Step separation

The semiclassical approach to QCMD, as introduced in [10], derives the QCMD equations within two steps. First, a separation step makes a tensor ansatz for the full wavefunction separating the coordinates x and q ... 

Remark The statement of Prop. 6 is also valid for the g-expectation (g) = ( ,g ) of the TDSCF solution. Consequently, TDSCF fails near the crossing, a fact, which emphasizes that the reason for this failure is connected to the separation step. 

In many molecular dynamics simulations, equilibration is a separate step that precedes data collection. Equilibration is generally necessary lo avoid introducing artifacts during the healing step an d to en su re th at the trajectory is aciii ally sim u laiin g eq u i librium properties. The period required for equilibration depends on the property of Interest and the molecular system. It may take about 100 ps for the system to approach equilibrium, but some properties are fairly stable after 1 0-20 ps. Suggested tim es range from. 5 ps to nearly 100 ps for medium-si/ed proteins. 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

Use of an excess of the halogenating agent results in halogenation at the 3-position of the oxindole[3,4]. The halogenation and hydrolysis can be carried out as two separate steps. One optimized procedure of this type used NCS as the halogenating agent and it was found that 70% phosphoric acid in 2-mcthoxycthanol was the most effective medium for hydrolysis[2]. If the halogenation is carried out in pyridine, the intermediate is trapped as an... 

Many reactions classified as dehydrogenations occur within the cells of living sys terns at 25°C H2 is not one of the products however Instead the hydrogens are lost m separate steps of an enzyme catalyzed process The enzyme indicated m the reaction... 

The E2 mechanism is a concerted process m which the carbon-hydrogen and carbon-halogen bonds both break m the same elementary step What if these bonds break m separate steps s... 

Figure 10 12 shows the interaction between the HOMO of one ethylene molecule and the LUMO of another In particular notice that two of the carbons that are to become ct bonded to each other m the product experience an antibondmg interaction during the cycloaddition process This raises the activation energy for cycloaddition and leads the reaction to be classified as a symmetry forbidden reaction Reaction were it to occur would take place slowly and by a mechanism m which the two new ct bonds are formed m separate steps rather than by way of a concerted process involving a sm gle transition state... 

Stage 2 The tetrahedral intermediate dissociates to give an amide The O—H and C—O bonds may break m the same step or m separate steps... 

The condensation leaves epoxy end groups that are then reacted in a separate step with nucleophilic compounds (alcohols, acids, or amines). Eor use as an adhesive, the epoxy resin and the curing resin (usually an aliphatic polyamine) are packaged separately and mixed together immediately before... 

Liquid chromatography is a separation method that is often applied to nonvolatile, thermally labile materials such as peptides, and, if their mass spectra are required after the separation step, then a mild method of ionization is needed. Since FAB/LSIMS is mild and works with a liquid matrix, it is not surprising that attempts were made to utilize this ionization source as both an inlet... 

The initial manufacture of mixed xylenes and the subsequent production of high purity PX and OX consists of a series of stages in which (/) the mixed xylenes ate initially produced (2) PX and/or OX are separated from the mixed xylenes stream and (3) the PX- (and perhaps OX-) depleted xylene stream is isomerized back to an equiUbtium mixture of xylenes and then recycled back to the separation step. These steps are discussed below. 

Acrylic Acid Recovery. The process flow sheet (Fig. 3) shows equipment and conditions for the separations step. The acryUc acid is extracted from the absorber effluent with a solvent, such as butyl acetate, xylene, diisobutyl ketone, or mixtures, chosen for high selectivity for acryUc acid and low solubihty for water and by-products. The extraction is performed using 5—10 theoretical stages in a tower or centrifiigal extractor (46,61—65). 

Vitamins. The preparation of heat-sensitive natural and synthetic vitamins (qv) involves solvent extraction. Natural vitamins A and D are extracted from fish Hver oils and vitamin E from vegetable oils (qv) Hquid propane [74-98-6] is the solvent. In the synthetic processes for vitamins A, B, C, and E, solvent extraction is generally used either in the separation steps for intermediates or in the final purification. 

Another use is in various extraction and absorption processes for the purification of acetylene or butadiene and for separation of aHphatic hydrocarbons, which have limited solubiHty in DMF, from aromatic hydrocarbons. DMF has also been used to recover CO2 from flue gases. Because of the high solubiHty of SO2 iu DMF, this method can even be used for exhaust streams from processes using high sulfur fuels. The CO2 is not contaminated with sulfur-containing impurities, which are recovered from the DMF in a separate step (29). 

In another extractor (Automatic Machinery and Electronics Inc. (AMC)) the individual fmits are cut in half as they pass a stationary knife. The halves are oriented in a vertical plane, picked up by synthetic mbber cups, and positioned across plastic serrated reamers revolving in a synchronized carrier in a vertical plane. As the fmit halves progress around the extractor turntable, the rotating reamers exert increasing pressure and express the juice. The oil and pulp contents in the juice increase with greater reaming pressure. The recoverable oil is removed in a separate step prior to juice extraction. Needle-sharp spikes prick the peel of the whole fmit, releasing oil that is washed away with water and recovered from the oil—water emulsion. 

When the recycle soot in the feedstock is too viscous to be pumped at temperatures below 93°C, the water—carbon slurry is first contacted with naphtha carbon—naphtha agglomerates are removed from the water slurry and mixed with additional naphtha. The resultant carbon—naphtha mixture is combined with the hot gasification feedstock which may be as viscous as deasphalter pitch. The feedstock carbon—naphtha mixture is heated and flashed, and then fed to a naphtha stripper where naphtha is recovered for recycle to the carbon—water separation step. The carbon remains dispersed in the hot feedstock leaving the bottom of the naphtha stripper column and is recycled to the gasification reactor. 

Some of the economic hurdles and process cost centers of this conventional carbohydrate fermentation process, schematically shown in Eigure 1, are in the complex separation steps which are needed to recover and purify the product from the cmde fermentation broths. Eurthermore, approximately a ton of gypsum, CaSO, by-product is produced and needs to be disposed of for every ton of lactic acid produced by the conventional fermentation and recovery process (30). These factors have made large-scale production by this conventional route economically and ecologically unattractive. 

In the most common method for chemiluminescent immunoassay (GLIA), after the immunological reaction and any necessary separation steps, the labeled compounds or complexes react with an oxidizer, eg, hydrogen peroxide, and an enzyme, eg, peroxidase, or a chelating agent such as hemin or metal... 

Uranium ores are leached with dilute sulfuric acid or an alkaline carbonate [3812-32-6] solution. Hexavalent uranium forms anionic complexes, such as uranyl sulfate [56959-61-6], U02(S0 3, which are more selectively adsorbed by strong base anion exchangers than are other anions in the leach Hquors. Sulfate complexes are eluted with an acidified NaCl or ammonium nitrate [6484-52-2], NH NO, solution. Carbonate complexes are eluted with a neutral brine solution. Uranium is precipitated from the eluent and shipped to other locations for enrichment. Columnar recovery systems were popular in South Africa and Canada. Continuous resin-in-pulp (RIP) systems gained popularity in the United States since they eliminated a difficult and cosdy ore particle/leach hquor separation step. 

Ore Dressing. The principal lead mineral, galena, in most cmde ores, is separated from the valueless components, or gangue. Other valuable minerals that are present in the ore may be recovered either together with the lead, or in a separate step (2,3). Occasionally, the ores are sufftciendy rich in lead and low in impurities to be smelted direcdy. 

The product stream contains gases and soflds. The soflds are removed by using either cyclones, filters, or both in combination. Cyclones are devices used to separate soflds from fluids using vortex flow. The product gas stream must be cooled before being sent to the collection and refining system. The ALMA process uses cyclones as a primary separation technique with filters employed as a final separation step after the off-gas has been cooled and before it is sent to the collection and refining system (148). As in the fixed-bed process, the reactor off-gas must be incinerated to destroy unreacted butane and by-products before being vented to the atmosphere. 

Mercurous Nitrate. Mercurous nitrate [10415-75-5] Hg2N20 or Hg2(N02)2, is a white monoclinic crystalline compound that is not very soluble in water but hydrolyzes to form a basic, yellow hydrate. This material is, however, soluble in cold, dilute nitric acid, and a solution is used as starting material for other water-insoluble mercurous salts. Mercurous nitrate is difficult to obtain in the pure state directly because some mercuric nitrate formation is almost unavoidable. When mercury is dissolved in hot dilute nitric acid, technical mercurous nitrate crystallizes on cooling. The use of excess mercury is helpful in reducing mercuric content, but an additional separation step is necessary. More concentrated nitric acid solutions should be avoided because these oxidize the mercurous to mercuric salt. Reagent-grade material is obtained by recrystaUization from dilute nitric acid in the presence of excess mercury. 

Because of its volatility, the cobalt catalyst codistills with the product aldehyde necessitating a separate catalyst separation step known as decobalting. This is typically done by contacting the product stream with an aqueous carboxyhc acid, eg, acetic acid, subsequently separating the aqueous cobalt carboxylate, and returning the cobalt to the process as active catalyst precursor (2). Alternatively, the aldehyde product stream may be decobalted by contacting it with aqueous caustic soda which converts the catalyst into the water-soluble Co(CO). This stream is decanted from the product, acidified, and recycled as active HCo(CO)4. 

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