Small disturbance

There are two types of measurement errors, systematic and random. The former are due to an inherent bias in the measurement procedure, resulting in a consistent deviation of the experimental measurement from its true value. An experimenter s skill and experience provide the only means of consistently detecting and avoiding systematic errors. By contrast, random or statistical errors are assumed to result from a large number of small disturbances. Such errors tend to have simple distributions subject to statistical characterization. 

Schemes to control the outlet temperature of a process furnace by adjusting the fuel gas flow are shown in Figure 13. In the scheme without cascade control (Fig. 13a), if a disturbance has occurred in the fuel gas supply pressure, a disturbance occurs in the fuel gas flow rate, hence, in the energy transferred to the process fluid and eventually to the process fluid furnace outlet temperature. At that point, the outlet temperature controller senses the deviation from setpoint and adjusts the valve in the fuel gas line. In the meantime, other disturbances may have occurred in the fuel gas pressure, etc. In the cascade control strategy (Fig. 13b), when the fuel gas pressure is disturbed, it causes the fuel gas flow rate to be disturbed. The secondary controller, ie, the fuel gas flow controller, immediately senses the deviation and adjusts the valve in the fuel gas line to maintain the set fuel gas rate. If the fuel gas flow controller is well tuned, the furnace outlet temperature experiences only a small disturbance owing to a fuel gas supply pressure disturbance.

Laminar flow ceases to be stable when a small perturbation or disturbance in the flow tends to increase in magnitude rather than decay. For flow in a pipe of circular cross-section, the critical condition occurs at a Reynolds number of about 2100. Thus although laminar flow can take place at much higher values of Reynolds number, that flow is no longer stable and a small disturbance to the flow will lead to the growth of the disturbance and the onset of turbulence. Similarly, if turbulence is artificially promoted at a Reynolds number of less than 2100 the flow will ultimately revert to a laminar condition in the absence of any further disturbance. 

Similarly it may be shown that, at the critical conditions, the flowrate is a maximum for a given value of the specific energy J. At the critical velocity, (ir/gD) is equal to unity. This dimensionless group is known as the Froude number Fr. For velocities greater than the critical velocity Fr is greater than unity, and vice versa. It may be shown that the velocity with which a small disturbance is transmitted through a liquid in an open channel is equal to the critical velocity, and hence the Froude number is the criterion by which the type of flow, tranquil or rapid, is determined. Tranquil flow occurs when Fr is less than unity and rapid flow when Fr is greater than unity. 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

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. 

When two immiscible fluids flow relative to each other along an interface of separation, there is a maximum relative velocity above which a small disturbance of the interface will amplify and grow and thereby distort the flow. This phenomenon is... 

In practice, small pressure waves (such as sound waves) propagate virtually isentropically. The reasons for this are that, being a very small disturbance, the change is almost reversible and, by virtue of the high speed, there is very little heat transfer. Thus the speed of sound c is equal to the speed at which a small pressure wave propagates isentropically, so from equation 6.69... 

What can be concluded about the dynamics of the system from this steady-state plot Imagine that a small disturbance causes the temperature to increase slightly above its steadystate value f. At the higher temperature the heat-... 

Now start bringing in integral action by reducing by factors of 2, making small disturbances at each value of i to see the eifect. 

Cbbied slowly and riord small disturbance can make... 

The O M program can be divided into three types of projects (a) those that are unlikely to involve any direct contact with ACM (b) those that may cause accidental disturbance of ACM and (c) those that involve relatively small disturbances of ACM. 

Norepinephrine is made in cells located in the brain stem, mostly in a group of cells called the locus coeruleus. These neurons send widespread projections throughout the brain. This distribution has functional consequences. Small disturbances in the locus coeruleus can have a large impact on many different brain areas at the same time, and thus influence many behaviors. Disorders of emotion and mood are similar in that they simultaneously impact many different behaviors. Therefore, it is reasonable to suggest that norepinephrine might affect emotion in some ways. 

If the deton wave is a shock wave initiating chem reaction and continuously supported by energy thus set free, then it must be protected against the rarefaction which will always follow. This is impossible, if the velocity of small disturbances behind the wave is greater than that of the wave itself. In other words, if (a ) is the velocity of sound at Xj relative to the fluid there, (which itself moves with velocity Wj), and if ai+Wf exceeds the wave cannot be steady but must loose velocity. If a + is less than D, the wave can apparently remain steady. However, the condition a + W < D must, by reason of continuity, persist some little way into Xj Xs, say up to a section X1 (not shown in Fig 5). Then the chem energy released within X Xp can have no influence on what happens ahead of X and is therefore ineffective from the point of view of supporting the wave front. 

From here a sufficiently small disturbance of an initial wave profile develops into a state that remains close to the appropriate propagating wave that is, the waves above are at least marginally stable. 2... 

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

The validity of Eqs.(4.10)(4.12) probably extends well beyond the Rouse model itself [characterized by the specific set of rt values in Eq. (4.5)1 and it seems likely that they will apply, at least for small disturbances, whenever the elements supporting the stress are joined by sufficiently flexible connectors and configurational relaxation is driven by simple Brownian diffusion. One might speculate further that these same forms would apply even in concentrated systems, with Eq.(4.10) expressed in a somewhat more general form because of intermolecular interactions ... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

Given the assumptions of vanishingly small u and p, terms that involve the products of small variables are negligible. For small disturbances the flow may be considered isen-tropic, which justifies the substitution for the sound speed. Also, pa/p 1. Consequently the system becomes... 

The dependence e vs a derived from Eq. (36) is plotted on Fig. 4 as a dotted line. Solutions of Eq. (36) and of Eq. (27) show good agreement only at small values (e < 0.05) as would be expected, because the assumptions used in Eq. (32) are valid only at sufficiently small amplitudes A /A -C 1. Note that the changes of the gas motion before and after the curved front were considered indirectly through value /(a) taken from the Landau theory for a small disturbance. 

We will discuss some preliminary results, which have been performed recently l01). In Fig. 39a the results for polymer No. 2d of Table 10 are shown, which were obtained by torsional vibration experiments. At low temperatures the step in the G (T) curve and the maximum in the G"(T) curve indicate a p-relaxation process at about 120-130 K. Accordingly the glass transition is detected at about 260 K. At 277 K the nematic elastomer becomes isotropic. This phase transformation can be seen only by a very small step in G and G" in the tail of glass transition region, which is shown in more detail in Fig. 39 b. From these measurements we can conclude, that the visco-elastic properties are largely dominated by the properties of the polymer backbone the change of the mesogenic side chains from isotropic to liquid crystalline acts only as a small disturbance and in principle the visco-elastic behavior of the elastomer... 

Mullins and Sekerka (88, 89) analyzed the stability of a planar solidification interface to small disturbances by a rigorous solution of the equations for species and heat transport in melt and crystal and the constraint of equilibrium thermodynamics at the interface. For two-dimensional solidification samples in a constant-temperature gradient, the results predict the onset of a sinusoidal interfacial instability with a wavelength (X) corresponding to the disturbance that is just marginally stable as either G is decreased... 

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