Circular averaging

Descriptions of the experimental scattering and microscopy conditions have been published elsewhere and are referenced in each section. Throughout this report certain conventions will be used when describing uncertainties in measurements. Plots of small angle scattering data have been calculated from circular averaging of two-dimensional files. The uncertainties are calculated as the estimated standard deviation of the mean. The total combined uncertainty is not specified in each case since comparisons are made with data obtained under... 

CCD data by circular averaging and expressed as a function of scattering angle in terms of the wave vector of momentum transfer q, with q = 4n sin (())//-, where 2d is the total scattering angle and 2 the X-ray wavelength. 

A semidilute solution [42] of high molecular weight deuterated polystyrene (Mw = 1.95 x 106 g/mole, Mw/M = 1.64) in dioctyl phthalate (DOP) at a volume fraction of 2.83% of polystyrene was measured by SANS at room temperature. A characteristic intensity behavior I(Q) was obtained after data correction (solvent incoherent scattering, empty cell scattering and usual background corrections, etc.) and was circularly averaged. The reduced I(Q) data was then fitted to the following form ... 

Table 8-13 also shows that hair from Ethiopians has a greater deviation from circularity (average D1ID2 of 1.75) than that of the other two races... 

The circularly averaged intensity of the patterns at lOQ C monotonicaUy decreases with scattering vector q (Fig. 7.7). The increase in intensity with time can be associated with the increases in the number and/or the anisotropy of the domains. In Fig. 7.8, the shape of the log-log plots of scattering profiles indicates that the domains are fractal objects. Kratky plots in Fig. 7.9 indicate that these domains are diffuse. For, the intensity profiles at high q region (if q I, where a the length scale of the scattering objects,) can be fitted to the Porod form [52-54] ... 

The long period, spacing between adjacent crystalline lamellae layers, can be estimated from a circular averaged 1-D SAXS data. The lamellar morphology of iPP in the blends was also determined from the position of the SAXS maxima. 

Because of the physical design of a broadband panel antenna, the cross-section is larger than the typical narrowband pole antenna. Therefore, as the operating frequencies approach the high end of the UHF band, the circularity (average circle to minimum or maximum ratio) of an omnidirectional broadband antenna generally deteriorates. 

The SANS experiments were performed with the SANS instrument at Japan Atomic Energy Research Institute. SANS data obtained from the mixtures were corrected for transmission, scattering from an empty ceU and incoherent scattering, and were calibrated to the absolute intensity using the scattered intensity of water. Since the sample has no orientation, SANS data were circularly averaged. 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

With the hydraulic diameter subsititued for D in/and Re, Eqs. (6-37) through (6-40) are good approximations. Note that V appearing in/and Re is the actual average velocity V = Q/A for noncircular pipes it is not ( /(7CD /4). The pressure drop should be calculated from the friction factor for uoucirciilar pipes. Eqiiations relating Q to AP and D for circular pipes may not he used for noncircular pipes with D replaced by because V Q/( KDh/4). 

Traversing for Mean Velocity Mean velocity in a duct can be obtained by dividing the cross section into a number of equal areas, finding the local velocity at a representative point in each, and averaging the results. In the case of rectangular passages, the cross section is usually divided into small squares or rectangles and the velocity is found at the center of each. In circular pipes, the cross section is divided into several equal annular areas as shown in Fig. 10-7. Read-... 

The graphical integration method is based on graphical presentation of the average flow profile. For a circular duct, the cross-section is virtually divided into several concentric ring elements. The spatial mean velocity of such an element is determined as an arithmetical mean of local velocities along the circumference of the corresponding radius. For a circular cross-section the flow rate can be expressed as... 

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