Correspondence length

Find the magnitude of the weight on bit and corresponding length of drill collars that result in second order of buckling. Drill collars 6 in. x 2 ft, mud density =12 Ib/gal. 

Some examples for NMR techniques that have been used for investigating catalytic reactors (most of which will be discussed in this chapter) and the corresponding length scales which they probe are as follows ... 

Considerations analogous to those for velocity apply to scalar fields as well, and lengths analogous to /k have been introduced for these fields. They differ from /k by factors involving the Prandtl and Schmidt numbers, which differ relatively little from unity for representative gas mixtures. Therefore, to a first approximation for gases, Zk may be used for all fields and there is no need to introduce any new corresponding lengths. 

There are several ways that dispersion (and the corresponding length scale) can be brought at the phenomenological level into the conservative part of the model. The two most well known examples of such theories are gradient (or van der Waals) models with energy... 

This expression has a maximum value at x = 3 and the corresponding length Lpr is called the predominant length... 

The systems may be similar powder compacts of the same powder material but, differing particle sizes, or they may be model systems such as those illustrated in Figs. 16.6 and 16.9 but with all corresponding length dimensions scaled similarly. 

If uSL(x) is not to change during a pass, the total amount of solute in the zone [which is of length l and uniform composition cL(a )] at each stage of the process must be equal to the amount of solute that was in the corresponding length l of the solid bar before it was overrun. Therefore, ... 

Geometric similarity is the most obvious requirement in a model system designed to correspond to a given prototype system. It is the similarity of shape. Two systems are geometrically similar when the ratio of any length in one system to the corresponding length in another system is everywhere the same. 

Figure 1 illustrates two geometrically similar objects. Here P and P are known as corresponding points, X and X, Y and Y are known as corresponding lengths. The ratio of corresponding lengths is known as the scale factor. 

Fig. 14.8 The case of nonmonotonous transport rate function 1 1. v, shown by a solid line in (a,e,h). From top to bottom different location of the cleavage centre D = 3, 15, 32 (shown the vertical line) with respect to two points where the condition of maximum holds (shown vertical lines) (b,f,i) the corresponding length distributions for the...

Fig. 3. Scale separation in a packed-bed reactor L and t are the corresponding length and time scales in meters and seconds, respectively.

If the quantity measure is number, Qfx) is called a cumulative number distribution. If it is length, area, volume, or mass, then the corresponding length [Qfx ], area [Qf,x ], volume, or mass distributions are formed [Qs(x), mass and volume are related by the specific density p. The index r in this notation represents the quantity measure (ISO 9276, Representation of Results—Part 1 Graphical Representation). The choice of the quantity measured is of decisive importance for the appearance of the PSD, which changes significantly when the dimension r is changed. As, e.g., one 100- lm particle has the same volume as 1000 10- lm particles or lOVl-pm particles, a number distribution is always dominated by and biased to the fine fractions of the sample while a volume distribution is dominated by and biased to the coarse. 

Glass fiber reinforced polymers and resins have found widespread application as light weight construction materials. In addition, the properties of interfaces and. interphases play a crucial role in determining the ultimate mechanical and other properties in these and related applications. Since the corresponding length scales can be on the far sub-micrometer level, a more detailed insight requires local analysis techniques which provide materials contrast. 

The corresponding length scale characterizing the inner layer, the viscous length scale, is defined as... 

Similarly, the fixed values of X2 define elliptical cones with vertex at the origin, axis along the x-axis, and elliptical cross sections — in vertical planes with a fixed value of x — and with major axis parallel to the z-axis and minor axis parallel to the y-axis, with the corresponding lengths as in the family of cones in the previous paragraph with the change of the index 1 to 2 in the coordinate x and parameter k, and the exchange of coordinates x and z. 

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