Saddle-point of the problem


The preceding discussion on site energy distributions has made the implicit assumption that each kind of site occurred in patches at least large enough that boundary effects could be neglected. In effect, each patch is treated as a small adsorption system within which lateral interaction between adsorbate molecules would occur. At the other extreme, sites of varying energy are randomly distributed on the surface, and only as lower energy sites b ame occupied would there be a developing probability of adjacent sites being occupied. A complete description of surface heterogeneity would have to include the distribution of site energies adjacent to a site of given energy. At this point there are probably too many variables to be extracted from adsorption data alone, although the comparison of isotherms for adsorbates of varying size may help (e.g., nitrogen versus butane—see Ref. 162). A formal statistical mechanical approach to the problem was made by Steel [166].  [c.660]

Unfortunately, there is no good theory that can be used to consistently predict the peak height and return time for a given set of physical and chemical parameters. The design of a flow injection analyzer for a particular analytical problem still occurs largely by a process of experimentation. Nevertheless, some general observations about the effects of physical and chemical parameters can be made. In the absence of chemical effects, sensitivity (larger peak height) is improved by injecting larger samples, increasing the flow rate, decreasing the length and diameter of the tubing in the mixing and reaction zone, and merging separate channels before the point where the sample is injected. Except for sample volume, an improvement in the sampling rate (smaller return time) is achieved by the same combination of physical parameters. Larger sample volumes, however, lead to longer return times and a decrease in sample throughput. The effect of chemical reactivity depends on whether the species monitored by the detector is a reactant or a product. For example, when the monitored species is a reactant, sensitivity is improved by selecting a combination of physical parameters that enables the sample to reach the detector more quickly. Adjusting the chemical composition of the carrier stream in a manner that decreases the rate of the reaction also improves sensitivity in this case.  [c.651]

We shall continue basing our discussion on the step-growth polymerization of the hypothetical monomer AB. In Sec. 5.7 we shall take a second look at this problem for the case of unequal concentrations of A and B groups. For now, however, we assure this equality by considering a monomer which contains one group of each type. In a previous discussion of the polymer formed from this monomer, we noted that remnants of the original functional groups are still recognizable, although modified, along the backbone of the polymer chain. This state of affairs is emphasized by the notation Ababab. . . abaB in which the a s and b s of the ab linkages are groups of atoms carried over from the initial A and B reactive groups. In this type of polymer molecule, then, there are n - 1 a s and 1 A if the degree of polymerization of the polymer is n. The a s differ from the A s precisely in that the former have undergone reaction while the latter have not. At any point during the polymerization reaction the fraction of the initial number of A groups which have reacted to become a s is given by p, and the fraction which remains as A s is given by 1 - p. In these expressions p is the same extent of reaction defined by Eq. (5.3).  [c.292]

Another issue affecting the architectural paint industry is the remediation of homes, buildings, and stmctures that contain lead-based paint. Lead poisoning in children has been linked to ingestion of paint dust or paint chips that contain lead pigments and this has resulted in U.S. government regulations to reduce the lead content in paint to no more than 0.06%. This restriction essentially bans the use of lead-containing materials in paints, with the exception of lead impurities in the water and minerals used in paints. Prior to these government regulations, most paint companies had already phased out their use of lead pigments. However, homes and other stmctures of older constmction (pre-1960) may still contain lead paint (16). Efforts in some states have required the removal of lead-containing paints in homes and buildings inhabited by young children (see Paint and finish removers). However, lead paint removal is expensive and in some cases has exacerbated the problem as lead-containing dust or chemicals are generated during the removal process. As a result, remediation efforts have begun to allow the use of lead paint encapsulants to coat over lead paint.  [c.547]

The possibihty of casting molten steel continuously into useflil shapes equivalent to conventional semifinished shapes, such as slabs and billets, and thus eliminating the ingot and primary-mill stages of roUed-steel production, led to continuous attempts to solve the problem by many investigators using a variety of machine designs. Because of the high melting point, high specific heat, and low thermal conductivity of steel, success did not come easily in the case of ferrous metals. For nonferrous metals, on the other hand, continuous-casting quickly proved practical.  [c.381]

Unfortunately, there is no good theory that can be used to consistently predict the peak height and return time for a given set of physical and chemical parameters. The design of a flow injection analyzer for a particular analytical problem still occurs largely by a process of experimentation. Nevertheless, some general observations about the effects of physical and chemical parameters can be made. In the absence of chemical effects, sensitivity (larger peak height) is improved by injecting larger samples, increasing the flow rate, decreasing the length and diameter of the tubing in the mixing and reaction zone, and merging separate channels before the point where the sample is injected. Except for sample volume, an improvement in the sampling rate (smaller return time) is achieved by the same combination of physical parameters. Larger sample volumes, however, lead to longer return times and a decrease in sample throughput. The effect of chemical reactivity depends on whether the species monitored by the detector is a reactant or a product. For example, when the monitored species is a reactant, sensitivity is improved by selecting a combination of physical parameters that enables the sample to reach the detector more quickly. Adjusting the chemical composition of the carrier stream in a manner that decreases the rate of the reaction also improves sensitivity in this case.  [c.651]

In 1985, ethylene oxide (EO) ignited in a 1 pint clip top glass sample bottle during sampling from a tank car. The /4-in. stainless steel liquid sample line from the car entered a sampling hood and EO flowed at relatively high velocity (4 m/s) into the bottle through a Teflon tube. The Teflon tube was used to prevent water condensation on the cold sample line, since if water dripped into the sample bottle it would affect turbidity evaluation. EO has a very small ignition energy in air and it was not possible to absolutely rule out ignition by a charged operator. Plausible mechanisms involved (1) EO charging during flow through the Teflon tube, or (2) induced charge on EO liquid from an externally charged sample bottle, or (3) induced charging of the wire clip on the bottle stopper by charged Teflon tube. All of these mechanisms could be avoided by bottom filling with a grounded stainless steel dip tube. The water condensation problem was avoided by fitting the dip tube with a thermally insulating Teflon sleeve, leaving a few millimeters of exposed metal at the outlet. Subsequent to this fire, operators were grounded using wrist bracelets.  [c.157]

To summarize the sizing to this point the inlet volume, an overall head, number of stages, head per stage, impeller tip speed, and impeller diameter have been established. The one parameter of interest still missing is the efficiency. To obtain an estimate of efficiency without empirical data, a generalized form may be used. As in the previous chapters, where estimates were involved, the data presented is just one way to approach the problem, and any other reasonable source such as specific vendor data may be used. To use the generalized curve. Figure 5-26, the volume for the first and last stage must be developed. The volume for the first stage is the inlet volume. The volume for the last stage, Qis, can be estimated by  [c.162]

Oversized Valve - "Pop action PR valves in vapor service open at the set point by the action of static process pressure on the valve disc, and move to full open position at only a small overpressure. Typically, a flow through the valve equal to at least 2% of its capacity is necessary to keep the disc in the open position. At lower rates, the kinetic energy of the vapor flow is insufficient to keep the valve open against the action of the spring and it returns to the closed position, only to reopen immediately since the static pressure within the system still exceeds the set pressure. Chattering results from continuous cycling in this manner. It can occur when a "pop" type PR valve is too large for the quantity of fluid being discharged. In most cases, the use of multiple PR valves with staggered set points may be appropriate to eliminate this problem.  [c.168]

There is a vast and often bewildering array of synthetic methods and reagents available to organic chemists today. Many chemists have their own favoured methods, old and new, for standard transformations, and these can vary considerably from one laboratory to another. New and unfamiliar methods may well allow a particular synthetic step to be done more readily and in higher yield, but there is always some energy barrier associated with their use for the first time. Furthermore, the very wealth of possibilities creates an information retrieval problem How can we choose between all the alternatives, and what are their real advantages and limitations Where can we find the precise experimental details, so often taken for granted by the experts There is therefore a constant demand for books on synthetic methods, especially the more practical ones like Organic Syntheses," "Organic Reactions.and "Reagents for Organic Synthesis." which are found in most chemistry laboratories. We are convinced that there is a further need, still largely unfulfilled, for a uniform series of books, each dealing concisely with a particular topic from a practical point of view—a need, that is. for books full of preparations, practical hints, and detailed examples, all critically assessed, and giving just the information needed to smooth our way painlessly into the unfamiliar territory. Such books would obviously be a great help to research students as well as to established organic chemists.  [c.197]

In this chapter, we have discussed the potential of membrane technology for chiral separations. As the focus of this chapter has been on large-scale separations, the various developments have been discussed from an engineering point of view. A significant effort is being put into the development of chiral polymer membranes by a number of research groups. Nevertheless, these membranes still are in an early stage of development, and for industrial use significant improvements on flux and/or enan-tioselectivity are required. Once these limitations have been overcome, the application of chiral membranes will be similar to reverse osmosis and ultrafiltration, which will allow a relatively rapid implementation. Similarly, a large number of chiral selectors have been developed for application in liquid membranes. A major issue for the application of liquid membranes on an industrial scale lies in the problem of staging. As a result of the many recycles in the process, selective liquid membranes will require a large number of storage vessels, which is undesirable from a process economics standpoint.  [c.147]

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as [c.194]

Thickness and uniformity Normally about 7S-125 fim on products, 25 ftm on sheet. Coatings up to 250 fim on products by prior grit-blasting. Very uniform —any discontinuities due to poor preparation of the steel are readily visible as black spots Thickness variable at will, generally 100-200 fim but coatings up to 250 pm or more can be applied. Uniformity depends on operator skill. Coatings are porous but pores soon fill with zinc corrosion products thereafter impermeable Thickness variable at will generally 2-25 pm. Thicker layers are possible but generally uneconomic. Uniform coating within limitations of throwing power of bath. Pores not a problem as exposed steel protected by adjacent zinc Usually about 12-40 fim closely controlled. Thicker coatings also possible. Continuous and very uniform even on threaded and irregular parts Up to 40 fim of paint (and more with special formulations) can be applied in one coat. Good uniformity—any pores fill with reaction products  [c.488]

In Chapter 5, this characteristic was applied to centrifugal compressors. The airplane wing analogy of stall was used, which is very directly applicable to the axial s airfoil-shaped blades. The incidence angle, described earlier, defines the onset of surge by stating that when the incidence exceeds the stall point, as developed from the cascade data, the foil ceases to produce a forward motion to the gas. When the gas cannot move forward, it moves in reverse, opposing the incoming flow. When the two collide, there is a noise, sometimes very loud. Recompression of the gas causes the temperamre to rise very high very quickly. There have been cases where, when the blades were sufficiently strong not to break from the unsteady forces, they melted. It is more normal with prolonged surge to experience catastrophic blade breakage. The axial can also exhibit a phenomenon referred to as rotating stall. Rotating stall (propagating stall) is generally encountered when the axial compressor is started or operated too near the surge limit. This is especially true for compressor with adjustable vanes with the vanes in their extreme open or closed position. Vane movement is limited in some cases to minimize this problem. A flow perturbation causes one blade to reach a stalled condition before tbe other blades. This stalled blade does not produce a sufficient pressure rise to maintain the flow around it, and an effective ilow blockage or a zone of reduced flow develops. This retarded flow diverts the flow around it so that the angle of attack increases or decreases on adjacent blades. These blades, with the increased angle of attack, stall and stay in a cell-like form. The cell then propagates around the stage or possibly two in which it occurs and at some fraction (40-75%) of rotor speed. Once begun, the cells continue to generate, causing inefficient perfonnance, and if not terminated, may continue until a blade failure occur - This is especially true if the cell s rotating speed coincides with  [c.237]

For centrosynnnetric media the spatially local contribution to the second-order nonlinear response vanishes, as we have previously argued, providing the interface specificity of the method. This spatially local contribution, which arises in the quantum mechanical picture from the electric-dipole tenns, represents the dommant response of the medium. Flowever, if we consider the problem of probing interfaces closely, we recognize that we are comparing the nonlinear signal originating from an interfacial region of monolayer thickness with that of the bulk media. In the bulk media, the signal can build up over a thickness on the scale of the optical wavelength, as dictated by absorption and phase-matching considerations. Thus, a bulk nonlmear polarization that is much weaker than that of the dipole-allowed contribution present at the interface may still prove to be significant because of the larger volume contributing to the emission. Let us examine this point in a somewhat more quantitative fashion.  [c.1279]

The first direct dynamics application of the spawning method was on the collision dynamics of sodium iodide [245,246]. This is a classic diatomic system in which the lowest two adiabatic electronic states change character on dissociation. Thus the neutral atoms approach each other on the ground-state, and when close enough together an electron is exchanged to form the ionic species. As a ID nuclear problem this system does not contain a conical intersection, but the non-adiabatic coupling is still sfiong where the surfaces come close together. This is the crossing point, where the electron transfer takes place.  [c.308]

It is, of course, possible to first form the above spin-pure P as a trial wavefunetion and to then determine the orbitals 1 s 1 s and 2s which minimize its energy in so doing, one is dealing with a spin-pure function from the start. The problem with carrying out this process, which is referred to as a spin-adapted Hartree-Fock calculation, is that the resultant Is and Is orbitals still do not have identical spatial attributes. Having a set of orbitals (Is, Is, 2s, and the virtual orbitals) that form a non-orthogonal set (Is and Is are neither identical nor orthogonal) makes it difficult to progress beyond the singleconfiguration wavefunetion as one often wishes to do. That is, it is difficult to use a spin-adapted wavefunetion as a starting point for a correlated-level treatment of electronic motions.  [c.463]

We have now completed the first stage of the problem we set out to consider. We have arrived at the probability that chains are capped at both ends by potential branch points. The second stage of the derivation is concerned with the reaction between these chain ends via the remaining f - 1 reactive A groups. By hypothesis, the mixture contains an excess of B groups, so there are still unreacted BB monomers or other polymer chain segments with terminal B groups which can react with the Aj- j groups we have been considering. Table 5.6 shows the connecting of such groups-by BB molecules for simplicity for several values of f. For each of the situations shown in Table 5.6, converting the boxed A BB A groups into a condensed abba sequence amounts to linking into a linear polymer the capped segments which had been separate until now. Since the capped ends have f - 1 remaining functional groups at this point, a linear condensation product results when any one of these groups reacts. Thus l/(f- 1) is the probability of this particular eventuality.  [c.318]

Air-cooled acid plants are characterized by a large refractory-lined combustion chamber from which waste heat is removed by radiation and convection. The combustion chamber is constmcted of graphite or of carbon steel lined with a single layer of high alumina refractory brick. Refractory units operate at cooler temperatures because of the poorer heat transfer properties of brick compared to graphite. Corrosion of the carbon steel is, surprisingly, not a serious problem as long as the combustion gas stream and refractory stay well-above the dew point of the azeotropic (92% P2O5) phosphoric acid. Air-cooled plants normally operate with about 1—200% excess combustion air to reduce the flame temperature (1000—1700°C) and carry waste heat to the hydrator—absorber, where it is removed by evaporation of water.  [c.327]

Polymer recycle has been practiced as part of the manufacturing process for nylon-6,6 (219) and nylon-6 (220) almost from the beginning of the iadustry. Acid hydrolysis by Du Pont and base hydrolysis by BASF and Rhc ne-Poulenc of relatively pure nylon-6,6 waste streams, followed by separation of iagredients, purification, and reuse, has been practiced for many years. Also, phosphoric acid-cataly2ed hydrolysis and steam distillation of caprolactam from pure nylon-6 is still used by BASF, Rhc ne-Poulenc, and SNIA. However, it is the challenge of recycling post-consumer waste that has generated the greatest activity siace 1990. Stimulated by more stringent governmental regulations for recycling plastic packagiag and automotive plastic components la Germany and the growing landfill problem ia the United States, the nylon iadustry has developed several technologies to address the issue of recycling post-consumer waste. One of the greatest economic challenges is the collection and separation of nylon from other wastes, including other polymers. The ultimate solution to this problem may await the development of a cost-effective waste handling iafrastmcture for all recycled materials, at which poiat relatively pure, high volume, low cost, post-consumer nylon will become available. Recycled nylon carpets constitute the largest single supply of potentially recyclable nylon. A patent has appeared for the preparation of a thermoplastic composite by remelting all the components of a nylon carpet and forming it iato bulk plastic parts (221). Unfortunately, because of the thermo- and photooxidative products formed ia nylon during manufacture and use, and the thermal degradation and thermooxidative products formed during further melt processiag, any direct remelt processiag of nylon results ia a low grade product, even with the use of currentiy available thermal stabilizers. Other approaches have focused on depolymerization and separation of the iagredients. Several patents and articles have appeared regarding the recovery of caprolactam from post-consumer waste nylon-6 via hydrolysis (222) or polymer pyrolysis (223), and the recovery of polymer via solvent dissolution of nylon-6 from nonpolyamide contamination (224) however, these technologies are limited to waste streams that contain nylon-6 as the only polyamide. In particular, nylon-6,6 significantly iaterferes with these processes. Several technologies have appeared which attempt to separate nylon-6,6 and nylon-6, and convert them simultaneously to usehil monomers (225,226). The most promising technology to date appears to be the ammonolysis of nylon-6,6—nylon-6 mixtures, which converts all three iagredients to bexametbylenedi amine (227,228).  [c.241]

The reaction of aniline with formaldehyde can be carried out in a single reactor (Fig. 2). However, most commercial processes probably use multiple reactors, which provide greater control of the MDA isomer distribution and oligomeric content of the final product (17—20). Use of hydrochloric acid and high reaction temperatures necessitates the use of corrosion resistant metallurgy. Normally the acid is first mixed with excess aniline, which causes an exotherm. Formaldehyde is then added, with efficient agitation and at low temperatures (<50°C), to the aniline—aniline hydrochloride solution. The reaction is usually staged to control the condensation and rearrangement steps. The final reaction temperatures are normally 80—120°C. After completion of reaction, the acidic PMDA is treated with aqueous sodium hydroxide to neutrali2e the excess acid. A large amount of salt is formed during this step thus the plants must be located near an oudet capable of handling the generated salt water (normally a seacoast). Processes that recycle the acid and eliminate the salt disposal problem have been patented (21,22). The organic layer is then washed with water and stripped to remove unreacted aniline and water. The unreacted aniline is recycled back to the beginning of the reaction. The product may be purified to isolate pure 4,4 -MDA, packaged for shipment, or treated with phosgene to produce the corresponding isocyanate. The 4,4 -MDA is normally sold in flaked or granular form in lined steel dmms. Depending on the MDA content, PMDA is sold as a waxy soHd or a yellow to brown viscous supercooled Hquid in steel dmms.  [c.249]

When an engineer sets up the model of a process with a recycle stream, at some point in the problem formulation, he or she will come across a cell (ie, variable) whose value depends on itself That is perfecdy normal. Starting with an initial guess for the variable (which may be 2ero), a value is calculated for that variable, and this calculated value is subsequently used as the initial guess for the next round of calculations. This method of direct substitution (also known as the Gauss-Seidel method) occurs every time the spreadsheet is recalculated, and recalculations are under the direct control of the user in all spreadsheet programs via the use of a specially designated key. After a number of these recalculations (iterations), the difference between the initial guess and the calculated value approaches 2ero (ie, the values become identical), and convergence is reached. An added benefit is the abiUty to observe the sensitivity of all the other parameters in the spreadsheet as a function of the convergence process (74).  [c.85]

The term SAXPY has arisen as a mnemonic for scalar alpha Xplus Y (2). This loop requires two operands and produces one result for each iteration of the loop. In 64 bit, or 8 byte, precision (8 bytes per floating-point number), this is a total of 24 bytes of data being consumed or produced per iteration. If the memory bandwidth were 240 megabytes per second, the memory subsystem could maintain this loop at 10 million iterations per second. Each loop iteration represents two floating-point operations, a multiplication and an addition thus mnning at 10 million iterations per second is only 20 MFLOPS, a small fraction of the peak performance of supercomputers. Most supercomputers have memory subsystems with much higher bandwidths, sometimes with separate pathways for read and write operations. Nevertheless, careful analysis of memory subsystem usage can be an important ingredient of any code optimization. In the preceding example, the system should look for subsequent operations to be performed on Z(I) while it is still near the CPU, before it is returned to main memory. The problem of optimizing CPU performance is not specific to supercomputers. However, given the enormous cost, a great deal more effort is devoted to optimizing codes on supercomputers than on other machines.  [c.89]

Residual tensile stresses, as we have seen, are a problem. But compressive residual stresses, in the right place, can be used to advantage. Toughened glass is made by heating the product above its annealing point, and then cooling rapidly. The surface contracts and hardens while the interior is still hot and more fluid it deforms, allowing the tensile stress in the surface to relax. Then the interior cools and contracts. But the surface is below its strain point it cannot flow, so it is put into compression by the contracting interior. With the surface in compression, the glass is stronger, because the microcracks which initiate failure in a glass are always in the surface (caused by abrasion or corrosion). The interior, of course, is in tension and if a crack should penetrate through the protective compressive layer it is immediately unstable and the toughened glass shatters spontaneously.  [c.201]

Other researchers have substantially advanced the state of the art of fracture mechanics applied to composite materials. Tetelman [6-15] and Corten [6-16] discuss fracture mechanics from the point of view of micromechanics. Sih and Chen [6-17] treat the mixed-mode fracture problem for noncollinear crack propagation. Waddoups, Eisenmann, and Kaminski [6-18] and Konish, Swedlow, and Cruse [6-19] extend the concepts of fracture mechanics to laminates. Impact resistance of unidirectional composites is discussed by Chamis, Hanson, and Serafini [6-20]. They use strain energy and fracture strength concepts along with micromechanics to assess impact resistance in longitudinal, transverse, and shear modes.  [c.345]

Other researchers have substantially advanced the state of the art of fracture mechanics applied to composite materials. Tetelman [6-15] and Corten [6-16] discuss fracture mechanics from the point of view of micromechanics. Sih and Chen [6-17] treat the mixed-mode fracture problem for noncollinear crack propagation. Waddoups, Eisenmann, and Kaminski [6-18] and Konish, Swedlow, and Cruse [6-19] extend the concepts of fracture mechanics to laminates. Impact resistance of unidirectional composites is discussed by Chamis, Hanson, and Serafini [6-20]. They use strain energy and fracture strength concepts along with micromechanics to assess impact resistance in longitudinal, transverse, and shear modes.  [c.345]

On-column injection of large volumes of aqueous samples has achieved considerable attention in the field of on-line reversed phase LC-GC. The main problem in direct introduction of water, as mentioned above, is the poor wettability of the uncoated precolumns. In 1989, Grob and Li (20) tested several fused silica and glass precolumns deactivated by using different methods and concluded that the transfer of aqueous solvents by retention gap techniques was not achievable because it was impossible to find a precolumn that at the same time was both water-wettable and inert. This problem was tentatively solved by using an organic solvent with a higher boiling point than that of water. In fact, these same authors (21) investigated the wettability of phenyl- and cyanosilylated precolumns with mixtures of organic solvents and water, by using the retention gap technique for transferring water and mixtures of water with organic solvents. Their results demonstrated that, depending on the organic solvent being used, mixtures of such solvents with high concentrations of water still wet the precolumns (e.g. 70% water and 30% 1-propanol), although water did not evaporate together with the organic component. Azeotropically boiling mixtures, e.g. 28% water and 72% of 1-propanol, demonstrated that wettability of such precolumns is possible and thus allows the introduction of water by the retention gap technique.  [c.28]

Cracking, flaking, scaling or blistering due to under-rusting (the latter often being accompanied by brown discoloration of the film) is, as has already been explained, due to mechanical action by the products of corrosion. This may at times pose the problem of whether the paint or the painting system was responsible for the corrosion, or whether, on the other hand, it was the corrosion (possibly residual) which was responsible for the unsatisfactory performance of the paintwork. The better the adhesion of the paint to the metal, the less damage there will be to the paint film, and the less premature corrosion will ensue. This is similarly the case with non-ferrous metals. Rough (especially blasted) steel surfaces which have received too thin a paint coverage will be indicated by the presence of pinpoint rust spots in the film surface, wherever the metal peaks have not been sufficiently protected.  [c.616]


See pages that mention the term Saddle-point of the problem : [c.1873]    [c.199]    [c.75]    [c.1035]   
See chapters in:

Analysis of cracks in solids  -> Saddle-point of the problem