Slope infinite

Similaily, the entice equilibiium curve is calculated and is plotted in Figure 10. The feed is at the boiFng point so the q line is drawn vertically with an infinite slope. 

Because the determinant is equal to zero, the (X X) matrix cannot be inverted, and a unique solution does not exist. An interpretation of the zero determinant is that the slope P, and the response intercept Po are both undefined (see Equations 5.14 and 5.15). This interpretation is consistent with the experimental design used and the model attempted the best straight line through the two points would have infinite slope (a vertical line) and the response intercept would not exist (see Figure 5.8). 

Figure 7 shows the temperature-pressure phase diagrams for STO 18-92 and SCT(0.007) [21]. The data show clear evidence that Tc for STO 18 vanishes with an infinite slope, i.e., dTc/dp - cx) as Tc 0 K. This is a requirement of the third law of thermodynamics for both first- and second-order phase... 

For the family of identity exchange reactions such as (109), a plot of log against log gives a straight line with infinite slope, i.e. the Bronsted... 

The book Gas Turbine Performance by Walsh and Fletcher [10] has excellent treatment on turbomachineiy maps. An example of a compressor map is shown in Figure 8.10. This map fully defines the pressure-flow-efficiency-rotational speed relationship of the compressor. Employing the Beta-line or R-line method, maps can be digitized into tabular form as described by Walsh and Fletcher [10], The Betaline method is helpful to ensure numeric stability with such maps that would be otherwise problematic because of the near zero and infinite slope at the ends of the constant speed lines. Note that the compressor model can either be a flow element or a pressure element. That is, from speed and pressure it is possible to obtain flow and efficiency from the map, or from speed and flow it is possible to obtain pressure and efficiency from the map. The choice depends on what is more convenient, that is, what type of elements are modeled upstream and downstream. 

The elastic scattering cross section must fall as the positron energy is increased above the threshold, and it will either rise or fall as the threshold is approached from below, depending on the value of l and on the phase shift at the threshold. Furthermore, because the s-wave contribution to crPs, considered as a function of the positron energy, has an infinite slope at the threshold energy EPs, equation (3.99), so too does energy dependence has the shape of either a cusp or a downward rounded step. All other partial-wave contributions to aei, however, continue through the threshold with no discontinuity of slope. 

It should be emphasized that for given parameters (Mo, i, and so on) the (6, t) behavior of the solution ensures that at most one integral curve from the cold-boundary singularity reaches the hot-boundary point in (cp, T, e) space all other integral curves from the cold boundary that pass through [( ==

boundary conditions. Thus, each solution curve shown in Figure 6.3 corresponds to a different value of some wave parameter (for example, the reaction-rate frequency factor... 

This result is not surprising, since this model predicts an unrealistic infinite slope at time zero. Nevertheless, to our knowledge, it is the first time that this theoretical weakness is demonstrated by experimental results. 

As in the fuse problem, we begin with the explanation of how the presence of defects can increase the local breakdown field. A defect is a local change in the properties of the sample. In an insulator, the defects are conducting parts of the sample. We consider again a spherical defect (circular in two dimensions) and we draw the equipotential surfaces or lines (in two dimensions). In a pure sample, these surfaces or lines are parallel to the electrodes (Fig. 2.12a) but in a sample with one defect they show distortions near it. For a two-dimensional sample, the new equipotential lines are shown in Fig. 2.12(b). One sees that in the vicinity of the defect there is an increase of the field. The sample will break at an applied voltage smaller than the one which is needed to break a pure sample. This is the enhancement effect identical to that of the fuse problem and consequently the curve Vb(p) will exhibit an infinite slope when p goes to zero. 

With a downward-opening concave shape as illustrated here, it starts by canceling only a part of the effect of ideal mixing then it more than cancels this effect and takes over in making the Gi iIRT curve switch from convex-upward to concave-downward. It is this switch in shape that indicates liquid/liquid behavior. The mixing term approaches its endpoints with an infinite slope so the GnJRT curve always starts out in the downward direction, no matter what model we use to estimate the nonideal behavior. 

Balakotaiah, Kodra, and Nguyen also studied the CSTR. They found that the boundary between the insensitive and runaway region is where there are two limit points (a point of infinite slope - there are two such limit points in Fig. 2) in the reaction path connecting the initial and final states. They foimd identical criteria to the batch reactor in the special limit of 7 —> oo and Og = 0. That is, the safe criteria under adiabatic conditions (a = 0) is given... 

Since the entropy ASm decreases with decreasing temperature and equals zero at (C, 7 ),the To-curve has exactly there an infinite slope. For a binary Zr-Ni system, C equals to 11.5 at. % Ni and T = 638 K [2.20]. 

The supercooled liquid catastrophe, if it exists, would necessarily be associated with diverging fluctuations in the structural order parameter F. This stems from the fact that the Y surface develops a vanishing curvature in the F direction as this endpoint is approached. Because the bicyclic octamer elements are bulky, fluctuations in their coiKentration amount to density fluctuations. Diverging density fluctuations then imply diverging isothermal compressibility. Furthermore the infinite slope of the metastable liquid locus at its endpoint implies the divergence of thermal expansion. Potential energy fluctuations remain essentially normal, so constant-volume heat capacity remains small. But the volumetric divergence creates an unbounded constant-pressure heat capacity. 

Morgenstem and Price, 1965 Janbu, 1973). The primary methods are (1) infinite slope analysis (Figure 11.8), (2) Swedish circle method, and (3) method of slices (Figure 11.9). The final evaluation of the stability or instability of the slope is made normally by the use of appropriafe safefy factors. 

Submarine slopes are commonly fairly gentle, uniform, and homogeneous over considerable horizontal distances (see Figure 11.8). Thus in many cases infinite slope limiting equilibrium methods are applicable. Assume an element of an infinite slope as shown in Figure 11.10. 

The equDlbrivun of an infinite slope under undrained conditions after Morgenstem, N. R., (a), drained conditions (b), and partially drained conditions (c). (Submarine slumping and the initiation of turbidity cturents. In Richards A.R ed. Marine Geotechnique. University of Illinois Press, Champaign, IL, 189-220,1967.) Reprinted with permission of Morgenstem, N.R... 

Factors of safety from infinite slope analysis for partially drained conditions, R = 0.25, ( ) = 25°. 

Again, the two-phase dome is shown as a broken line. The larger compressibility factor is the vapor and the smaller for the liquid. Again, they meet at the critical point, which is a compressibility of 0.2746. The maximum temperature on this plot is 183 °C, which corresponds to a reduced temperature of 1.5, and the maximum pressure is 369 bar, = 5. The plot also shows the critical isobar (P = 1) note the infinite slope at the critical point and the steep slope in the vicinity of the critical. 

Figure 4.23 shows the feed lines and operating lines for a saturated feed aud for a saturated vapor feed. The feed line for liquid comes with an infinite slope as a vertical line starting from the feed location on the diagonal line on a McCabe—Thiele graph. On the other hand, the slope of the feed line for a saturated... 

Although initially the mole fraction of solute in the vapor phase falls rapidly as the pressure is increased, this process reverses as the critical region is approached. At the critical end point temperature. Fig. 8c, the solubility increases strongly, with an infinite slope dx/dP for the isotherm. It likewise decreases with infinite slope dx/dT on an isobar passing through... 

Lowing the temperature above some critical value Fc may lead to a variation as depicted in Figure 8.1b. Since the laws of dilute solutions require G ix ) to have an infinite slope at both ends of the X2 axis, and negative at V2 = 0 and positive at V2 = 1, there must be two positively curved portions of the Gj x2) curve surrounding a negatively curved portion, separated by two points of inflection at the compositions X2si and X2s2- Between these two compositions a... 

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