Double interpolation

In max mixed flow, employ double interpolation for x as tabulated. 

Isentropic expansion of steam from the initial conditions to this pressure results in a slightly superheated vapor, for which by double interpolation in Table F.2 ... 

The saturation pressure at Point 4 from Table F.l is 5.318 kPa. We must find in Table F.2 the enthalpy (Point 3 ) at this pressure and at the entropy S2. This requires double interpolation. The pressure lies between entries for pressures of 1 and 10 kPa, and linear interpolation with P is unsatisfactory. Steam is here very nearly an ideal gas, for which the entropy is linear in the logarithm of P, and interpolation must be in accord with this relation. The enthalpy, on the other hand, changes very little with P and can be interpolated linearly. Linear interpolation with temperture is satisfactory in either case. 

La solution num rique de ces equations est extremement fastidieuse, mais on peut les simplifier par des formules d approximation satisfaisantes. Au lieu de celles-ci nous avons prefere reunir ici dans le Tab. A 3 quelques valeurs de fTfb correspondant aux valeurs usuelles de 8t et de j8. Dans ce domaine on peut en deduire aisement le parametre fTjb par une double interpolation. 

However, reading a chart can be difficult and time-consuming, and using tables often requires double interpolation. 

Use the tables for the American engineering units in the pocket in the back of the book. You niust employ double interpolation to get the specific enthalpies, AH, relative to the reference for the table ... 

LK compressiblity factor Since vapor phase values are needed, the appropriate values from Tables 2-351 and 2-352 that can be used to double-interpolate are... 

Figure 2-5 Linear interpolation (a) simple interpolation (b) double interpolation.

In thermodynamics we are usually dealing with functions of two variables, for example,/(x, y). If point (x, y) is such that the value y is found in the table but the value of x falls between tabulated values, the above equation may be used to interpolate with respect to x. If both x and y fall between tabulated entries, a double interpolation is necessary. The procedure involves three simple interpolations, and can be outlined as follows fFigure 2-j bI first interpolate between the tabulated values at (x., y.) and (x., y ) to obtain the value of/at point C do the same between points (X2, y.) and (x, y ) to obtain/at point C. Finally, interpolate between C and C to obtain the value at the desired point B. Alternatively, interpolate to obtain points A and A followed by interpolation between A and A to obtain B—the result is the same. These steps can be combined into a single equation which takes the form ... 

In this problem, pressure (1.25 bar) and temperature (185 °C) both lie between the tabulated values. Therefore, a double interpolation is required. We do the calculation first by successive interpolations, then by applying equation (2.14) for double interpolations. 

Method II—double interpolation To apply the double-interpolation formula, we take x to be temperature and y to be pressure, and/. to be the specific volume at temperature x and pressure y. The factors a and b are calculated from eg. (2.1.0 ... 

We must interpolate between these values for pressure and temperature, given the internal energy and volume. The reverse problem would be easier If we knew pressure and temperature and were asked to calculate internal energy and volume, we would perform a double interpolation. Using x for temperature and y for pressure, the corresponding equations are... 

We note that this state is found to lie somewhat farther away from the points used in the interpolation (the pressure in fact lies outside the pressure interval of the interpolation). If a more accurate answer is needed, one should use the above solution to guess that the answer must lie between 400 °C and 450 °C in temperature, and between 8 bar and 9 bar in pressure. A refinement could then be obtained by double interpolation between the four states. This is... 

If neither of the two independent properties used to specify the system falls on a value reported in the steam tables, we must first interpolate with respect to one property. Then we use those interpolated values to interpolate for the other property. This process is termed double interpolation. Determination of volume using double interpolation is illustrated in Example 1.3. 

In the course of the work it was found that the value assumed five years ago for the carbon double-bond covalent radius (obtained by linear interpolation between the single-bond and the triple-bond radius) is 0.02 A. too large in consequence of this we have been led to revise the double-bond radii of other atoms also. 

TABLE 7.18 Interpolated Apparent and Membrane Permeabilities Determined from Double-Sink Conditions 20% Soy Lecithin in Dodecane... 

Several trials are made. A double linear interpolation is made on the sketch with the results... 

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 three models are compared in the figure below (note the double logarithmic scale). As predicted by Eq. 3 of Box 21.5, if the mean input concentration, CNTAin, is much smaller than C - 4.0xl0 1°M, then C(fXAoo CNTAin that is, NTA behaves like a conservative compound. In contrast, for CNTAin C, C (A 3 increases as the square root of the input. Note that for the three models an interpolation from the measured input rate (13 mol d 1) to other input rates predict very different CNTAoo values. 

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. 

The interpolation processor has a double buffered memory that permits updates during processing. Note that interpolation happens on every sample, thereby avoiding zipper noise due to coefficient quantization. This is an expensive strategy, however it always works. 

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