Oscillation unstable

Method Stable and No Oscillation Stable and Oscillation Unstable... 

A jet emerging from a nonciicular orifice is mechanically unstable, not only with respect to the eventual breakup into droplets discussed in Section II-3, but, more immediately, also with respect to the initial cross section not being circular. Oscillations develop in the Jet since the momentum of the liquid carries it past the desired circular cross section. This is illustrated in Fig. 11-20. 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

Galerkin method becomes unstable and useless. It can also be seen that these oscillations become more intensified as a becomes larger (note that the factor affecting the stability is the magnitude of a and oscillatory solutions will also result using large negative coefficients). 

The main limitation of the pilot-operated regulator is stability. When the gain in the pilot amplifier is raised too much, the loop can become unstable and oscillate or hunt. The two-path pilot regulator (see b) is also available. This regulator combines the effects of self-operated and the pilot-operated styles and mathematically produces the equivalent of proportional plus reset control of the process pressure. 

Unstable liquid oscillations on a tray have received only limited examination when compared to perhaps tray weeping, flooding and froth build-up. Biddulph [87] pro-... 

As more air was added to the channel, the slug flow became unstable, the slug bubble broke down, and eventually the churn flow occurred in the channel. As shown in Fig. 5.3d, the most significant feature of flow characteristics in the churn flow is that the pressure oscillated at a relatively high amplitude, since the gas plug and liquid bridge flowed through the test section alternatively. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

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. 

This equation continues to conserve mass but is no longer stable. The original upset grows exponentially in magnitude and oscillates in sign. This marching-ahead scheme is clearly unstable in the presence of small blunders or round-off errors. 

There are many variations on this theme. Fed-batch and continuous emulsion polymerizations are common. Continuous polymerization in a CSTR is dynamically unstable when free emulsifier is present. Oscillations with periods of several hours will result, but these can be avoided by feeding the CSTR with seed particles made in a batch or tubular reactor. 

Steady states may also arise under conditions that are far from equilibrium. If the deviation becomes larger than a critical value, and the system is fed by a steady inflow that keeps the free energy high (and the entropy low), it may become unstable and start to oscillate, or switch chaotically and unpredictably between steady state levels. 

Model instability is demonstrated by many of the simulation examples and leads to very interesting phenomena, such as multiple steady states, naturally occurring oscillations, and chaotic behaviour. In the case of a model which is inherently unstable, nothing can be done except to completely reformulate the model into a more stable form... 

Figure 3.17. Phase-plane representations of reactor stability. In the above diagrams the point -I- represents a possible steady-state solution, which (a) may be stable, (b) may be unstable or (c) about which the reactor produces sustained oscillations in temperature and concentration.

Thus it is possible for continuous stirred-tank reactor systems to be stable, or unstable, and also to form continuous oscillations in output, depending upon the system, constant and parameter, values. 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

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