Tie point

Pumps are operated in parallel to divide the load between two (or more) smaller pumps rather than a single large one, or to provide additional capacity in a system on short nodce, or for many other related reasons. Figure 3-35 illustrates the operational curve of two identical pumps in parallel, each pump handling one half the capacity at the system head conditions. In the parallel arrangement of two or more pumps of the same or different characteristic curves, the capacities of each pump are added, at the head of the system, to obtain the delivery flow of the pump system. Each pump does not have to carry the same How but it will operate on its own characteristic curve, and must deliver the required head. At a common tie point on the discharge of all the pumps, the head will be the same for each pump, regardless of its flow. 

FIGURE 5.7 Phase separation in styrene-butadiene-styrene (SBS) triblock copolymer. The isolated spherical styrene domains form the hard phase, which act both as intermolecular tie points and filler. The continuous butadiene imparts the elastomeric characteristics to this polymer. MW = molecular weight. (From Grady, B.P. and Cooper, S.L., Science and Technology of Rubber, Mark, J.E., Erman, B., and Eirich, F.R. (eds.). Academic Press, San Diego, CA, 1994. With permission.)... 

In TPE, the hard domains can act both as filler and intermolecular tie points thus, the toughness results from the inhibition of catastrophic failure from slow crack growth. Hard domains are effective fillers above a volume fraction of 0.2 and a size <100 nm [200]. The fracture energy of TPE is characteristic of the materials and independent of the test methods as observed for rubbers. It is, however, not a single-valued property and depends on the rate of tearing and test temperature [201]. The stress-strain properties of most TPEs have been described by the empirical Mooney-Rivlin equation... 

Figure 4.12 Magnification of the region near Lq in Figure 4.11. L is the Lorentz point (the Lane point corrected for the mean refractive index). A is one of the tie-points selected on the dispersion surface, and o and are the deviation parameters at that point (shown on branch 2)...

In the cases where no tie-points are selected, no wavefields at all (other than a very rapidly decaying evanescent wave) are generated inside the crystal. The X-rays are effectively excluded from the crystal and the reflectivity, with a zero-absorption crystal, is 100%. This is the range of total reflection. 

Figure 4.14 The selection of tie-points on the dispersion surface for the transmission (Lane) case, using the construction of Figure 4.13 (branch 2 is on the left, branch 1 is on the right)...

Figure 4.15 The energy flow through the crystal and the geometry of diffracted and forward-diffracted beams, for the case shown in Figure 4.14. P and P g are the Poynting vectors associated with the tie-points A and B...

Instead of considering the dispersion surface as a variable and the reciprocal lattice as invariant, it is usually easier to consider the reciprocal lattice as the variable. Then equation (8.30) determines the variation of the amplitude ratio of the reflected and transmitted components as the wavefield propagates through the crystal. The ratio R characterises a particular tie-point on the dispersion surface and if R varies the tie-point must migrate along the dispersion surface branch. This results in a change in the intensity of the transmitted and diffracted... 

When the distortion is high, we may obtain qualitative insights by retaining the concepts of the perfect crystal and treating the defective region as a crystal boundary. We then find that this leads to the concept of tie-points jumping from one branch to another when we match the wavefields across the interface. 

As the fibers absorb water, the chains in the amorphous areas are pushed apart the crystallites act as tie points to hold the chains together... 

The Skillful rating was given to 8% of the responses to this topic. In such responses, students used detail and elaboration in parts of the response, with transitions to connect ideas. In the response shown above, the student specifies who will be the narrators of the show and the order in which information will be presented The show is about four teenagers, around the ages of fourteen to seventeen who travel around the world. In each show they travel to two cities. When they arrive in the city they will first talk about the city s history and what it is like now in the present. The student also uses the example of Paris as the subject for one show. The student uses complex sentences and transitions (such as When they arrive in the city. .. , For example. . . . ) to tie points together and lead the reader through the essay. 

But 0 Sg are related by Eq. (4.23). Therefore, the selection of the point A must satisfy this equation. Because the spheres about O and G can be represented as planes in the neighborhood of Q, the locus of the points A are hyperbolic sheets with these planes as asymptotes. These hyperbolic sheets are called the dispersion surfaces. A more detailed view of the neighborhood around Q is shown in Figure 4.3. The two branches of the dispersion surface are called a and jS, the one closer to the L point being a. A point on the dispersion surface is called a tie point. The arbitrarily selected tie points (A and A2) and the directions of their associated wavevectors are shown. Note that for the a branch, 0 and Jg are both positive, but for the branch, they are both negative. 

We now show that a tie point on the dispersion surface also characterizes the ratio of the amplitudes of the waves inside the crystal associated with the tie point. From the first dispersion equation, Eq. (4.19a), we have... 

Now it can be seen from Figure 4.3 that for tie points such as A, o tends to zero. Therefore, from Eq. (4.28a), also tends to zero so there is... 

Selection of active tie points boundary conditions at entrance face of the crystal... 

In the discussion of the dispersion surface in Section 4.3, tie points on the dispersion surface were arbitrarily selected in order to show how the tie points characterize the allowed waves in the crystal. We must now examine how the tie points are selected in an actual experiment, but first we consider the boundary conditions that exist at the boundary between vacuum and the entrance face of the crystal. 

Figure 4.4. Construction in reciprocal space showing the selection of tie points A and B on the two branches of the dispersion surface.

We see therefore that the wavevectors of the two waves can differ only by a vector normal to the boundary. In other words, the tangential components of the wavevectors must be equal. There is no such restriction on the normal components in fact, they are not equal. In any particular experiment, the tie points are selected by the geometry of the experiment, that is, by the angle of incidence to the crystal face and the orientation of the diffracting planes to this face. The tie points for a given geometry can be found by a simple construction in reciprocal space. We now describe this construction and justify it by showing that it is consistent with the boundary conditions just discussed. 

To select the tie points, draw a vector through P parallel to n. It cuts the dispersion surface at two points, A on the a branch and B on the branch A and B are the required tie points. We see, therefore, that the outside incident wave gives rise to four waves inside the crystal Koc and Ko(3 in the forward direction (transmitted waves) and two strongly diffracted waves Kgo, and Kg. This is precisely what we would expect from the splitting of the energy at the Brillouin zone boundary. 

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