1 (tv2 ca. 1.2 y) at pH 4. Iso-CsA (i.e., the O-peptide) had a much greater chemical stability than CsA (i.e., the A-peptide) under acidic conditions, in contrast to other findings where the opposite was true. Interestingly, O-acclyl-CsA did not yield iso-CsA and exhibited a much greater stability than CsA, consistent with the nucleophilic mechanism mentioned above. 

Prom the results presented in this chapter, it has been shown that the first step in the control problem of a CSTR should be the use of an appropriate mathematical model of the reactor. The analysis of the stability condition at the steady states is a previous consideration to obtain a linearised model for control purposes. The analysis of a CSTR linear model is carried out trough a scaling up reactor s volume in order to investigate the difference between the reactor and jacket equilibrium temperatures as the volume of the reactor changes from small to high value. 

Second method consists of a straightforward discretization method first order (Euler) explicit in time and finite differences in space. Both the time step and the grid size are kept constant and satisfying the Courant Friedrichs Lewy (CFL) condition to ensure the stability of the calculations. To deal with the transport part we have considered the minmod slope limiting method based on the first order upwind flux and the higher order Richtmyer scheme (see, e.g. Quarteroni and Valli, 1994, Chapter 14). We call this method SlopeLimit. 

Stability of an enzyme is defined as its ability to retain catalytic activity under specified conditions (Martinek and Berezin, 1978). The quantitative criterion for stability is the value of the first-order rate constant of monomolecular inactivation, king k-. 

A characteristic manifestation of the coexistence of two gel phases and hence of the first-order phase transition in a swollen network consists of the van der Waals loop which appears in the dependence of the swelling pressure P (or of the chemical potential of the solvent plf see Eq. (1)) on 0. The composition of coexisting gel phases at the collapse (values

2) is given by the condition of equality of the chemical potentials of the solvent px and polymer p2 in both phases... 

First suppose that the stability condition (X.3.4) is obeyed. Then there is only one stationary macrostate s. It is related to the stationary mesostate PS(X) in the sense that the latter consists of a sharp peak around Q(/)s, which in the limit Q-+ oo tends to a delta function at 2s. Moreover, thanks to (X.3.4) it is possible to relate with every time-dependent macrostate ( ) a time-dependent mesostate P(X, t) consisting of a sharp peak around (j)(t) and moving along with it. This relation is neither unique nor precisely defined to each ( ) there are many P(X, t) with these properties and there is no precise way of telling how sharp they have to be, except that the width must not be larger than order Q1/2. 

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 ... 

Storage stability over time under fixed conditions of temperature, pH value and concentration of additives often can be expressed by a first-order decay law (analogous to radioactive decay) [Eq. (2.19)]. 

In the reactions discussed and exemplified above, reactants, transient species and products are related by linear sequences of elementary reactions. The transient species can be regarded as a kinetic product and, if isolable, subject to the usual tests for stability to the reaction conditions. Multiple products, however, may also occur by a mechanism involving branching. Indeed, the case shown earlier in Fig. 9.5b, where the transient is a cul de sac species, is the one in which the branching to the thermodynamic product P and kinetic product T occurs directly from the reactant. In the absence of reversibility, the scheme becomes as that shown in Scheme 9.8a, where the stable products P and Q are formed as, for example, in the stereoselective reduction of a ketone to give diastereoisomeric alcohols. The reduction of 2-norbornanone to a mixture of exo- and cndo-2-norbornanols by sodium borohydride is a classic case. The product ratio is constant over the course of the reaction and reflects directly the ratio of rate constants for the competing reactions. The pseudo-first-order rate constant for disappearance of R is the sum of the component rate constants. 

Gutsche et al. observed that a lipophilic dioxime (the ligand in 4) accelerates by a factor of 60 to 140 the hydrolysis of acetyl phosphate in the presence of an equimolar amount of metal (M = Cu, Zn, Ni) under comicellar conditions with cetyl trimethylammonium bromide (CTAB) at pH 11.5 [22], The hydrolysis reaction was followed by pseudo-first-order kinetics at [ligand] = [M] = 2.5 mM and [CTAB] = 250 mM. Although a possible complex 4 was proposed to exist in the hydroxamate form, there were no experimental data for the stability and structure of metal complexes at pH 11.5 (e.g., it is not known whether the... 

---