Isochronous interpolation

Kinetic Light-Scattering Method. Isochronous Interpolation. When high-activity samples of endocellulase are used, the reaction proceeds so quickly that, since the measurements of scattered intensity at different angles are not performed at the same extent of reaction, the extrapolations to zero angle and the subsequent calculations are erroneous. For this reason Kratochvil et al. (27) have proposed an isochronous interpolation method, whereby the Kc/Re values are plotted against sin21 (0/2) + kft. As in the double-extrapolation method of Zimm, the value of k may be chosen arbitrarily in order to space the experimental data. 

Figure 3. Computer plot obtained by isochronous interpolation of the experimental light-scattering data (O) of HEC during endocellulase attack. The Langrangian interpolation functions are given by the horizontal curves ana the isochronous interpolated Kc/Re values by A. The quadratic least squares extrapolations to zero angle ( ) are given by the...

Figure 4. Changes in Mw as a function of the enzymic hydrolysis times, calculated from the light-scattering data by isochronous interpolation and by subsequent extrapolation to zero concentration using Equation 17.

Figure 11.23 shows the isochron obtained by Marshall and De Paolo (1982) for the granite batholith of Pikes Peak (Colorado). The effectiveness of the double-spike technique is evident, especially when we see that aliquot-spiked samples do not fall on the best-fit interpolant (York s algorithm York, 1969). The obtained age (1041 32 Ma) is consistent with that previously obtained with Rb-Sr whole rock analyses (1008 13 Ma see Marshall and De Paolo, 1982, for references). The initial ratio ( Ca/ Ca)o of 151.0 is identical, within the range of uncertainty, to upper mantle values, indicating negligible contamination by old crust components the relative K/Ca abundance in the earth s mantle is about 0.01, a value too low to alter the primordial (" Ca/" Ca)o composition. 

The interpolated values of Kc/Re at constant time are calculated or read from the graph, and then the extrapolation of the isochronous Kc/Re values to zero angle is made in the same manner as mentioned for the Zimm plot, by using quadratic least squares regression analysis. 

The optimum time step in a FEM-CVA simulation is the one that fills exactly one new control volume. Once the fill factors are updated, the simulation proceeds to solve for a new pressure and flow field, which is repeated until all fill factors are 1. While the FEM-CVA scheme does not know exactly, where the flow front lies, one can recover flow front information in post-processing quite accurately. One very common technique is for the simulation program to record the time when a node is half full, / = 0.5. This operation is performed when the nodal fill factors are updated if the node has fk <0.5 and fk+1 >0.5 then the time at which the fill factor was 0.5 is found by interpolating between tk and tk+l. These half-times are then treated as nodal data and the flow front or filling pattern at any time is drawn as a contour of the corresponding half-times, or isochronous curves. 

For simple material comparisons, determine the stress to produce 1% strain in 1000 h. Select several loads to produce strains in the approximate range of 1% strain and plot the 1000-h isochronous stress-strain curve from which the stress to produce 1% strain may be determined by interpolation. 

One convenient method is to combine the information available from either isochronous stress-strain curves or stress-time curves obtained on the same materials at different temperatures. For example, suppose the performance criterion for a particular application is that the total strain should not exceed 2% in 1000 hours. Using the 1000 hour isochronous stress-strain curve for each temperature, and erecting an ordinate at the 2% point on the strain axis, the individual working stresses for each temperature can be obtained. Alternatively, by erecting an ordinate at the 1000 hour point on the stress-time curve for 2% strain for each temperature investigated, the individual working stresses can be similarly obtained. From these interpolated results the stress-temperature curve can be drawn. 

Figure 10.1 Tensile creep of polypropylene at 60 °C. The stress and time dependence are approximately separable and therefore creep curves at intermediate stresses can be interpolated from a knowledge of two creep curves ( ) and the isochronous stress-strain relationship (X). (Reproduced with permission from Turner, Polym. Eng. Sci., 6, 306 (1966))...

The data in Figure 2.9 demonstrate the marked influence of density on the creep behavior of polyethylene. The curves in Figure 2.9 are relevant to a total strain of 1%, but similar plots for other permissible strains can be readily derived from the isochronous stress-strain curves. The linear relationship between creep and density for polyethylene at room temperature, irrespective of the melt index over the range investigated (i.e., 0.2-5.5), has enabled the stress-time curve of Figure 2.10 to be interpolated for the complete range of polyethylene. In this case, the data have been based on a permissible strain of 2%, but as previously explained, data for other permissible strains can be similarly interpolated from the creep curves. 

Interpolation is used to connect all points of similar potential at any given instant. Usually the maps are calculated and represented at 10 msec steps with voltage resolution of 10 Vm. Isochrones may also be constructed if so desired. 

Because many geotextile reinforcement design codes specify a limit to allowable creep strain, it is convenient to present the loads at which these strain limits are reached. In this context, the creep data may be presented in an isochronous plot consisting of an array of load—strain curves, similar to the one from a tensile test, but with each curve representing a different duration. Each isochronous (ie, constant time) curve is created by taking load and strain levels from each creep curve at a given constant time and plotting them to form an isochronous curve. Isochronous curves are not an extrapolation tool, but instead are an interpolation tool. 

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