Unsteady behavior

The purpose of these examples is to establish the following points  

To specify the long-term behavior, we do not need any knowledge of the springs. 

To specify the transient early behavior we do need knowledge of the springs (except that, as regards order of magnitude, neither force in string p or string q is likely to exceed 20 newtons). 

To estimate the time needed for the early excess force in string q of (16 — 5) or 11 newtons to diminish to half its original value, or 1/10 or 1/e or any desired fraction, one has to construct a differential equation and then solve it the answer depends on the properties of both the springs and the dashpots (see Appendix 20A). 

Earlier chapters contain very few references to experimental work. The reason is that relevant experiments are very difficult to perform the distance scales are so small that keeping track of either a stress profile or a composition profile as it evolves presents great difficulties. To illustrate the difficulties, a set of experiments by A. Y. Sane and A. R. Cooper (1987) will be briefly described. These were conducted with great ingenuity, skill, and care they provided estimates of compressive stress at sites as little as 2 pm apart and (indirectly) followed the change of these stresses through time, and yet they did not capture data that can be effectively compared with ideas from Chapter 16. (Of course, that comparison was not the objective of the 

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Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

Figure 55.4 compares the Raman spectra of the two samples spectra were recorded at 380°C in a 15% O2/N2 stream, on equilibrated catalysts downloaded after reaction. Catalyst VN 1.06 was not oxidized in the air stream, whereas in the case of catalyst PA 1.00 bands typical of a phosphate, ai-VOP04, appeared in the spectrum. These bands were not present in the spectmm of the equilibrated catalyst recorded at room temperature. Indeed, the spectra of the two equilibrated catalysts were quite similar when recorded at room temperature. This result confirms that the surface of catalyst VN 1.06 is less oxidizable than that of catalyst PA 1.00. Therefore, the latter is likely more oxidized than the former one under reaction conditions. A treatment in a more oxidant atmosphere than the reactive n-butane/air feed modifies the surface of catalyst VN 1.06, and leads to the unsteady behavior shown in Figure 55.1. The same treatment did not alter the surface of the equihbrated catalyst P/V 1.00 that was already in an oxidized state under reaction conditions.

The formulation of combustion dynamics can be constructed using the same approach as that employed in the previous work for state-feedback control with distributed actuators [1, 4]. In brief, the medium in the chamber is treated as a two-phase mixture. The gas phase contains inert species, reactants, and combustion products. The liquid phase is comprised of fuel and/or oxidizer droplets, and its unsteady behavior can be correctly modeled as a distribution of time-varying mass, momentum, and energy perturbations to the gas-phase flowfield. If the droplets are taken to be dispersed, the conservation equations for a two-phase mixture can be written in the following form, involving the mass-averaged properties of the flow ... 

Control and monitoring of the chemical reactor play a central role in this procedure, especially when batch operations are considered because of the intrinsic unsteady behavior and the nonlinear dynamics of the batch reactor. In order to meet such requirements, the following fundamental problems must be solved ... 

The major purpose of this paper is to present experimental results for the emulsion polymerization of vinyl acetate (VA) and methyl methacrylate (MMA) in a single CSTR. Both steady state and transient results will be presented and discussed. Possible causes for prolonged unsteady behavior will be outlined and several techniques for achieving steady operation with a CSTR will be described. 

Figures 3 through 6 show conversion-time data for a number of vinyl acetate runs. The start-up procedure for these experiments consisted of filling the reaction vessel with degassed water prior to introducing any feed streams. Periodic samples were taken and the monomer conversion measured gravimetrically. As can be seen, some of the conversion transients did not reach a steady state. Tendency toward unsteady behavior and the magnitude of the oscillations seemed to increase with increasing initiator concentration and mean residence time. The influence of changing the emulsifier concentration is not clear.

The unsteady behavior of single CSTR s (Figures 3, 4, 5, 6, 10 and 11) is probably caused by a combination of particle formation phenomena and the gel effect. 

In a double shock wave with chemical reactions, unsteady behavior can lead to a p-v space path that is not necessarily well described by Rayleigh lines. However, we assume here that for a given period of time the p-v space path can be transiently approximated by a set of Rayleigh lines. This description is valid when the timescale of the pressure change at point B in Figure 2 is less than the time required for a material element to progress from the initial state to the final shocked state. A more quantitative version of this statement is formulated in the remainder of this section. 

Substituting in terms of a, 3. and T, we arrive at two coupled differential equations describing the unsteady behavior of the tracer that must be solved simultaneously. 

Consistent with steady-state solution discussed earlier, we will consider the surface reaction (QS) approximation for unsteady behavior. Accordingly, the chemical heat release term is removed from the energy differential equation and placed in the surface boundary condition and a fraction of the absorbed radiant flux (1-/ ) is deposited in the surface layer ... 

Flux expressions (3.4.72), (3.4.76) and (3.4.81a) for a diffusing gas species i through a membrane of thickness dm describe the observed behavior at steady state achieved after an initial unsteady period. The initial unsteady behavior begins when the gas containing the permeating species i at concentration Cif (partial pressure p,y) is introduced at time t = 0 to the z = 0 surface of the membrane, the feed side. The rate of penetration of the membrane by species i is governed by the unsteady state diffusion equation (3.4.79), where is governed by Fick s flrst law... 

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