Inflexion region

A problem for the modeller is that all the normal flow equations, liquid and gas, contain a point of inflexion at flow reversal, when the derivative takes an infinite value. Thus all the normal flow equations possess an inflexion region from which, once there, the algorithm will emerge only slowly. 

Figure 18.4 demonstrates that the curve contains an inflexion region centered on Pk/Pj = 1, the point of flow reversal. 

The problem caused by inflexion regions may be understood by evaluating equation (18.28) for a sample set of numerical values. The situation is simplified by taking the common case where the liquid temperature and hence specific volume are the same throughout the network ... 

It is clear from the figure that the inflexion regions contained in all of the individual flow functions have been carried across to the flow balance function more or less without modification. One inflexion region occurs for each flow in the flow balance equation, in this case at the pressure ratios ... 

Fig. 8. Reversible-thermodynamic regions of the calibration curves from Fig. 7 drawn in detail up to the point of inflexion (ordinate is not normalized here, the intersections with the ordinate are equal to —In r0(T), obtained by a spline extrapolation)...

If the pH of the titrand is monitored throughout a titration, a graph.of pH against amount of titrant added may be constructed. The characteristics of this curve are important in the selection of suitable titration conditions and indicators. Of particular importance are the position of the point of inflexion representing the neutralization point, the slope of the curve in the end point region, and the size of the end point break . The influences of concentration and the strength of the acid or base are summarized in figures 5.1, 5.2 and 5.3. 

An alternative, simpler, procedure for improving the inflexion in the neutralization of an amino-acid is to add formaldehyde to the solution although this does not affect the acid-titration curve, the one for alkaline titration is changed, as seen in Fig. 107. The effect of the formaldehyde is to increase the strength of the ammonium ion acid which is being titrated, and so the pH inflexion at the equivalence-point becomes much more obvious. This is the basis of the formol titration of amino-acids discovered by Sorensen (1907) approximately 10 per cent of formaldehyde is added to the solution which is then titrated with standard alkali using phenolphthalein as indicator. In the presence of thii concentration of formaldehyde the pH-neutralization curve has a sharp inflexion in the region of pH 9, and so a satisfactory end-point is possible with the aforementioned indicator. 

The shape of adsorption-desorption isotherms gives the first pieces of information on a porous solid texture. Existence of an inflexion point in the low-pressure region of the high-resolution isotherm is a sign of the microporous character of a solid [6]. NaY zeolite appears microporous (type I isotherm) regardless of probe molecule and temperature. Similarly, USY zeolite is mainly microporous,but a small hysteresis loop at high P/Po values reveals the presence of some mesopores created during dealumination of Y zeolite [8,9]. 

Fig. 3.2. Profiles of the charge density along the C2 symmetry axis for the dissociation of the water molecule. In (a) p decreases monotonically from a maximum value at the oxygen nucleus. As Ro is increased, an abrupt change occurs in the form of the curve when a point of inflexion appears (b). This unstable critical point exists only for this single configuration X, of the nuclei and a further increase in Rq results in Its bifurcation to yield a maximum, corresponding to the formation of a bond critical point and a minimum, corresponding to the formation of a ring critical point a.s typified by (c). The system has entered the region of the ring structure.

Fig. 3. A sigmoidal dependence of rate upon substrate concentration. As in the case of a hyperbolic dependence, the curve is asymptotic to an upper limit. There is a point of inflexion in the curve, however, and in the middle region of the curve a tangent would cross the abscissa axis well to the right of the origin. This means that over that region a given fractional increase in [A] results in a greater fractional increase in V. Such amplification over a critical concentration range may often be of physiological value.

Fig. 4-209 gives the curves of TGA and DTA versus temperature for the oxidation experiments of bitumen B80. The TGA curve has at least four points of inflexion representing the different regions of reaction. The DTA curve displays four pronounced peak maxima. Even the DTG curve in Fig. 4-210 displays more than one maximum of the reaction rate against temperature. The maximum temperatures should be adjusted in order to fit the regression line in the plot of In (heating rate) versus 1 000/Kel vin. As mentioned in chapter 3.4.1.2, the kinetic parameters have to be calculated externally using those peak maximum temperatures. The kinetic parameters are presented in Table 4-209, No. 2 and No. 3. 

According to this, the formation of two liquid phases can only occur if the curve shape of the Gibbs energy as a function of the composition shows an inflexion point, that is, there must be a region where the following condition is valid ... 

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