Five-point method

Figure 15.2. Region of interest for computing potential based on Laplace or Poisson equations, where (a) a complete rectangular grid is established to cover the region, which may be adapted to finite-difference techniques using (b) a five-point method, or (c) a finite-element approach based on sampling functions.

A typical example of an efficient search technique by group experiments is known as the Five-Point Method and is explained in the following. The basis of this method is first to select the overall range of the surface to be examined and then to determine the values of the objective function at both extremes of the surface and at three other points at equally spaced intervals across the surface. Figure 11-13 shows a typical result for these initial five points for a simplified two-dimensional case in which only one maximum or minimum is involved. 

Illustration of Five-Point Method for group-experiment exploration of response surface. 

The measurements of external and internal specific surface area have already been discussed in Chapter 1, Section 1.1.3. The principles and the isotherm equation of the BET method to measure external specific surface area, including macro- and mesopores, have been presented in Chapter 1, Section 1.3.4.1.5. The external specific surface area is usually determined by nitrogen gas adsorption at the temperature of liquid nitrogen. Both static (one-point) and dynamic (five-point) methods are applied. The calculations are made by Equation 1.75 (Chapter 1), using one or five different pressure values. The external specific surface area is calculated from the maximum number of surface sites, that is, monolayer and the cross-sectional area of nitrogen molecules. 

The first five measure the activity of the solvent and the last five measure the activity of the solute. The boiling point method is generally not included in evaluations for two reasons there are little data from these measurements or the thermal data are not adequate to apply an accurate correction to obtain an activity at 298 K. 

LALS) in THF and the intrinsic viscosity of five of the samples was measured by the single point method. Combining the LALS data with the intrinsic viscosity work allows one to obtain the Mark-Houwink constants, and a, by plotting log[77] vs logMw. For PIPTBK, and a in THF at 25° were found to be 5xlO-3 ml/g and 0.75 respectively. 

Remember the basic rules organic methods typically require a five-point calibration trace element AA methods use a calibration blank and three standards the ICP-AES instruments are calibrated with one standard and a calibration blank inorganic analysis methods use three- to five-point calibration curves. 

In addition to obtaining correlograms, a large battery of methods are available to smooth time series, many based on so-called windows , whereby data are smoothed over a number of points in time. A simple method is to take the average reading over five points in time, but sometimes this could miss out important information about cyclicity especially for a process that is sampled slowly compared to the rate of oscillation. A number of linear filters have been developed which are apphcable to this time of data (Section 3.3), this procedure often being described as convolution. 

Each type of smoothing function removes different features in the data and often a combination of several approaches is recommended especially for real world problems. Dealing with outliers is an important issue sometimes these points are due to measurement errors. Many processes take time to deviate from the expected value, and a sudden glitch in the system unlikely to be a real effect. Often a combination of filters is recommend, for example a five point median smoothing followed by a three point Hanning window. These methods are very easy to implement computationally and it is possible to view the results of different filters simultaneously. 

Calculate the first derivative at each wavelength and each point in time. Normally the Savitsky-Golay method, described in Chapter 3, Section 3.3.2, can be employed. The simplest case is a five point quadratic first derivative (see Table 3.6), so that... 

The space/time over which the problem is formulated is covered with a mesh of points, often referred to as nodes . At each point, the derivatives in the material balance equation are approximated as differences of the concentrations at the given and surrounding points. This leads to a set of linear equations (based on a five-point stencil in two dimensions - each node is related to its four nearest neighbours) which can be solved to give the solution to the PDE. The methods are well suited to simulations in rectangular regions, which is often compatible with an electrochemical cell. These are by far the most popular methods for electrochemical simulations and will therefore be the focus of the remainder of this section. 

To calculate the rates of styrene polymerization with the help of experimental data on conversion-time the method of the digital differentiation using five points (16) was applied. In table V the values of Kp/Kt for different conversions are represented, the values have been obtained during the styrene polymerization in the presence of different initial concentrations of BzjOa at temperature of 70° C, Up to conversion of - 40% the ratio of Kp/Kt somewhat increases smd does not depend on the initial concentration of initiator. But at more high stages of polymerization the difference of MW of polymer being formed reveals itself and the ratio of Kp/Kt increases as quickly as the initial concentration of initiator decreases. Dependences of Kp/Kt on the con-... 

The Czechoslovak Standard CSN 640214 recommends that the test core method should be employed for determining the flowability of aminoplasts. The test core is stepped in shape. The flowability criterion is the time required to fill up the mould. The five-point scale for assessing the flowability is used. The average pressure on the material amounts to 54 1 MPa, the free-stroke velocity of the crosspiece of the press is 20 5 mm/s. 

This set of ODEs can be solved by any convenient method. The computational template is shown in Eigure 16.3. Five points, centered around (x, y, t) on thex-y plane, are used to project ahead to one point, V (x, y, t + At), at the new time. 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

The relative merits of these different methods can be compared by differentiating a known mathematical function. The model we will use is y = x + x /2), X = 0. .. 4, at X = 2. Various levels of noise are imposed on the signal y (Table 3.2). The resulting derivatives are shown in Table 3.3. As the noise level reduces and tends to zero, the derivative results from applying the five-point polynomial (Equation 3.4) converge more quickly towards the correct noise-free value of 3 for the first derivative, and 1 for the second derivative (Equation 3.5). As with polynomial smoothing, the Savitzky-Golay differentiation technique is available with many commercial spectrometers. 

Sestak (43) compared the kinetic results calculated by five different methods for a system corresponding to the dehydration of -CaS04 0.5H2O. The five methods evaluated mathematically were (1) Freeman and Carroll (83) (2) Doyle (84) (3) Coats and Redfern (85) (4) Horowitz and Metzger (88) and (5) Van Krevelen et al. (87). From these calculations it was found that the deviations of computed values oF E did not differ by more than 10%. Thus, all the methods appear to be satisfactory for the calculation of E within the limits of accuracy required. The errors of each method due to the inaccuracy of visual deduction of values from the TG curves were also calculated. These errors, % and e (errors in calculation of E or n, respectively), were as follows (1) Freeman and Carroll method, eE = 4% and e = 12% (2) Horowitz and Metzger method, ee = 2% (when the correct value of n is assumed) (3) Doyle method. eE = 4%. However, the magnitude of this error depends primarily on the position of the point on the TG curve on which the calculations are being performed. In the case of differential methods, me most accurate data are calculated from the medium-steep parts of the curve. For the approximation method, the accuracy depends on the determination of the curve inflection point temperature. 

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