Sufficient closeness

Note that, due to the small size of the water molecules, a large number of them is sufficiently close to the solute to allow efficient interaction. 

These models, though necessarily idealized, are sufficiently close to the actual systems found in practice to enable useful conclusions to be drawn from a given Type IV isotherm as to the pore structure of a solid adsorbent. To facilitate the discussion, it is convenient to simplify the Kelvin equation by putting yVJRT = K, and on occasion to use the exponential form ... 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

Diamond is supreme among natural gemstones ia H, RI, and DISP. Table 3 shows the steady improvement ia the sequence of diamond imitations, the aim being to produce a colorless, adequately hard material having closely matching optical properties. The iatroduction of synthetic cubic 2irconia ia 1976 brought about a sufficiently close match. 

To obtain a metallurgical bond between two metals, the atoms of each metal must be brought sufficiently close so that their normal forces of interatomic attraction produce a bond. The surfaces of metals and alloys must not be covered with films of oxides, nitrides, or adsorbed gases. When such films are present, metal surfaces do not bond satisfactorily (see Metal surface treatments). 

For combustion of simple hydrocarbons, the oxidation reactions appear to foUow classical first-order reaction kinetics sufficiently closely that practical designs can be estabUshed by appHcation of the empirical theory (8). For example, the general reaction for a hydrocarbon ... 

This states that/(x) can be calculated as close to N as desirable by making x sufficiently close to a. This does not put any restriction on fix) when x = a. Alternatively, for any given positive number , a number 6 can be found such that 0 < b — xl < 6 implies that IN —fix) < . The following operations with hmits (when they exist) are valid ... 

Successive Substitutions Let/(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F x), then an iterative scheme can be set up in the form Xi + = F xi). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writings = F x), of which there will usually be several forms. However, if 7 is a root of/(x) = 0, and if IF ( 7)I < I, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xi + is proportional to the first power of the error in xi for large k. 

At this point in the inside-out method, the revised column profiles of temperature and phase compositions are used in the outer loop with the complex SRK thermodynamic models to compute updates of the approximate K and H constants. Then only one inner-loop iteration is required to obtain satisfactory convergence of the energy equations. The K and H constants are again updated in the outer loop. After one inner-loop iteration, the approximate K and H constants are found to be sufficiently close to the SRK values that overall convergence is achieved. Thus, a total of only 3 outer-loop iterations and 4 inner-loop iterations are required. 

In inaccessible regions where an impressed current installation is not sufficiently close to a low voltage supply, the protection current can be supplied from batteries, thermogenerators, and if there is sufficient radiation from the sun, solar cells. Wind generators and diesel units, on the other hand, are less suitable because of the maintenance necessary for continuous operation. 

Direct interception occurs when the fluid streamline carrying the particle passes within one-half of a particle diameter of the filter element. Regardless of the particle s size, mass, or inertia, it will be collected if the streamline passes sufficiently close. Inertial impaction occurs when the particle would miss the filter element if it followed the streamline, but its inertia resists the change in direction taken by the gas molecules and it continues in a... 

Most MO methods find a bond alternation pattern in the minimum-energy structure, but calculations that include electron correlation lead to a delocalized minimum-energy structure. Thus, although the n system in 1 is not completely planar, it appears to be sufficiently close to provide a delocalized 10-electron Ji system. A resonance energy of 17.2 kcal has been obtained on the basis of an experimental heat of hydrogenation. ... 

These values are sufficiently close to the previous values so the die exit dimensions are... 

When the pressures to induce shock-induced transformations are compared to those of static high pressure, the values are sufficiently close to indicate that they are the same events. In spite of this first-order agreement, differences between the values observed between static and shock compression are usually significant and reveal effects controlled by the physical and chemical nature of the imposed deformation. Improved time resolution of wave profile measurements has not led to more accurate shock values rather. 

The variational energy principles of classical elasticity theory are used in Section 3.3.2 to determine upper and lower bounds on lamina moduli. However, that approach generally leads to bounds that might not be sufficiently close for practical use. In Section 3.3.3, all the principles of elasticity theory are invoked to determine the lamina moduli. Because of the resulting complexity of the problem, many advanced analytical techniques and numerical solution procedures are necessary to obtain solutions. However, the assumptions made in such analyses regarding the interaction between the fibers and the matrix are not entirely realistic. An interesting approach to more realistic fiber-matrix interaction, the contiguity approach, is examined in Section 3.3.4. The widely used Halpin-Tsai equations are displayed and discussed in Section 3.3.5. 

Form the new density matrix. If it is sufficiently close to the previous density matrix, we are done, otherwise go to step (4). 

The advantage of the NR method is that the convergence is second-order near a stationary point. If the function only contains tenns up to second-order, the NR step will go to the stationary point in only one iteration. In general the function contains higher-order terms, but the second-order approximation becomes better and better as the stationary point is approached. Sufficiently close to tire stationary point, the gradient is reduced quadratically. This means tlrat if the gradient norm is reduced by a factor of 10 between two iterations, it will go down by a factor of 100 in the next iteration, and a factor of 10 000 in the next ... 

US) and 12 (English metric) are not exactly equivalent but are sufficiently close for practical valve sizing purposes. 

Consider a fluid flowing between two sections, 1 and 2 (Figure 6.12), which are sufficiently close for friction losses to be negligible between the two sections they are a sufficient distance apart, however, for the presence of a small surface at right angles to the direction of flow at section 2 to have negligible effect on the pressure at section 1. These conditions are normally met if the distance between the sections is one pipe diameter. 

Equations 6.11 and 6.12 can be used for the calculation of the fluid velocity and the impact pressure in terms of the static pressure a short distance upstream. The two sections are chosen so that they are sufficiently close together for frictional losses to be negligible. Thus Pi will be approximately equal to the static pressure at both sections and the equations give the relation between the static and impact pressure—and the velocity — at any point in the fluid. 

This value is sufficiently close for practical purposes to the value of 0.0906, calculated previously. If necessary, a second iteration may be carried out. 

---