Unimodal surface

A unimodal surface has been chosen because we have no way of dealing sequentially with a surface that has two or more peaks or valleys. The only reasonable approach is to start the sequential procedure at a number of widely disparate points and to determine whether the paths converse toward the same optimum. If two or more different peaks or valleys are indicated, the investigator must find the optimum for each possible peak or valley and then select the best. There is no way of knowing that all peaks or valleys have been explored except to map the whole surface finely using some factorial design. 

Method of steepest descent applied to unimodal surface. 

The mode of distribution is simply the value of the most frequent size present. A distribution exhibiting a single maximum is referred to as a unimodal distribution. When two or more maxima are present, the distribution is caUed bimodal, trimodal, and so on. The mode representing a particle population may have different values depending on whether the measurement is carried out on the basis of particle length, surface area, mass, or volume, or whether the data are represented ia terms of the diameter or log (diameter). 

To understand the method of steepest accent, consider a hiker who wishes to reach the summit of a volcanic island (assumed unimodal) by taking the shortest path. The shortest path is the steepest. Suppose our hiker is a mathematician, and rather than use his eyesight he decides to determine mathematically the best direction in which to proceed. Further, he knows that if he goes 20 ft to the north of his current position he will rise 20 ft and if he goes 20 ft to the east he will fall 10 ft. If the surface were a plane, he could approximate it by the following equation ... 

Figure 14-5 A response surface that is not strongly unimodal... 

Figure 12.28 shows the particle surface area size distribution before the Mount Pinatubo eruption (Fig. 12.28a), inside the main aerosol layer several months after the eruption (Fig. 12.28b), and almost two years after the eruption (Fig. 12.28c). (See Chapter 9.A.2 for a description of how particle size distributions are normally characterized.) Prior to the eruption, the surface area distribution is unimodal, with typical radii of 0.05-0.09 /xrn and a number concentration of l-20 particles cm 1. In the main stratospheric aerosol layer formed after the eruption, the distribution is bimodal...

An ICI-Joyce Loebl Disc Centrifuge MK III, a photosedimento-meter, was used to measure the latex particle size distribution. The latex had a unimodal particle size distribution with a diameter of 1.05 micrometers (surface area average). The methods of separating latex particles by a centrifugal field and detecting the size distribution by a photocell may be found in the literature. 

By laser diffractometry, Okechukwu and Rao (1996a) found that ungelatinized cowpea starch granules had a unimodal distribution with a mean of 19 pm. In an unrelated method, Chuma et al. (1982) used photography, a digitizer, and a microcomputer to calculate the size, surface area, and volume of grains and soybeans. 

The adsorption-desorption reaction in Eq. 4.3 has been applied to soils in an average sense in a spirit very similar to that of the complexation reactions for humic substances, discussed in Section 2.3.11 Although no assumption of uniformity is made, the use of Eq. 4.3 to describe adsorption or desorption processes in chemically heterogeneous porous media such as soils does entail the hypothesis that effective or average equilibrium (or rate) constants provide a useful representation of a system that in reality exhibits a broad spectrum of surface reactivity. This hypothesis will be an adequate approximation so long as this spectrum is unimodal and not too broad. If the spectrum of reactivity is instead multimodal, discrete sets of average equilibrium or rate constants—each connected with its own version of Eq. 4.3—must be invoked and if the spectrum is very broad, the sets of these parameters will blend into a continuum (cf. the affinity spectrum in Eq. 2.38). 

The commercial catalyst used in this work contains 12 wt% Ni and 83 wt% a-Al203. It has a BET total surface area of 3.4m /g and a unimodal pore size distribution with volume 0.155 cc/g, mean pore radius 1600 A and void fraction 0.362. Its activation required a reduction which was carried out under atmospheric pressure in situ, for 72 hrs at 850°C by means of a pure dried hydrogen flow of lOO Nl/hr. These severe reduction conditions were required because 20 wt% of the Ni was present as NiAl204-spinel phase, which could only be reduced above 770°C. It led to a very active catalyst, with a specific Ni-surface area of 0.68 m Ni/g.cat. 

For the optimization situation in which two or more independent variables are involved, response surfaces can often be prepared to show the relationship among the variables. Figure 11-12 is an example of a unimodal response surface with a single minimum point. Many methods have been proposed for exploring such response surfaces to determine optimum conditions. 

The method outlined in the preceding obviously can become very tedious mathematically, and a computer solution is normally necessary. The method also has limitations based on choice of scale and incremental steps for the variables, extrapolation past the region where the straight line approximates the surface, and inability to handle surfaces that are not unimodal. 

In addition to the method of steepest ascent and descent, many other strategies for exploring response surfaces which represent objective functions have been proposed. Many of these are based on making group experiments or calculations in such a way that the results allow a planned search of the surface to approach quickly a unimodal optimum point. 

A surface with only one side, such as a Moebius strip, unimodal... 

This relatively large surface area and the fact that the isotheim is of Type IV indicate that this film has porosity in the mesoporous range. A pore size distribution using the desorption branch of the isotherm is shown in Figure 4.16 (page 190). A fairly unimodal distribution is obtained with a median pore diameter of... 

The limestone used in the AFBC was deeply characterized. The limestone SEM-EDX analysis shows Ca as the unique identified element and its XRD analysis shows calcite as the only crystalline chemical species. After heat treatment at 100"C, calcite showed an apparent surface area of 19 m /g, 48.5% porosity with unimodal small pores of 5.5nm. The coal burned in the AFBC plant was a low-rank coal with 0.5-1 imn particle size. 

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

One further difference exists between HDS and HDM. Bridge [37] has shown, very clearly, that HDS is not limited by diffusion while HDM is. Using a nickel-molybdate based catalyst with a unimodal microporous size distribution, the demetalation of Arabian heavy atmospheric residuum was found to be affected by catalyst particle size, while HDS was not. As the diameter of the pore was decreased, the maximum in the metals deposition profile moved closer to the external surface of the pellet, agmn indicating difiusional limitations for FIDM. 

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