Fuzzy set theory


For feed FI, the upper edge of the reachable composition region is along the MEK—MIPK (water-free) face of the composition triangle and part of the lower edge is along the MEK—water (MIPK-free) face. There exist conditions under which both the water in the bottoms MIPK product can be driven to low levels (high product purity) while the MIPK in the distillate is also driven to low levels (high product recovery), although achieving such operation depends on having adequate stages and reflux ratio.  [c.447]

This result is important, since in practice we should not focus exclusively on the single sequence with the lowest overall cost. Rather, because of the uncertainties in the calculations and the fact that other factors need to be considered in a more detailed evaluation, the best few sequences should be evaluated in more detail. Thus there is no need to solve the separation sequence and heat integration problems simultaneously. Rather, decouple the two problems and tackle them separately, simplifying considerably the overall task. It must be emphasized strongly that the decoupling depends on the absence of significant constraints limiting the heat integration potential within the distillation sequence. For example, there may be limitations on the pressure of some of the columns due to product decomposition, etc. that limit the heat integration potential. In these  [c.143]

The ultimate reference in guiding process changes to reduce utility costs and utility waste is the plus/minus principle. However, process changes so identified prompt changes in the capital/energy tradeoff and utility selection. Using the total cost targeting techniques described in Chaps. 6 and 7, it is possible to effectively screen a wide range of options using relatively simple computation. In the next three chapters we shall focus in detail on heat integration of reactors and heat-driven separators.  [c.323]

Ultrasonic testing of austenitic weld metals is a mayor NDE problem in the nuclear industry. The elastic anisotropy of the material is due to large elongated crystals, with their main axes in the direction of the cooling gradient, forming a characteristic texture, se figure 1. Also, the large crystals will themselves act as scatterers, leading to scattering and mode conversion of the ultrasonic waves at crystal interfaces. The scattering, including back scattering, will increase the noise level and the propagating beam will be attenuated. An operator will face problems like e.g. beam-skewing, structural noise and attenuation, quite different from when the material is isotropic. This will lead to problems in detecting, locating and sizing defects.  [c.221]

To consider the geometrical magnification of the pipe image on film or IP in the tangential exposure technique according to the set-up shown in fig. 4 a correction of the measured wall thickness (w ) must be performed. The true wall thickness (w) depends in a rather complicated way on the film-focus-distance (f), the radius of the pipe (r) and the radius of the insulation (R) as shown in equation 2  [c.520]

In order that the data acquisition system can obtain information about the spatial location and orientation of the probe, a four-channel incremental encoder interface board is installed. Three channels are used to define position in three-dimensional space, while the fourth monitors the skew of the probe (skew is defined as rotation about an axis normal to the probe face). Although six measurements are required to completely define the location and orientation, it is assumed that the probe remains in contact with the inspection surface.  [c.768]

While professional awareness has increased, there are still hindrances to universal access in the workplace. Today almost all companies have access to the Internet, but company policy may limit usage during business hours or on company equipment Also, many countries still face prohibitive online access charges, although free market competition will take care of that in time, and Canada s no-charge policy may well be the wave of the future  [c.974]

Typical eddy-current inspections are carried out with the inspector moving the probe on the test-piece while viewing a digital read-out, usually an impedance display The inspector is frequently glancing from display to test-piece and from test-piece to display. This dual role of the inspector makes it harder to carry out an inspection than it needs to be -the inspector has to see two events at the same time which of course is physically impossible. Devices are used to help the operator. Triggers can be set with audible tones that sound when signal amplitudes exceed pre-set levels, allowing the inspector to focus on moving the probe smoothly and accurately without the distraction of the screen. Painting devices work in a similar way, spraying paint on areas that have exceeded pre-set tolerances. While inexpensive and simple, these techniques suffer from calibration issues and both turn what is often 16 bits of signal into 1 bit, on or off, signal. It would be better to sample, collect the data as a sequence of position/signal pairs, and then view collections of data all at once. By sampling, unusual areas are immediately evident and their location has already been stored. The question is how to collect this spatial information. Two sampling techniques are readily available.  [c.1015]

The consequences of undetected corrosion in a bulk carrier are particularly serious with the vertical side shell frame end brackets posing the greatest risk. General corrosion of face plates and web plates of the end brackets, possibly compounded by grooving corrosion at weld toes, will weaken the frame to the point of failure if not detected. The problem is progressive and failure can be rapid since neighbouring frames are likely to fail through the imposition of transferred loads. Loss of the support given by these frames is liable to result in horizontal fractures in the side shell plating at the hinge formed with either tire topside tank or the hopper tank. The ingress of sea water followed by the consequent sloshing effect can cause further degradation of the hull structure. In cases where corrosion has weakened the inner bottom structure or the transverse bulkhead structure the combined loading of cargo and seawater can result in progressive flooding and the buoyancy of the ship will be threatened. If this sequence of events takes place in the Nol forward hold the effects are particularly serious and the loss of the ship can occur within minutes.  [c.1048]

There has been tremendous activity in the study of spread films with many different experimental techniques. Here we limit our focus to the more basic methods giving classic information regarding film pressure, electrical potential, viscosity (through viscometry and dynamic light scattering), thickness, and molecular arrangement. We restrict our attention to methods used to study spread films transferred or Langmuir-Blodgett film techniques are covered in Chapter XV.  [c.113]

Given the set of surface free energies for the various crystal planes, draw a set of vectors from a common point of length proportional to the surface free energy and of direction normal to that of the crystal plane. Construct the set of planes normal to each vector and positioned at its end. It will be possible to find a geometric figure whose sides are made up entirely from a particular set of such planes that do not intersect any of the other planes. The procedure is illustrated in Fig. VII-2 for a two-dimensional crystal for which 710 is 250 ergs/cm and 7ii is 225 ergs/cm. Note that the optimum shape makes use of both types of planes for these surface tensions there is a free energy gain in truncating the comers of a (11) sided square crystal to reduce its perimeter (note Problems VII-2-VII-6). A statement of Wulff s theorem is that for an equilibrium crystal there exists a point in the interior such that its perpendicular distance h, from the jth face is proportional to 7,-. This is, of course, the basis of the construction in Fig. VII-2.  [c.261]

There are many studies involving the Wulff construction, including an analytical proof of the theorem [29], a determination of the shapes of face-centered cubic (FCC) and body-centered cubic (BCC) crystals [30], and investigations of the finite radius of curvature of edges [31] and the possibility of cusp formation [32]. Minimum free energy polyhedra do exist in nature, in rock salt [33], metals [34], and in other small crystals [35]. Wulff constructions have been worked out for crystals in a gravitational field [36], on a surface, or in a comer [37]. Combination of the Wulff construction with density functional theory of the solid (see Section IX-2) shows the lack of faceting in systems having a very short-range attraction [38].  [c.261]

Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface.  [c.264]

Looking at the multivesicular liposome in Fig. XV-11, estimate the average number of edges per face, q, and use your value of q to determine the average number of faces per polyhedra, F, using the Euler relation V - E + F = 2, where V is the number of vertices and E is the number of edges. Tetrahedral coordination restricts 2V = 2E = qF. Compare your result with the experimental finding that there are 5.1 faces per pair of edges.  [c.562]

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is  [c.52]

A concomitant increase in the coverage of environmental issues is evidenced in the Eniyclopedia. There are new articles, ie. Groundwatermonitoring, Hazard analysis and riskassessp nt. Hazardous waste treati nt, and Waste reduction, which have environmental topics as the primary focus. There is also a growing tendency to merge environmental issues and economics in decision making within the chemical industry (see CATALYSTS,REGENERATION Economic evaluation Fuels from waste Herbicides Recycling Solvents, industrial Wastes, industrial). Moreover, because environmental issues ate pervasive in the chemical industry, these concerns have brought about change in chemical technology. In many fields, environmental concerns and regulations ate driving technological development (see CoATING PROCESSES Dyes, ENViRONPffiNTAL CHEMISTRY Exhaust control, automotive Exhaust control, industrial).  [c.281]

Consider the recovery of MIPK from an MEK-MIPK-water mixture. The bow-tie approximation of reachable compositions for several feeds are shown in Fig 13-61 7 and the exact reachable compositions are shown in Fig. 13-6IZ . From Feed F3, which is situated in a different distiUation region than the desired product, pure MIPK cannot be obtained at aU. With the upper edge of the bow-tie region for Feed FI along the MEK-MIPK (water-free) face of the composition triangle, and part of the lower edge along the MEK-water (MIPK-free) face, there are conditions under which both the water in the bottoms MIPK product can be driven to low levels (high-product purity) and MIPK in  [c.1296]

Most alloy systems are more complicated than the lead-tin system, and show intermediate phases compounds which form between components, like CuAlj, or AljNi, or FojC. Their melting points are, usually, lowered by alloying also, so that eutectics can form between CuAlj and A1 (for example), or between AljNi and Al. The eutectic point is always the apex of the more or less shallow V formed by the liquidus lines.  [c.347]

Collecting information on complaints is an essential part of the investigation. Occupant data falls into two categories complaints of discomfort or other symptoms (e.g., teary eyes, chills) and perceptions of building conditions (e.g., odors, draftiness). Investigators can gather valuable information about potential indoor air problems by listening to occupants, and use that information for defining the complaint area within the building suggesting directions for further investigation, either by identifying other events that seem to happen at the same time as the incidents of symptoms or discomfort, or by identifying possible causes for the types of symptoms or discomfort that are occurring indicating potential measures to reduce or eliminate the problem. If there is a record of occupant complaints, a review of that record can help to define the location of the lAQ problem and identify people who should be interviewed as part of the investigation. Information about the history of complaints could also stimulate theories about potential causes of the problem. The most obvious way to collect information from building occupants is to talk to them in person. If it is not possible to interview everyone who has complained about building conditions, the investigator should attempt to interview a group of individuals that reflects the concerns of the affected areas. The investigation may also include occupant interviews with building occupants who do not have complaints. Then conditions in the complaint area can be compared to conditions in similar building locations where there are no complaints. Many events occur simultaneously in and around a complex building, and it can be very difficult to judge which of those events might be related to the lAQ complaints. In trying to resolve stubborn problems, professional investigators sometimes ask occupants and facility staff to keep day-by-day records. Occupants are asked to record the date and time of symptoms, where they are when the symptoms appear, and any other information that might be useful. Such information could include observations about the severity and duration of symptoms and comments on weather conditions, events, and activities that are happening at the same time. Facility staff are asked to record the date and time of events such as maintenance work, equipment cycles, or deliveries. If symptoms seem to occur at particular times of day, staff can focus their attentions on recording events that  [c.216]

On a so-called vicinal face there are many steps running in parallel with almost the same separation or terrace width in between. At a finite temperature, these steps also fluctuate. But due to the high energy cost for the formation of overhangs on the crystal surface, steps cannot cross each other. This non-crossing condition suppresses the step fluctuation.  [c.872]

The first electrolytic theoiy was expounded in 1805 by Christian Grothuss who postulated that an electric field rotates the molecules so that their positive and negative components ( ions ) face their  [c.231]

Whatever the method used to screen possible sequences, it is important not to give exclusive attention to the one that appears to have the lowest vapor load or lowest total cost. There is often little to choose in this respect between the best few sequences, particularly when the number of possible sequences is large. Other considerations such as heat integration, safety, and so on also might have an important bearing on the final decision. Thus the screening of sequences should focus on the best few sequences rather th2in exclusively on the single best sequence.  [c.142]

The term vitamin K2 was applied to 2-methyl-3-difarnesyl-l,4-naphthoquinone, m.p. 54 C, isolated from putrefied fish meal. It now includes a group of related natural compounds ( menaquinones ), differing in the number of isoprene units in the side chain and in their degree of unsaturation. These quinones also appear to be involved in the electron transport chain and oxidative phosphorylation.  [c.423]

The cetane number of heavy fuels is generally between 30 and 40, but its determination is inexact considering the difficulties encountered when testing such products in a CFR engine. One possible way to operate is to dilute the fuel with diesel fuel or home-heating oil and estimate a blending index. There are also formulas to predict the auto-ignition qualities of heavy fuels using their physical characteristics (Fiskaa et al., 1985). We cite as an example the relation giving the CCAl (Calcuiated Carbon Aromaticity Index)  [c.240]

It cannot be expected that IP s with their different characteristics in comparison to the film can be applied in the same way. Special filters must be configured to optimize the exposure arrangement. The comparison to the film should be performed generally under optimized conditions for both systems. The image quality achieved by the NDT-film system classes is the basis for the comparison and evaluation of the CR by IP s. The exposure process and the handling of the IP s is optimized with the goal to achieve the same or better image quality than film systems. Similar factors influence the image quality due to optical scattering, limitations of data processing and monitor properties. For the performed measurements systems from AGFA and Fuji-Film were tested.  [c.467]

The laser beam has almost parallell light and the photodetector is only interested in the amount of flourescent light coming in. There is no imaging or similar complication requiring an adjustment in focus or any other adjustments in the optics. The only light received originates from illumination by the tiny laser spot and this only lights where it is told to. These properties provide a system with minimal sensitivity to variations in distance to the test piece. This is also very important. If you think of the situation when testing a crankshaft the problem becomes obviuos  [c.640]

In order to predict the acoustic field generated by a given transducer configuration, a model based on Huygens principle has been employed [17]. This assumes that each finger may be represented as a series of point sources, the field at any point on the structure being obtained by summing the components from all the sources on the transducer. One particular interest has been to design interdigital transducers (IDTs) which have all round 360" vision of the surrounding structure. Fig 5(a) shows a typical sub-transducer, designed to excite the a, mode at 1.1 MHz In 1.2 mm thick steel plate. The complete transducer would have six such subtransducers arranged in a circle with a common focus, the idea being that each sub-transducer would generate a uniformly diverging beam to interrogate the 60" sector of structure in front of it. Fig 5(b) shows the Huygens prediction of the field from the sub-transducer. To investigate the beam divergence, an angular cross section is taken along the dotted line and plotted in Fig 5(c), where a comparison with experimental measurements is also shown. The agreement is not as good as has been obtained in other cases [17], but this is thought to be due to uneven bonding of the experimental IDT. However, both the experimental and predicted results indicate that the beam is uniform over around 40" and not the required 60", hence this particular transducer configuration was not suitable for the required purpose. Further work is in progress designing integrated IDTs which have a greater degree of overlap between the fields of their constituent sub-transducers.  [c.719]

Maybe we should regard the efforts of Commission V of the IIW in the seventies and eighties, to establish Fitness For Purpose approaehes, as being far ahead of their time. Fihiess For Purpose criteria cannot exist in combination with NDT methods that simply do not provide the necessary information. But nowadays, we are in a much more comfortable situation.  [c.948]

Fig. XV-11. Electron micrograph of a freeze fracture replica of a region inside a mul-tivesicular liposome. Note the tetrahedral coordination nearly every vertex has three edges, and each face is connected to three others. The average number of edges per face is 5.1. (From Ref. 77.) Fig. XV-11. Electron micrograph of a freeze fracture replica of a region inside a mul-tivesicular liposome. Note the tetrahedral coordination nearly every vertex has three edges, and each face is connected to three others. The average number of edges per face is 5.1. (From Ref. 77.)
One specific factor that is important in catalytic activity is the precise nature of the atomic spacings of the catalyst sites. This point was mentioned in Section XVIII-4A, and there are many specific examples of activity depending on what crystal planes are present face-centered-cubic metals tend to be more active than body-centered ones, and for the former, (111) planes are more active than (110) or (100) ones. Somoijai has discussed a number of such cases, as in Ref. 215. On the theoretical side, transition state structures have been calculated for metal atom insertion into methane CH bonds, leading to predicted orders of activity for Fe and Ni, (100) planes now being the most active [216]. The importance of the morphology of a catalyst surface has been recognized for a long time, of course, and some early ideas are those of Balandin [217] and Taylor [218]. A contemporary example is the effect of faceting of the Pt/W(l 11) surface on the product ratios in the catalyzed n-butane hydrogenolysis [219].  [c.723]

The rather complicated issues raised in the preceding paragraph are central to the subject of quantum mechanics, and their resolution fonus the basis of one of the most unportant postulates associated with the Schrodinger fomuilation of the subject. In the example above, discussion focuses entirely on the eigenvalues of the momentum operator. What significance, if any, can be attached to the eigenfiinctions of quantum-mechanical operators In the interest of simplicity, the remainder of this subsection will focus entirely on the quantum mechanics associated with operators that have a finite number of eigenvalues. These are said to have a discrete spectrum, in contrast to those such as the linear momentum, which have a continuous spectrum. Discrete spectra of eigenvalues arise whenever boundaries limit the region of space in which a system can be. Examples are particles in hard-walled boxes, or sofi-walled shells and particles attached to springs. The results developed below can all be generalized to the continuous case, but at the expense of increased mathematical complexity. Readers interested in these details should consult chapter 1 of Landau and Lifschitz (see additional reading).  [c.8]

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the n-electron wavefiinction of the atom or molecule can be written as a Slater detenninant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical fonn of these orbitals is not known (especially m molecules), then the resulting problem is highly nonlinear and fonnidably difficult to solve. However, as mentioned in subsection (A1.1.3.2). a connnon approach is to assume that the orbitals can be written as linear combinations of one-electron basis fiinctions. If the basis fiinctions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar fonn of the trial fiinction occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefiinction can be written as a linear combination of Slater detenninants (see equation (A1.1.104)). In the conceptually simpler latter case, tire objective is to minimize an expression of the fonn  [c.37]

There is one notable difference between the Hartree-Fock summation and the Flartree siumnation. The Flartree-Fock sums include the =j tenns in equation Al.3,13. This difference arises because the exchange temi corresponding to =j cancels an equivalent temi in the Coulomb smmnation. The =j temi in both the Coulomb and exchange temi is interpreted as a self-screening of the electron. Without a cancellation between Coulomb and exchange tenns a self-energy contribution to the total energy would occur. Approximate fonns of the exchange potential often do not have this property. The total energy then contains a self-energy contribution which one needs to remove to obtain a correct Flartree-Fock energy.  [c.92]

Since the pseudopotential does not bind the core states, it is a very weak potential. Simple basis functions can be used to describe the pseudo-wavefiinctions. For example, a simple grid or plane wave basis will yield a converged solution [25]. The simplicity of the basis is important as it results in an unbiased, flexible description of the charge density. Also, since the nodal structure of the pseudo-wavefunctions has been removed, the charge density varies slowly in the core region. A schematic model of tire pseudopotential model is illustrated in figure Al.3.10. The pseudopotential model describes a solid as a sea of valence electrons movmg in a periodic background of cores (composed of nuclei and inert core electrons). In this model many of the complexities of all-electron calculations, calculations that include the core and valence electrons on an equal footing, are avoided. A group IV solid such as C with 6 electrons per atom is treated in a similar fashion to Sn with 50 electrons per atom since both have 4 valence electrons per atom. In addition, the focus of the calculation is only on the accuracy of the valence electron wavefunction in the spatial region away from the chemically inert core.  [c.108]


See pages that mention the term Fuzzy set theory : [c.158]    [c.463]    [c.292]    [c.80]    [c.118]    [c.256]    [c.296]    [c.247]    [c.214]    [c.723]    [c.730]    [c.32]    [c.33]    [c.35]    [c.54]    [c.91]    [c.196]   
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Advanced control engineering  -> Fuzzy set theory


Advanced control engineering (2001) -- [ c.326 ]