96 - exact values

The exact values of E and 5E / 5n are in general unknown and the Kirchhoff or physical optics method consists in approximating the values of these two quantities on the surface and then evaluating the Helmholtz integral. We shall approximate the field at any point of the surface by the field that would be present on a tangent plane at the point. With this approximation, the field on the surface and its normal derivative are... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

Let us compare computations of the effectiveness factor, using each of the three approximations we have described, with exact values from the complete dusty gas model. The calculations are performed for a first order reaction of the form A lOB in a spherical pellet. The stoichiometric coefficient 10 for the product is unrealistically large, but is chosen to emphasize any differences between the different approaches. 

Note, that the eigenfunctions of H are /n with eigenvalues En. Compare this "exact" value with that obtained by perturbation theory in part a. 

During a molecular dynamics simulation, HyperChem stores the current positions, Tj (t), and the mid-step velocities, Vj (t - 1/2 At). Since the algorithm provides mid-step velocities, but not velocities, Vj (t), for the positions at time t, HyperChem calculates approximate values of Ej-qt (O- This results in slightly larger fluctuations in Ej-ot (t) than an algorithm that calculates exact values of... 

These calculations lend theoretical support to the view arrived at earlier on phenomenological grounds, that adsorption in pores of molecular dimensions is sufficiently different from that in coarser pores to justify their assignment to a separate category as micropores. The calculations further indicate that the upper limit of size at which a pore begins to function as a micropore depends on the diameter a of the adsorbate molecule for slit-like pores this limit will lie at a width around I-So, but for pores which approximate to the cylindrical model it lies at a pore diameter around 2 5(t. The exact value of the limit will of course depend on the actual shape of the pore, and may well be raised by cooperative effects. 

Table 2 summari2es the key physical properties of siUca aerogels. A range of values is given for each property because the exact value is dependent... 

The octane value of an unknown fuel sample is determined by comparing its knocking tendency to various primary reference fuels. Its measured octane is equal to the octane of the PRE which has the same knocking intensity. Knock intensity is controlled to an average value by varying the compression ratio of the CER engine. In practice, the exact value of a fuel s octane number is determined to the nearest 0.1 octane number by interpolation from two PREs that are no more than two octane numbers apart. 

It is assumed that process conditions and physical properties are known and the following are known or specified tube outside diameter D, tube geometrical arrangement (unit cell), shell inside diameter D shell outer tube limit baffle cut 4, baffle spacing and number of sealing strips N,. The effective tube length between tube sheets L may be either specified or calculated after the heat-transfer coefficient has been determined. If additional specific information (e.g., tube-baffle clearance) is available, the exact values (instead of estimates) of certain parameters may be used in the calculation with some improvement in accuracy. To complete the rating, it is necessary to know also the tube material and wall thickness or inside diameter. 

Typical coil coefficients are listed in Table 11-2. More exact values can be calculated by using the methods for natural convection or forced convection given elsewhere in this section. 

There are numerous solubility data in the literature the standard reference is by Seidell (loc. cit.). Valuable as they are, they nevertheless must be used with caution because the solubihty of compounds is often influenced by pH and/or the presence of other soluble impurities which usually tend to depress the solubihty of the major constituents. While exact values for any system are frequently best determined by actual composition measurements, the difficulty of reproducing these solubility diagrams should not be underestimated. To obtain data which are readily reproducible, elaborate pains must be taken to be sure the system sampled is at equihbrium, and often this means holding a sample at constant temperature for a period of from 1 to 100 h. While the published cui ves may not be exac t for actual solutions of interest, they generally will be indicative of the shape of the solubility cui ve and will show the presence of hydrates or double salts. 

The above values give only a preliminary idea of the no-load data of a motor. The exact values for a particular motor may be obtained from the manufacturer. 

All values are approximate and are I oi reference only. For exact values consult the manufacturer. 

GD during a light start, considering the GD of the coupling (impeller) as equal to the motor (the exact value should be obtained from the manufacturer). 

Note These values are only indicative and may vary with the quality of mix and process of curing etc., and differ from one manufacturer to another. For exact values, contact the manufacturer. 

Note The exact values will depend upon the surface condition of the material and the field experience and skill of the operator. For more accurate details consult the chemical manufacturer. 

The results for the tunneling splitting calculated with the use of some of the earlier proposed reaction paths for a single PES (4.40) (with the parameters adopted here) are collected by Bosch et al. [1990]. All of them underestimate by at least an order of magnitude the numerically exact value 10.6 cm which is also given in that paper. The parameters C and Q hit the intermediate region between the sudden and adiabatic approximations, described in sections 2.5 and 4.2, and neither of these approximations is quantitatively applicable to the problem. 

The second reason for modification of the displaced volume is that in real world application, the cylinder will not achieve the volumetric performance predicted by Equation 3.4. It is modified, therefore, to include empirical data. The equation used here is the one recommended by the Compressed Air and Gas Institute [1], but it is somewhat arbitrary as there is no universal equation. Practically speaking, however, there is enough flexibility in guidelines for the equation to produce reasonable results. The 1.00 in the theoretical equation is replaced with. 97 to reflect that even with zero clearance the cylinder will not fill perfectly. Term L is added at the end to allow for gas slippage past the piston rings in the various types of construction. If, in the course of making an estimate, a specific value is desired, use, 03 for lubricated compressors and. 07 for nonlubricated machines. These are approximations, and the exact value may vary by as much as an additional. 02 to. 03... 

As earlier discussed, the dominant factor in the near-surface region is the particle detection system. For a typical silicon surface barrier detector (15-keV FWHM resolution for Fle ions), this translates to a few hundred A for protons and 100— 150 A for Fle in most targets. When y rays induced by incident heavy ions are the detected species (as in FI profiling), resolutions in the near-surface region may be on order of tens of A. The exact value for depth resolution in a particular material depends on the rate of energy loss of incident ions in that material and therefore upon its composition and density. 

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