Disturbance amplification

Brewster, R.A. and Gebhart, B., Instability and Disturbance Amplification in a Mixed-convection Boundary Layer , J. Fluid Mechanics, Vol. 229, pp. 115-133. 1991. 

Brewstar R.A. and Gebhart B. (1991). Instability and disturbance amplification in a mixed-convection boundary layer. J. Fluid Mech, 229 115-133. 

Some factors that can mitigate disturbance amplification in a column train are... 

Figure 19.13 Disturbance amplification in a column train, (a) No heat interaction (6) with heat interaction, poor control (c) with heat interaction, good control. (Parts b and c based on Dak E. Lupfer, Proc. 53rd Ann. Convention of the Gas Processors Assoc., March 25-27,1974. Reproduced courtesy of the Gas Processors Association.)...

Beware of disturbance amplification via a feed-bottom interdianger. Some control schemes handle such disturbances better than others. 

The model and parametric uncertainties are represented by a differential operator A and can be properly treated as a disturbance to the plant, Ws = A(xp), which physically represents the energy amplification from input to output. Its global behavior is characterized by the L2 gain as follows ... 

From the development of a comprehensive model of the factors leading to instabilities in molecular discharges, Haas [87] has found that these systems are prone to several modes of instability. Of these modes, the conditions leading to the development of ionization and neutral particle energy transfer modes of instabilities have been found to be most easily satisfied under laser conditions. Ionization instabilities are produced when conditions enhance the temporal amplification of an imbalance between electron production and loss processes. The characteristic growth time for this type of instability that has been found to be independent of pressure and power density is in the 10-8-10 4 sec range [53, 78, 87], In contrast, neutral particle energy transfer instability modes that occur due to amplification of disturbances in translational... 

Let us therefore discuss about spatial instability of parallel flows, mainly the flow past a flat plate at zero angle of attack- a problem that enjoys a canonical status for instability analyses. For the spatial instability problem associated with two-dimensional disturbance held of two-dimensional primary flows, the disturbance quantities will have the appearance of Eqn. (2.3.28) with /3 = 0. Thus for a fixed Re, one would be looking for complex a when the shear layer is excited by a fixed frequency source of circular frequency, lvq- If we define Re in terms of the displacement thickness S as the length scale, then Re = and the results obtained will be plotted as contours of constant amplification rates Oj in Re — lvo)— plane, as shown in Fig. 2.2. 

The Reynolds number at observed transition location (defined as a location where the intermittency factor is about 0.1 i.e. the flow is 10 % of time turbulent and rest of the time it is laminar) for zero pressure gradient flat plate boundary layer is of the order of 3.5 X 10 . This corresponds to Re = 950. The distance between the point of instability and the point of transition depends on the degree of amplification and the kind of disturbance present with the oncoming flow. This calls for a study of local and total amplification of disturbances. The following description is as developed in Arnal (1984) for two-dimensional incompressible flows. 

Figure 2.5 Sketch of local and total amplification of the disturbance field...

The amplification rate suffered by the disturbances within the neutral curve is shown in the middle panel of Fig. 2.5. Note the sign of the plotted rate, with negative values plotted along the positive ordinate direction. For two-dimensional disturbances in two-dimensional mean flow, the amplification rate can be expressed as. 

Following the path-breaking experiments of Schubauer Skramstad (1947), there have been sustained efforts to link the stability theory in predicting transition. Michel (1952) reported first that his compiled data showed the transition to be indicated when the total amplification of TS waves corresponded to A/Ao 10, where Aq is the disturbance amplitude at the onset of instability. This motivated Smith Gamberoni (1956) and van Ingen (1956) to use temporal theory results to show that at transition the total amplification is given by,... 

Convecting free-stream disturbances within this speed range are likely to trigger strong sustained instability, because of the high amplification rate that such modes would experience. Also for c > 0.4f/oo, the convecting disturbances would create damped wave packets. 

In critical cases it may well be worthwhile to make a complete analysis of stability. In many cases, however, enough can be learned by studying what Bilous and Amundson (B7) called parametric sensitivity. These authors derived formulas for calculating the amplification or attenuation of disturbances imposed on an unpacked tubular reactor originally in a steady state, with the idea that if the disturbances grow unduly the performance of the reactor is too sensitive to the conditions imposed on it, that is, to the parameters of the system. The effect of feedback from a control system was not considered. As pointed out by the authors, it would be a much more complicated task to apply their procedure to a packed reactor, but it still would entail far less computation than a study of the transient response. 

A pH meter is a voltmeter that converts the unknown voltage to a current that is amplified and read out. These are high-input impedance devices. (Impedance in an ac circuit is comparable to resistance in a dc circuit. These devices convert the signal to an ac signal for amplification.) Because of their high-input resistance, very little current is drawn, typically 10 to 10 A, and so chemical equilibrium is not greatly disturbed. A voltmeter must be used for irreversible reactions that do not return to the prior state when disturbed by appreciable drawing of current. 

The physical reason why a slow-moving liquid jet breaks up into drops at some distance below the nozzle lies in the interaction between small-amplitude disturbances on the jet and surface tension, with subsequent high-gain amplification of the capillary perturbation. The initial disturbances may be a result of random excitations, such as jet friction or nozzle roughness, or they may be impressed on the jet. 

The wave number of the fastest-growing disturbance can be obtained by differentiating Eq. (10.4.32) with respect to a for = 0 and setting the result to zero. This calculation was made by Rayleigh, and he found the amplification factor to have a maximum value as a function of a given by... 

Snowball effect designates the situation in which small variations of a flowsheet variable, generally a flow rate at the inlet or outlet of a process with recycles, generates large variations of streams around some units inside the process (Fig. 13.7). It is worthy to note that snowball is essentially a steady state effect, and not a dynamic one, because at finite disturbance the amplification remains finite. However, some units cannot tolerate large fluctuations in flows, particularly the distillation columns. Therefore, the designer should avoid snowball effects already at the conceptual stage. 

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