Surface averaging

Since is the surface-averaged part of 8p from equation (A3.3.36h is the appropriately surface-... 

Freshwater with less than 500 ppm (or 0.05%) dissolved soHds is generally considered to be potable. Rain is the source of freshwater, and its precipitation of >1.3 x 10 m /d over the earth s surface averages about 1.05 m (depth) per year. Extremes range from almost 2ero ia North Chile s desert borderiag the Pacific Coast to > 25.4 m ia some tropical forests and on some high slopes where the high, cold mountains condense floods from the clouds. 

When a liquid is dispersed into droplets the surface area is increased, which enhances the rates of heat and mass transfer. For a particular liquid dispersed at constant concentration in air the MIE varies with approximately the cube of surface average droplet diameter, hence the MIE decreases by a factor of about 8 when the surface average diameter D is halved (A-5-1.4.4). Ease of ignition is greatly enhanced for finely divided mists with D less than about 20 /rm, whose MIE approaches that of the vapor. Below 10 /rm a high flash point liquid mist (tetrahydronaphthalene) was found to behave like vapor while above about 40/rm the droplets tended to burn individually [ 142]. Since liquid mists must partially evaporate and mix with air before they ignite, the ease with which a liquid evaporates also affects MIE (Eigure 5-1.4.4). 

Cp a = specific heat of air at constant pressure AT jj = temperature rise for stoichiometric combustion D = surface average particle diameter Pa = air density Pf = fuel density

equivalence ratio B = mass transfer number... 

The surface-averaged Gaussian curvature, K y, introduces the length scale, describing an average radius of curvature of the single,... 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

The average particle size and surface average particle size are given by (/n =... 

Free-fall Surface Average Temperature Density Area(b) Pore Limit ( C) (g/cm ) (m /g) Diameter (nm) ... 

Cathode surface-average Raman spectra were produced with the Raman microscope. Raman spectra were collected from multiple 52x75pm sections of the cathode surfaces at 0.7pm resolution and averaged into one representative spectrum for each cathode. MicroRaman surface-average spectra of the cathode from the virgin cell, and cathodes removed from cells,... 

Figure 8.6. The ratio of the surface-averaged intensity to the incident intensity for the same particle as in Figure 8.4.

The excitation spectrum proves even more useful near the surface. Since anisotropic molecules at the surface of a liquid tend to orient relative to the surface tangent, one might expect the excitation spectrum to be sensitive to such orientation. For example, suppose we take the extreme case in which molecules at the surface are oriented with their transition moments perpendicular to the surface tangent. Then the only field component which can excite these molecules is the radial field at the surface. When one recalls that only the N type vector field has radial components, one expects that a calculation of the excitation spectrum of such a molecular layer will yield half as many resonant features as shown in Figure 8.4. Indeed this is the case. Figure 8.7 shows the calculated surface average of the square modulus of the radial component of the local electric field, < E er 2>J, where sr is the radial unit vector. 

The surface-averaged final diameter Dsf can be easily controlled by adjusting the amount of particles. Because the solid particles are irreversibly adsorbed at the oil-water interface, the inverse average droplet diameter varies linearly with the amount of particles according to the simple equation ... 

Figure 5.12 represents the evolution in time of the surface-averaged droplet diameter for different amounts of solid particles. The kinetic curves confirm the qualitative evolution previously described. The droplet growth is initially rapid but the coalescence rate progressively decreases until the average diameter reaches an asymptotic value. Figure 5.13 shows the change in the droplet size distribution... 

The 5 wt% Pt-on-alumina was a commercial catalyst (Engelhard 4462). The metal dispersion (D = 0.30) was determined from TEM pictures. Different fields were examined and about 1000 particles were counted and their size determined. The degree of dispersion was calculated from the surface average diameter (76). 

Table 5.2. Surface average diameter determined by TEM and H2 adsorption.

Surface average ds (dT)1/2 Square of diameter Number in class... 

The surface average diameter is always larger than the direct average of the diameters for a polydisperse system since the larger diameters contribute relatively more to the sum of the squares than they would if totaled directly. Example C. 1 illustrates such a calculation. 

Solids of practical interest often are mixtures of a range of particle diameters, but, for convenience, correlations are expressed in terms of a single size which is almost invariably taken as the surface average diameter given by... 

A second kind of polymer, a colloidal aqueous dispersion, was reported by Renfrew (1950) who used bis- (/ -carboxypropionyl) peroxide as the polymerization initiator, and later described in more detail by Lontz and Happoldt. The specific surface of dispersion polymer is on the order of 12 m2/g, and the equivalent surface average diameter for dense spheres is about 0.2 fi. This is a good check with the observed size seen in the electron micrograph of Fig. lb and indicates that the primary dispersion particles have little, if any, porous structure. 

Dpg = surface average diameter of particles = Avogadros number... 

Other possible geometrical diameters can be used to determine the mean particle diameter of a polydisperse system. Examples are the surface average, ds, and volume average diameters, dv where ds is defined as the diameter of a sphere having the same surface area as the particle and dw is the diameter of a sphere having the same volume as the particle. These are given by ... 

---