Point of tangency

Use the ethanol curve similar to Figure 8-37, or refer to the data of Reference 133 the point of tangency of the line from the distillate composition of the diagonal is xj) = 0.80 andyv = 0.80. Thus the minimum internal reflux is set by this tangent line ... 

Fig. 123.—(a) Phase diagram calculated for three-component systems consisting of nonsolvent [1], solvent [2], and polymer [3] taking Xi==X2=l and Xz equal to 10 (dashed curve), 100 (solid curve), and °° (dotted curve) xi2 = xi3 = 1.5 and X23 =0. All critical points (O) are shown and tie lines are included for the xs = 100 curve. (Curves calculated by Tompa. ) (b) The binodial curve for a 3 = 100 and three solvent ratio lines. The precipitation threshold is indicated by the point of tangency X for the threshold solvent mixture. 

This point of tangency can be determined, assuming that the equation of state P = P( V, E) of the products is known. The chemical composition of the products changes with the thermodynamic state, so thermochemical codes must solve for state variables and chemical concentrations simultaneously. This problem is relatively straightforward, given that the equation of state (EOS) of the fluid and solid products are known. 

The lines of constant closedloop log modulus L, are part of the Nichols chart. If we are designing a closedloop system for an L specification, we merely have to adjust the controller type and settings so that the openloop B curve is tangent to the desired line on the Nichols chart. For example, the G B curve in Fig. 13,11b with X, = 20 is just tangent to the +2 dB line of the Nichols chart. The value of frequency at the point of tangency, 1.1 radians per minute, is the closedloop resonant frequency aif. The peak in the log modulus plot is clearly seen in the closedloop curves given in Fig. 13.12. 

Find the new resonant frequency (the frequency at the point of tangency). If it has changed appreciably, repeat steps 3 to 6. 

This vel is fixed in the case of deton by the slope of the straight line thru point 0 to point of tangency C on the Hugoniot of Fig 2. In the case of a deflagration, the vel is fixed by the slope of the straight line thru point 0 to the point of tangency D on the Hugoniot. 

P = Pex. If, for simplicity, we assume that the impacted sample is at Tm at t = 0, then (in accordance with Eq la) its T—t history will be given by curve I in the diagram. For each T there is a corresponding adiabatic explosion time Tad- Three typical T—Tad pl°ts (curves II, III IV) are shown in the diagram. Curve II does not intersect curve I, therefore there can be no explosion at t = 0. Curve III intersects curve I twice, therefore explosion should have occurred before it actually did, which is contrary to the assumption of explosion at f = 0. Curve IV, which has a point of tangency with curve I, is thus the only possible curve that satisfies the requirements of the problem under consideration. This condition of tan-... 

If the reaction is not sufficiently exothermic, so this inequality is not satisfied, then the reaction curve R is much flatter, as shown in Fig. 7.3(c). There are now no points of tangency as the residence time, and hence the gradient of L, varies. There is only ever one intersection and hence only ever one stationary state for any given rres (Fig. 7.3(d)). 

The second locus, DH2, emerges from the point of tangency between the double-zero eigenvalue curve A and the hysteresis line at p0 = k2 = 34/45. 

The quantity D reaches a minimum in the state B, which corresponds in the drawing to the point of tangency of the line AB, drawn from A, (p0, v0). Chapmann asserts that it is just this minimum value of the detonation velocity that is realized in experiment. Comparison with measurements fully confirms this hypothesis. In this same way definite values are established for the volume and pressure of the products. The substance is compressed by almost two times, and a temperature is achieved which slightly exceeds the explosion temperature. The reaction products acquire a velocity close to D/2. 

With stereographic projection, the elements on the lower half of the sphere are projected onto a flat surface from a point opposite the point of tangency. 

The solubility of carbon in austenite. If there is true equilibrium with graphite, the solubility is lower because the point of tangency on the austenite free energy curve is lower than for metastable equilibrium with cementite. 

Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

The characteristics of the primary surfaces and the derived surface for the solid-liquid equilibria and the solid-vapor equilibria are identical to those for the liquid-vapor equilibria with the exception that no critical phenomena have ever been observed in these two equilibria. Thus, a plane, tangent to the solid and liquid surfaces, may be rolled along these surfaces. The projections of the loci of the points of tangency are represented by the lines al and bs in Figure 5.5. If the points of tangency are connected by a straight line lying in the plane at any position of the plane, a ruled surface is derived... 

As illustrated in Figure 1, the surrogate worth trade-off method provides a means of locating the preferred (or optimal) solution by determining the point of tangency between the trade-off curve (function) and the so-called indifference curve (function). 

At minimum reflux, the pinch occurs at the intersection of the component balance line and the g-line when the equilibrium curve has no inflection points (Fig. 2,11c), This would be expected because the component balance lines intersect on the q-line. When the equilibrium curve has a point of inflection (Fig. 2.12), the pinch between the equilibrium curve and the component balance line may occur at the point of tangency instead of the intersection of the g-line and the component balance line. This condition is termed tangent pinch. 

The straight line CD in Fig. 11-5 starts at a distance equivalent to 6C on the left of the plot origin. The slope of this straight line is Q/(0b + 8C), with the values of Q and 6b determined by the point of intersection between line CD and curve OB. The maximum value of Q/(8b + 0C) occurs when line CD is tangent to the curve OB, and the point of tangency indicates the optimum value of the boiling time per cycle for conditions of maximum amount of heat transfer. 

Prove that the common tangent construction is equivalent to the equality of chemical potentials of the phases whose compositions are given by the points of tangency. 

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