Property-yield curves

In order to draw the property-yield curves for gasolines , it suffices to choose the initial point, which coilild be or 20°C, the end point being variable and situated between the end point of the heaviest gasoline cut which can be produced (200-220°C) and about 350°C. 

Unlike the property-yield curves, calculations are not necessary for determining the properties of a cut. 

There is considerable scope in the selection of the end point for Arrhenius analysis which can yield different predictions. This is particularly the case if the shape of the property-time curves is complex. An end point representing a smaller change gave better predictions that one representing a larger change. 

To belabor this point, let us consider in more detail a simple case, Refs. [78, 79], where the bound states of the Coulomb potential, through successive switching of a short-range barrier potential, becomes associated with resonances in the continuum. The simplicity of the problem demonstrates that resonances have decisively bound state properties, yields insights into the curve-crossing problem, and displays the tolerance of Jordan blocks. The potential has the form... 

As can be seen by this equation when /t = 1 the post yield curve is a straight line with a slope that is a function of the elastic and plastic material properties. For n > 1 the slope is a function of a i.e., not a straight line. 

From empirical investigations we know that the correlation should converge to unity as the difference in its maturities approaches zero. One the other hand, the correlation should vanish as the difference in the maturities goes to infinity. Another empirical implication is the relative smoothness of the observed forward rate curved Hence, we are able to separate the class of RF models according to the existence or absence of this smoothness property. Obviously, the non-differentiable class leads to non-smoothed forward rate curves, whereas the T-differentiable Random Fields enforces smoothed yield curves. Even if we restrict the number of admissible RF models to the non-differentiable Field dZ t,T) and the r-differentiable counterpart dU we obtain a new degree of freedom to improve the possible fluctuations of the entire term structure. 

This reflects the properties of the spline curve, including the fact that forward rates are described by a series of segments that are in effect connected together. This has the effect of localising the influence of individual yield movements to only the relevant part of the yield curve it also allows the curve to match more closely the observed yield curve. The goodness of the spline-based method is measured using Equation (5.23) ... 

In deriving the swap curve, the inputs should cover the complete term structure (i.e., short-, middle-, and long-term parts). The inputs should be observable, liquid, and with similar credit properties. Using an interpolation methodology, the inputs should form a complete, consistent, and smooth yield curve that closely tracks observed market data. Once the complete swap term structure is derived, an instrument is marked to market by extracting the appropriate rates off the derived curve. 

A bond may be valued relative to comparable securities or against the par or zero-coupon yield curve. The first method is more appropriate in certain situations. It is suitable, for instance, when a low-coupon bond is trading rich to the curve but fair compared with other low-coupon bonds. This may indicate that the overpricing is a property not of the individual bond but of all low-coupon bonds. 

Stress versus Strain curves are usually encountered in the measurement of Tensile Strength and Compressive Strength in polymers with the properties Yield Stress, Ultimate Strength, Elongation, and Modulus also being determeined in these tests. 

For those simple kinetic patterns where the algebra does not become too cumbersome, we will consider the three basic quantities, cell survival, mutation yield, and mutation frequency [equations (4)-(6)], as functions both of mutagen dose and of lethal hits. And in particular, we will focus on the following properties of the yield curves (1) slope at the origin (2) position of the maximum yield x (3) magnitude of the maximum yield Fmax = T(ii ) and (4) the integral under the yield curves. 

Curves of the proi>erties of the fractions vs. per cent distilled (mid per cent curves) or the average properties of a series of fractions vs. percentage yield (yield curves), by which realizations of yields can be prepared. Among common property curves are... 

The determination of properties for each cut enables curves to be obtained for yields and properties as well as curves for iso-properties that are useful in the economic analyses of crude oils. 

These curves are drawn for all properties having for the ordinate axis an appropriate scale of the property, and for the abscissa the yield in volume or weight. 

The comparison of curve 1 and 2 in Fig. 3 yields, that the convergence with respect to the number of projections k is not strongly influenced by noise because of the properties of the reconstruction algorithm. Nevertheless, the noise increases the asymptotic value of o(n)/0 ... 

The t and a.-methods, the nature of which was explained in Chapter 2, may be used to arrive at a value of the micropore volume. If the surface of the solid has standard properties, the t-plot (or a,-plot) corresponding to the isotherm of the nonporous powder in Fig. 4.11(a) will be a straight line passing through the origin (cf. curve (i) of Fig. 4.11(6)) and having a slope proportional to the specific surface of the powder. For the microporous powder which yields the isotherm (iii).of Fig. 4.11(a), the t-plot (or Oj-plot) will have the form of curve (iii) of Fig. 4.11(6) the linear branch of this curve will be parallel to curve (i), since it corresponds to the area of the outside of the particles which is identical with that of the nonporous parent particles. 

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

Fig. 41. Typical stress—strain curve. Points is the yield point of the material the sample breaks at point B. Mechanical properties are identified as follows a = Aa/Ae, modulus b = tensile strength c = yield strength d = elongation at break. The toughness or work to break is the area under the curve.

Regardless of the procedure used, certain initial steps must be taken for the determination or specification of certain product properties and yields based on the TBP distillation curve of the column feed, method of providing column reflux, column-operating pressure, type of condenser, and type of side-cut strippers ana stripping requirements. These steps are developed and ilhistrated with several detailed examples by Watkins (op. cit.). Only one example, modified from one given by Watkins, is considered briefly here to indicate the approach taken during the initial steps. 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

The mechanical properties can be studied by stretching a polymer specimen at constant rate and monitoring the stress produced. The Young (elastic) modulus is determined from the initial linear portion of the stress-strain curve, and other mechanical parameters of interest include the yield and break stresses and the corresponding strain (draw ratio) values. Some of these parameters will be reported in the following paragraphs, referred to as results on thermotropic polybibenzoates with different spacers. The stress-strain plots were obtained at various drawing temperatures and rates. 

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