Invariant process

A time-invariant process has time-independent parameters. Therefore, a time-invariant process is that for which both V and k are invariant in time. From the three previous relationships, the only time-independent situation occurs in the exponential empirical model when k (t) = j3. In this case, from (7.11) one has... 

Design of Control Systems for Mu Invariable Processes Plant Control... 

A distinction is often drawn between labour-intensive and capital-intensive processes. The former term is self-explanatory and alongside it the nature of the latter is also clear. Almost invariably, process plants which are costly to build and subsequently operated by relatively few staff are in the capital-intensive category. Thus with regard to a particular project a chemical company has to spend a lot of money before any product is produced for sale. Many are familiar with the long lead times of major electricity generating plants five years for conventional oil and coal fired plants and at least seven (often ten or more) for nuclear plants. In view of the massive amounts of money which are invested in such projects it is not... 

In particular, the enthalpies of invariant processes (melting of pure substances, eutectic and peritectic points, extreme phase transition temperature, etc.) can be measured with great precision. 

With respect to a set of functions li[t) that is orthonormal amd complete over the interval (0,00), it is known that an arbitrary function h t) has a formal expansion analogous to a Fourier expansion (Wyhe, 1960). Such an expansion has been widely used in numerical analysis for the approximation of functions in differential and integral equations. The idea behind using Laguerre functions to represent a linear, time invariant process is to take h t) to be the unit impulse response of the process, where h t) can be written as... 

It follows that the isothermal degree of conversion equals the non-isothermal degree of conversion only for invariant processes, where Aa, = 0. The difference of these two degrees of conversion is well revealed in Fig. 45., where we easily observe that in the isothermal measurement the system reaches a final, but incomplete value of Z (= Zpr), which corresponds to the given temperature, Tp, whereas on continuous increase of temperature the system always attains the ultimately finishing value of measured property,, corresponding to the definitive completion of reaction (Fig. 45.). Practical application of is shown in the functional case of non-stoichiometry interactions of Mn304 (see Fig. 46.). 

For the process with the stable initial states, we can distinguish invariant process, where changes in sudden break from 0 to 1, while when the change proceeds gradually over a certain temperature interval, from T to Tp, we talk about the variant process ofa continual character. If the transformation consists of both the stepwise (invariant) and continuous (variant) changes, we deal with the so-called combined process [370]. It is necessary to stress out again that these three types of changes are valid only under ideal accession to phase transition whereas under real experimentai conditions a certain temperature... 

This work focused on non linear time invariant processes. An approach to deal with time varying model was proposed by Genceli and Nikolau [16] using a finite impulse response (FIR) model that was updated in real time. Recently, Alexandridis and Sarimveis [17] proposed an adaptive model predictive control using a non linear... 

When water is produced along with oil, the separation of water from oil invariably leaves some water in the oil. The current oil-in-water emission limit into the sea is commonly 40 ppm. Oily water disposal occurs on processing platforms, some drilling platforms, and at oil terminals. The quality of water disposed from terminals remains an area of scrutiny, especially since the terminals are often near to local habitation and leisure resorts. If the engineer can find a means of reducing the produced water at source (e.g. water shut-off or reinjection of produced water into reservoirs) then the surface handling problem is much reduced. 

The condition that the process a(t) is a stationary process is equivalent to the requirement tiiat all the distribution fimctions for a t) are invariant under time translations. This has as a consequence that W a, t) is independent of t and that 1 2(0, t 2, 2) depeirds on t = 2 -1. An even stationary process [4] has the additional requirement that its distribution fimctions are invariant under time reflection. For 1 2, this implies fV2(a 02> t) = 2 2 1 caWcd microscopic reversibility. It means that the quantities are even... 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

Adsorption is invariably an exothermic process, so that, provided equilibrium has been established, the amount adsorbed at a given relative pressure must diminish as the temperature increases. It not infrequently happens, however, that the isotherm at a given temperature Tj actually lies above the isotherm for a lower temperature Ti. Anomalous behaviour of this kind is characteristic of a system which is not in equilibrium, and represents the combined effects of temperature on the rate of approach to equilibrium and on the position of equilibrium itself. It points to a process which is activated in the reaction-kinetic sense and which therefore occurs more rapidly as temperature is increased. 

The toxic nature of mercury and its compounds has caused concern over environmental pollution, and governmental agencies have imposed severe restrictions on release of mercury compounds to waterways and the air (see Mercury). Methods of precipitation and agglomeration of mercurial wastes from process water have been developed. These methods generally depend on the formation of relatively insoluble compounds such as mercury sulfides, oxides, and thiocarbamates. MetaUic mercury is invariably formed as a by-product. The use of coprecipitants, which adsorb mercury on their surfaces facihtating removal, is frequent. 

This catalyst system is used in about 70% of the -xylene oxidations, and the percentage is increasing as new plants almost invariably employ it. Process conditions are highly corrosive owing to the acetic acid and bromine, and titanium must be used in contact with some parts of the process. 

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