Numerical sufficiency

Combinations of Kelvin and Maxwell units with different characteristic time constants (tk or Jm) are usually required. Although models using Maxwell or Kelvin units have been reported, it is numerically sufficient to combine either Maxwell or Kelvin units into line spectra.Such line spectra are easier to handle, but require care in the choice of the properties of each individual element. If the experimenter cannot relate the elements chosen to structural features in the sample, such modeling is of little value. Examples of appropriate use of line spectra can be found in the literature. ... 

Thus, the use of a hybrid construction scheme may provide an alternative method for a numerical sufficiency condition. Validation of the AR might then only be applicable to the specific system computed, but this is nevertheless valuable, for at least a theoretically reliable estimate of the AR may be established. 

The implementation of very effective devices on vehicles such as catalytic converters makes extremely low exhaust emissions possible as long as the temperatures are sufficient to initiate and carry out the catalytic reactions however, there are numerous operating conditions such as cold starting and... 

At sufficiently high frequency, the electromagnetic skin depth is several times smaller than a typical defect and induced currents flow in a thin skin at the conductor surface and the crack faces. It is profitable to develop a theoretical model dedicated to this regime. Making certain assumptions, a boundary value problem can be defined and solved relatively simply leading to rapid numerical calculation of eddy-current probe impedance changes due to a variety of surface cracks. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

The last approximation is for finite At. When the equations of motions are solved exactly, the model provides the correct answer (cr = 0). When the time step is sufficiently large we argue below that equation (10) is still reasonable. The essential assumption is for the intermediate range of time steps for which the errors may maintain correlation. We do not consider instabilities of the numerical solution which are easy to detect, and in which the errors are clearly correlated even for large separation in time. Calculation of the correlation of the errors (as defined in equation (9)) can further test the assumption of no correlation of Q t)Q t )). 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

The numerous reactions that yield aldehydes and ketones discussed m earlier chapters and reviewed m Table 17 1 are sufficient for most syn theses... 

This new edition, the hfth under the aegis of the present editor, remains the one-volume source of factual information for chemists, both professionals and students—the hrst place in which to look it up on the spot. The aim is to provide sufficient data to satisfy all one s general needs without recourse to other reference sources. A user will hnd this volume of value as a time-saver because of the many tables of numerical data which have been especially compiled. 

If the surface of the adsorbent is energetically heterogeneous rather than homogeneous each step of the isotherm will be replaced by an assortment of steps, corresponding to the completion of a monolayer on the different homogeneous patches of the surface. If the steps are sufficiently numerous... 

The concepts behind the analysis are not difficult. The piping system is simply a stmcture composed of numerous straight and curved sections of pipe. Although, for straight pipe, elementary beam theory is sufficient for the solution of the problem, it is not adequate for curved pipe. However, by the iatroduction of a flexibiUty factor, to account for iacreased flexibiUty of curved pipe over straight pipe, and a stress intensification factor, /, to account for... 

The hydrolysis and polycondensation reactions initiate at numerous sites within the TMOS/H2O solution as mixing occurs. When sufficient intercoimected Si—O—Si bonds are formed in a region, the material responds cooperatively as colloidal (submicrometer) particles or a sol. The size of the sol particles and the cross-linking within the particles, ie, the density, depends on the pH and R ratio, where R = [H2 0]/[Si(0R)4]. 

Biphenyl has been produced commercially in the United States since 1926, mainly by The Dow Chemical Co., Monsanto Co., and Sun Oil Co. Currently, Dow, Monsanto, and Koch Chemical Co. are the principal biphenyl producers, with lesser amounts coming from Sybron Corp. and Chemol, Inc. With the exception of Monsanto, the above suppHers recover biphenyl from high boiler fractions that accompany the hydrodealkylation of toluene [108-88-3] to benzene (6). Hydrodealkylation of alkylbenzenes, usually toluene, C Hg, is an important source of benzene, C H, in the United States. Numerous hydrodealkylation (HDA) processes have been developed. Most have the common feature that toluene or other alkylbenzene plus hydrogen is passed under pressure through a tubular reactor at high temperature (34). Methane and benzene are the principal products formed. Dealkylation conditions are sufficiently severe to cause some dehydrocondensation of benzene and toluene molecules. 

The separation of Hquid crystals as the concentration of ceUulose increases above a critical value (30%) is mosdy because of the higher combinatorial entropy of mixing of the conformationaHy extended ceUulosic chains in the ordered phase. The critical concentration depends on solvent and temperature, and has been estimated from the polymer chain conformation using lattice and virial theories of nematic ordering (102—107). The side-chain substituents govern solubiHty, and if sufficiently bulky and flexible can yield a thermotropic mesophase in an accessible temperature range. AcetoxypropylceUulose [96420-45-8], prepared by acetylating HPC, was the first reported thermotropic ceUulosic (108), and numerous other heavily substituted esters and ethers of hydroxyalkyl ceUuloses also form equUibrium chiral nematic phases, even at ambient temperatures. 

Polyarylether Ketones. The aromatic polyether ketones are tme thermoplastics. Although several are commercially available, two resins in particular, poly ether ether ketone [31694-16-3] (PEEK) from ICI and poly ether ketone ketone (PEKK) from Du Pont, have received most of the attention. PEEK was first synthesized in 1981 (20) and has been well studied it is the subject of numerous papers because of its potential use in high performance aircraft. Tough, semicrystalline PEEK is prepared by the condensation of bis(4-fiuorophenyl) ketone with the potassium salt of bis(4-hydroxyphenyl) ketone in a diaryl sulfone solvent, such as diphenyl sulfone. The choice of solvent is critical other solvents, such as Hquid HE, promote the reaction but lead to premature low molecular-weight crystals, which do not exhibit sufficient toughness (21). 

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). 

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