Interpolative reasoning

D — and D oo the smooth and generic dimension-dependence of the integrals enables one to interpolate reasonably accurate D = 3 values (rms error 1%) from the dimensional limit results. The interpolated integrals can be used either on their ovm, or in conjunction with an integral equation approximation which sums some subset of the required integrals exactly (such as the hypemetted-chain or Percus-Yevick methods) the combination methods are invariably better than either dimensional interpolation or integral equations alone. Interpolation-corrected Percus-Yevick values can be computed quite easily at arbitrary order however, errors in higher-order values are... 

Is finding the y-intercept here interpolation, reasonable extrapolation, or reckless extrapolation ... 

The Newton iteration is initiated by providing reasonable guesses for all unknowns. However, these can be generated from guesses of just T, Tv, and one interstage value of Fj or Lj. Remaining values of Tj are obtained by linear interpolation. By assuming constant molal over-... 

Table 21-5 indicates screw-conveyor performance on the basis of material classifications as listed in Table 21-4 and defined in Table 21-3. Table 21-6 gives a wide range of capacities and power requirements for various sizes of screws handling 801 kg/m (50 lb/ fU) of material of average conveyabihty. Within reasonable limits, values from Tables 21-5 and 21-6 can be interpolated for preliminary estimates and designs. 

Useful yield provides an overall measure of the extent to which the sputtered material is used for analysis. It is a quantity employed to estimate the sensitivity of the mass spectrometric method. Values of Y (X (A)) for elements typically range from 10 to 10 in TOF SIMS. The number of sputtered particles A per incident primary ion (sputtering yield) can be measured from elemental and multielemental standards under different operational conditions and can, therefore, by judicious interpolation between standards, be estimated with reasonable accuracy for the material being analyzed. 

It has been shown throughout this chapter that the properties of plastics are dependent on time. In Chapter 1 the dependence of properties on temperature was also highlighted. The latter is more important for plastics than it would be for metals because even modest temperature changes below 100°C can have a significant effect on properties. Clearly it is not reasonable to expect creep curves and other physical property data to be available at all temperatures. If information is available over an appropriate range of temperatures then it may be possible to attempt some type of interpolation. For example, if creep curves are available at 20°C and 60°C whereas the service temperature is 40°C then a linear interpolation would provide acceptable design data. 

Strigle [82] and Kister [93] point out the importance of evaluating data where available to reduce the need for interpolating the GPDC charts. The question of reasonably accurate ( 10 to 15%) flooding pressure drop data has been studied by Kister [93], and the results suggest that the establishment of flooding pressure drop curves... 

Simulation by means of the digital computer has become an extremely useful technique (see Section 3.7) that goes far beyond classical interpolation/ extrapolation. The reasons for this are fourfold ... 

Example 55 An interpolation for 35°C on the assumption that the same sampling plan would apply as for the 80°C results yields an implausibly short shelf-life of 50 days (90% level) the reason is that both Xmean and Sxx are... 

Such relationships have been applied to solubility, vapor pressure, Kow, KAW, KOA, Henry s law constant, reactivities, bioconcentration data and several other environmentally relevant partition coefficients. Of particular value are relationships involving various manifestations of toxicity, but these are beyond the scope of this handbook. These relationships are valuable because they permit values to be checked for reasonableness and (with some caution) interpolation is possible to estimate undetermined values. They may be used (with extreme caution ) for extrapolation. 

---