Linear inverse interpolation

As it will be discussed, while three maxima of the first derivative are observed, the second one is a consequence of the applied numerical method. Using the second derivative values in the last column, local inverse linear interpolation gives V = 3.74 ml and V = 7.13 ml for the two equivalence points. We will see later on how the false end point can be eliminated. 

Using inverse linear interpolation the two titration equivalence points are obtained as the zero-crossing points of the second derivative at V = 3.78 ml and V = 7.14 ml. On Fig. 4.4 the second derivative curve of the interpolating spline (SD = ) and that of the smoothing spline (SD = 8.25) are shown. The false zero-crossing of the second derivative present at interpolation is eliminated by smoothing. 

A big advantage of this type of interpolation is that the matrices in the linear system of equations of coefficients,always have an inverse. A drawback is that in order to obtain the coefficients of the interpolation, we must solve very large systems of equations with full matrices. The most common radial functions are... 

This allows us to represent partial differential equations as found in the balance equations using the collocation method. Equation (11.47) is a solution to a partial differential equation represented by a system of linear algebraic equations, formed by the interpolation coefficients, oij, and the operated radial functions. The interpolation coefficients are solved for using matrix inversion techniques to approximately satisfy the partial differential equation... 

As mentioned above, equilibrium data can be presented and used in a variety of forms isotherms (loading vs. concentration at constant temperature), isosteres [partial pressure (or dewpoint, or some other form of concentration) vs. inverse absolute temperature at specific degrees of loading], and isobars [loading as a function of temperature for given partial pressures (or some other concentration)], listed in order of decreasing prevalence. The object of isosteres and isobars is to plot data on coordinates for which approximate linearity is expected, to make interpolation and extrapolation easier. 

Figure 10.2 shows a near linear trend between the effective pore diameter and gap for all LAD meshes and screen styles. Meanwhile, there is also an interesting correlation between the fineness of the screen (square root of the number of holes per screen area) and the pore diameter, as shown in Figure 10.3. In general, the inverse relationship from Figure 10.3 can be used to interpolate effective pore diameters for new, coarser screens where data is not yet available ...

The molar volume, v, is proportional to T. The first linear interpolation of Example 1.3 is consistent with this result, so again we have Vt=333 = 0.18086 [mVkg]. However, the ideal gas law shows molar volume is inversely proportional to pressure, so it is better to interpolate in (1/P). Thus, the second interpolation becomes ... 

---