Tangent point

O.D. of Tube, In. Flare Diameter, In. Radius of Flare, In. Tangent Point to Tubesheet, In. 

Intermlttency Manneville [mann80] showed that for the special case of a generic intermittency threshold in which the tangent point lies at the endpoint of the interval (in the case of a one dimensional iterative map of an interval to itself), the resulting chaotic dynamics has a power spectrum S f) 1/ (/(log/) ) for low /. Miracky, et. al. were able to modify the map to obtain an exact 1// behavior [mirack87]. Because the result depends on the fine-tuning of an external parameter, however, it does not so mucdi explain the generic appearance of flicker noise phenomena as beg the obvious question, why do systems typically sit at whatever... 

A tangent to the curve is then drawn from the point on the time axis at —(15 x 60) = —900 s. The reaction time at the tangent point is 1050 s, at which the fractional conversion is 0.68. 

Horizontal and vertical lines are drawn through the initial condition point, as well. These lines, of course, represent the conditions of constant pressure and constant specific volume (1/p), respectively. They further break the curve into three sections. Sections I and II are further divided into sections by the tangent points (J and K) and the other letters defining particular points. 

Figure 11.14 Flight radii and tangent points in a screw channel. The pushing flight R, in this case is estimated at 0.5 the depth of the channel (H) while the radius R2 at the trailing side of the channel is estimated at 1.5 times the channel depth...

Only the tangent point B satisfies both conditions at once it may be attained in the process of chemical reaction of a gas which is compressed by a shock wave (the jump HY and the drop YB), and at the same time the state B is compatible with the conditions of expansion of the detonation products upon completion of the reaction (the detonation velocity in state B is exactly equal to the velocity of propagation of a perturbation in the reaction products). 

Disruption of the combustion regime when the blast rate is increased and the residence time of the reacting substance in the vessel is decreased is quite remarkable. It is obvious that this disruption occurs when the intersection point C moves to the tangent point D (see Fig. 4) in the direct vicinity of point M. 

We reach, finally, the T-line which is tangent to the I/-curve at the point C at the value ip = ipc. Beginning from this value of

critical value (pc which corresponds to the tangent point determines the condition of ignition of the c-phase, and allows us to calculate the amount of heat and the rate of heating required. A calculation of [Pg.352]

We tacitly assumed that the point S, describing the steady regime, is higher than the tangent point C, as is shown in the drawing. 

Thus, for all values of T0 one and the same diagram with a single [/-curve and family of T-lines is appropriate. Only the location of the point S, which always lies on the U-curve, depends on T0. The relation between cs and T0 is elementary. Steady combustion is possible only at T0 such that cs > To i.e., is larger than the temperature corresponding to the tangent point. A powder or secondary EM cooled to a lower temperature is not capable of burning steadily and, when ignited, produces only flare-ups, as described above. Let us calculate the coordinates of the point C at which the T- and [/-lines come into contact. 

As is obvious from the previous section, the Chapman-Jouguet conditions, in accord with experiment, select the tangent point B of the line ABC drawn from the point representing the initial state to the dynamic adiabate. 

Wendlandt [10] emphasizes the analogy between the overcompressed detonation wave on the branch BFD and a simple compression shock wave without chemical reaction which is also overtaken and weakened from behind by rarefaction waves. In contrast, a detonation wave at the tangent point, for which the Chapman-Jouguet condition is satisfied, is similar to a sound wave and is transformed into a sound wave when the thermal effect of the reaction goes to zero. 

It is interesting that the temperature maximum on the Todes line is shifted somewhat to the left (see Fig. 1 or 5) with respect to the tangent point B so that between the maximum and the point B there is a paradoxical region in which exothermic reaction and heat release are accompanied by a decrease in the temperature as a result of the simultaneous expansion of the material. Heat release in this region is accompanied by a growth in entropy. Tentative calculations show that the maximum temperature exceeds the temperature at the point B at the end of the reaction by about 50-100°. 

Elements A and B are completely miscible if Q is negative. However, the solubility is limited if G is positive. Figure 7.1 is a plot of Equation (7.14) for a positive value of Q. The solubilities correspond to the tangent points for a line tangent to both minima. 

When the operating line and the equilibrium curve intersect, an infinite number of stages is required to achieve the separation (Fig. 11). The intersection point is called the pinch point and may occur at the bottom (Fig. 11a), at the top (Fig. lib), or at a tangent point (Fig. lie). The solvent rate leading to this intersection is the minimum solvent flow required to absorb the specified amount of solute. 

First, the experimental points are smoothed by a continuous line, with regular curvature. Parallel chords AjBj, A2B2,... of middle points M, M2,.. . are drawn. The intersection of the locus of points Mx, M2,.. . with the smoothed curve provides a tangent point M. The tangent MT is parallel to chords. 

In the metastable state, i.e. between the binodal and the spinodal, phase separation only occurs in the presence of nuclei. Spontaneous phase separation only occurs at temperatures below the spinodal, where the solution becomes unstable. The binodal and spinodal are obtained by constructing curves of AGm vs.

various values of y (i.e. for various values of temperature) and subsequently plotting the temperature vs. the tangent points and the inflection points to obtain the binodal and spinodal, respectively (see Fig. 7.7). 

When the proper form of the shift function has been determined, a computer program is then used to shift the raw data with respect to the chosen reference temperature. A polynomial least squares is then used to find the best fit through the shifted modulus and loss tangent points. The resulting curves and shifted points are then stored on floppy disks for plotting or recall by various modeling programs. 

Figure 12.1. Interpretation of multi point distances for a curve and a surface [19]. (a) Three-point distance for curve. Given any three distinct points, r is the radius of the unique circle that contains the points. When the points are from the same neighborhood on a curve, such as points 1, 2, and 3, r is close to the local radius of curvature. When points, such as 1, 2, and 4, are taken from two different neighborhoods of the curve that are close to intersection, r approximates (half of) the distance of closest approach of the curve to itself, (b) Tangent-point distance for a surface. Given two distinct points 1 and 2 on a surface, p is the radius of the unique sphere that contains both points and is tangent to the surface at point 1. When the points are neighbors on the surface, p approximates the absolute value of the local normal radius of curvature in the direction defined by the two points (not illustrated). When points are taken from different neighborhoods that are close to intersection, p approximates (half of) the distance of closest approach.

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