Calculations precision

If failure is initiated by an increase in internal pressure in combination with a malfunctioning of the pressure relief, the pressure at failure will equal the failure pressure of the vessel. This failure pressure is usually the maximum working pressure multiplied by a safety factor. For carbon-steel vessels, this safety factor can be taken as four. Mote precise calculations ate possible if the vessel s dimensions and material parameters ate known. 

Book II investigates the dynamical conditions of fluid motion. Book III displays the law of gi avitatioii at work in the solar system. It is demonstrated from the revolutions of the six known planets, including Earth, and their satellites, though Newton could never quite perfect the difficult theory of the Moon s motion. It is also demonstrated from the motions of comets. The gravitational forces of the heavenly bodies are used to calculate their relative masses. The tidal ebb and flow and the precession of the equinoxes is explained m terms of the forces exerted by the Sun and Moon. These demonstrations are carried out with precise calculations. 

That is the input power required for one set of impellers. Collection factors for non-geometrical similarity are required to include the effect of known factors in precise calculations. 

These approximate calculations reveal that the concentrations are many orders of magnitude greater than those permitted for such compounds in most situations (permitted levels are typically less than 10 mg-m-3). More precise calculations could be performed with an equation of state (e.g. Peng-Robinson Equation of State, see Chapter 4). 

As alluded to previously, numerical issues actually create a more complex situation than that just been described. For starters, the density of states is almost never calculated directly, as it typically spans many orders of magnitudes. This, in turn, would quickly overwhelm standard double-precision calculations in personal computers. This is easily remedied by working instead with the dimensionless entropy / = In Q, which for the purposes of this chapter will inherit all of the same notation used for the density of states in Chap. 1 - subscripts tot, ex, etc. 

The combination of high temperatures and densities at early times leads to the existence of a phase close to thermal equilibrium, when the signihcant particle reaction rates dominated over the expansion rate. This enables precise calculations to be made. 

The thermal expansion of the more common grades of steel used for piping may be determined from Table PL-2.5.2. For materials not included in Table PL-2.5.2 or for more precise calculations, reference may be made to authoritative source data. 

Logarithmic scales and power laws make extrapolation look too easy. The wide confidence limits are often overlooked and particular care should be exercised because small deviations can result in large changes in predicted lifetime. Precise calculations can lose their value when the extent of the confidence limits is noticed ( predicted life 10 years, upper limit 600 years, lower limit two months ). 

If chemistry was characterized in the nineteenth century by the precise measurement of the products of chemical combustion and combination, as well as by the precise calculation of elementary combining proportions or atomic weights, physics, too, came increasingly to be identified not just with experimentalism but with precise measurement and the "last decimal place." As Maxwell put it shortly before his death in 1879,... 

Although the harmonic ZPVE must always be taken into account in the calculation of AEs, the anharmonic contribution is much smaller (but oppositely directed) and may sometimes be neglected. However, for molecules such as H2O, NH3, and CH4, the anharmonic corrections to the AEs amount to 0.9, 1.5, and 2.3 kJ/mol and thus cannot be neglected in high-precision calculations of thermochemical data. Comparing the harmonic and anharmonic contributions, it is clear that a treatment that goes beyond second order in perturbation theory is not necessary as it would give contributions that are small compared with the errors in the electronic-structure calculations. 

For a more precise calculation of intensities of infrared bands it is necessary to take into account the variation of the dipole function with intemuclear distance,... 

Deviations from ideality in real solutions have been discussed in some detail to provide an experimental and theoretical basis for precise calculations of changes in the Gibbs function for transformations involving solutions. We shall continue our discussions of the principles of chemical thermodynamics with a consideration of some typical calculations of changes in Gibbs function in real solutions. 

However, for more precise calculations, it is necessary to consider that the mobility (hence, the conductance) of ions changes with concentration, even when dissociation is complete, because of interionic forces. Thus, Equation (20.20) is oversimplified in its use of Aq to evaluate a, because at any finite concentration, the equivalent conductances of the and Ac ions, even when dissociation is complete, do not equal Aq. 

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