Logarithmic behavior

Referring to the thin-gap solution on the left of Fig. 4.4, it is apparent that the P = 0 case has a nearly linear velocity profile. When the gap is very thin, the problem and the solution approach the planar situation. In this case the logarithmic behavior (seen in Eq. 4.8) is diminished and nearly eliminated. For the wide-gap case, however, the logarithmic effect is clearly important. This behavior illustrates the need for the extra parameter, which is not needed for planar problems. 

A second consideration with logarithmic amplifiers is that because of the complex way in which they operate (3), quantitative data often require reconversion to a linear scale before further calculations can be done. This process requires information about the exact number of channels/decade, and any deviation from true logarithmic behavior (4). Such deviations are sometimes found, particularly in the lowest decade. 

The estimation of the optimal pressure was previously discussed by taking into account the possible pressure dependence of [2] as well as the interrelation of pressure and temperature defined under isokinetic conditions [3], The relationship (Eq. (10.1)) underlines that the rate constant increases exponentially with pressure. The logarithmic behavior is illustrated in Fig. 10.1 which shows the variation of the rate constant ratio fep/ko with pressure at 25 °C. As an example, let us consider a pressure of 300 MPa which is usually an upper limit for large commercial pressure vessels. At that pressure the value of fep/fco approaches 10-40 for pressure-... 

Effects of surface roughness are also evident in the boundary layer mean velocity profiles shown in Fig. 6.46. The profiles still exhibit a near-wall logarithmic behavior, but with a dependence on the roughness Reynolds number k = ksu lv. The law of the wall for a rough surface may be written as... 

This profile relation may be viewed as a correction to the logarithmic law. For an almost neutral atmosphere, L is a large positive number, and the relationship between velocity and height is logarithmic. As stability increases, the positive L decreases and the deviation from the logarithmic behavior increases. Finally, for very stable conditions, L approaches zero, the second term in (16.78) dominates, and the velocity profile becomes linear. 

We may also find breaks which slow down corrosion, such as if a scale crystallizes only after some critical thickness or time is reached without disintegration. A parabolic-logarithmic behavior may follow. Note that this break is not apparent to the naked eye unless very long times of observation are involved. 

Another case of logarithmic behavior appears when air is sterilized (filtered) by fibrous media, and the log-penetration law is valid. This law relates filter effectiveness (N/Nq) to the filter thickness L and a factor Kp... 

Unfortunately, since the logarithmic behavior of V (or l/V) is involved in initiating and proceeding with the foregoing outline of the calculation, the point where V, = 0 is precluded as a starting point. The option therefore is to express the changes in terms of the reject rate L and its composition. 

In genaal, the kinetic behavior of a speciQc oxide scale may be different from the most observable ones cited above and therefore, the exponent n may take a different value, which must be determined experimentally. Also, a logarithmic behavior is possible for thin layers at relatively low temperatures [2]. For a logarithmic behavior, the weight gain may be (Mined by... 

---