Volume surface-weighted mean

Volume surface-weighted mean diameter d,s = di = S3 /S2 Volume moment-weighted mean diameter d,m = dt 3 = S4/S3... 

Table 5.3. Fat globule size (volume surface-weighted mean diameter, dys) in milk...

Arithmetic Length Surface 2 m/3/2 nd1 Volume-surface or surface-weighted mean, dvs... 

Example 1-3 The optical microscope analysis given in Table 1-5 was performed for a particulate sample of ground Pittsburgh seam coal that was to be fed to a combustor. Determine the length, surface, volume, and weight mean diameters. Using the equations in Table 1-4, one finds... 

Length-number mean Surface-number mean Volume-number mean Volume-surface mean Weight-moment mean... 

For a mixture of particles of several sizes, one evaluation of a mean diameter is the volume surface mean, or Sauter mean. When ws is the weight or volume fraction of particles of diameter ds the mean is... 

All equipment designed to measure surface area, adsorption-desorption isotherms or pore volume by adsorption actually determines the quantity of gas condensed on a solid surface at some equilibrium vapor pressure. The surface area or pore volumes and pore sizes are then calculated by means of an appropriate theory used to treat the adsorption and/or desorption data. Depending on the apparatus employed, the adsorbed quantity is measured as volume or weight. The accuracy of an adsorption apparatus is, therefore, dependent upon its ability to correctly measure either of these quantities. 

Because the inequalities among the various mean diameters are usually strengthened when the drop sizes are widely dispersed, the ratio of some higher-order moment to a lower-order moment is often useful as a measure of the dispersion of the drop sizes. For example, the coefficient of variation for the surface-weighted size distribution is a function of the ratio of the weight-weighted mean drop size to the volume-surface mean drop size. The variance of the drop-size distribution may also be expressed in terms of the moments of the unweighted size distribution. 

Results of this type have proved of value in experimental investigations involving surface-volume relations. Of particular interest is the fact that specific surface is inversely proportional to the first moment of the surface-weighted size distribution, and this moment, in turn, is equal to the harmonic mean of the volume-weighted size distribution. 

Table 9 shows the results of the sieve analysis obtained for a sample of a FCC1 powder. The values reported in the first two columns of the table are the standard diameters of the sieve apertures. From these the mean diameter dp is obtained for each two adjacent sieve sizes and the values are reported in the third column. From the mass fraction of powder in each sieve (values in the fourth column) the weight percentage is obtained and reported in the fifth column. Thus, the sum of the mass fraction over the mean diameter allows the calculation of the volume-surface mean particle diameter of the distribution using Equation (1) ... 

Volume = Volume enclosed by van der Waals radii Mass = molecular weight of nonionized amino acid minus that of water both adopted from Creighton (1993) HP scale = degree of hydrophobicity of amino acid side chains, based on Kyte Doolittle (1982) Surface Area = mean fraction buried, based on Rose et al. (1985) and Secondary structure propensity = the normalized frequencies for each conformation, adopted from Creighton (1993), is the fraction of residues of each amino acid that occurred in that conformation, divided by this fraction for all residues. 

For an open circuit ball mill 5 liters in volume operating at a flow rate of 1 liter per hr the product is a Z1O2 suspension with a Ros-lin-Rammler size distribution with a weight mean size of 0.5 jam and a volume to surface mean diameter of 0.7 jam. Determine the feed distribution to the mill assuming the value of j3 the grinding selectivity factor is 1.0 and k is 0.5 (hr/jam) . ... 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

A summary of these different types of averages can be found in Table 3. Example 5. Using the particle count data given in Example 4 and Table 2, compute the statistical mean diameters for the number-length, number-surface, number-volume, length-surface, length-volume, surface-volume, volume-moment and weight-moment mean diameters. 

Here, Pcz>/ P>gd/ o are constant for the whole pattern (taken at zero value of the wave vector Sq of the first peak). The integral breaths found from the experimental diffraction profiles allow calculation of the surface-weighted (Ds) and volume-weighted (Dy) domain size and a mean-square (Gaussian) strain, s, which is the total strain averaged over infinite distance ... 

In most cases d 2 (the volume/surface average or Sauter mean) is used, while the width of the size distribution can be given as the variation coefficient. The latter is the standard deviation of the distribution weighted with d divided by the corresponding average d. Generally C2 will be used which corresponds to d 2 An alternative way to describe the emulsion quahty is to use the specific surface area A (the surface area of all emulsion droplets per unit volume of emulsion). 

Droplet-size distributions and average droplet sizes (volume-surface mean diameter d32 and weighted average mean diameter d43) of O/W emulsions were... 

From the extent of reduction and the surface area of the metal, as calculated from the extent of chemisorption, the mean size of the metal can be calculated. The broadening of the maxima in the X-ray diffraction pattern also measures a mean particle size. Usually the mean particle size calculated from the X-ray line-broadening is larger than that calculated from the extent of reduction and the surface area of the metal particles. The difference is because X-ray line-broadening provides a weight-mean particle size, Ln d l Ln- di, whereas the extent of chemisorption and the extent of reduction result in a volume-surface mean diameter, Lnid /Lrijdj. ... 

In Equation (48) the ratio A" He/x from an individual sample has been taken as representative for the entire lake. If several measurements are available the volume weighted mean values of A He and x could be applied. This would correspond to the two box model (Eqn. 47) with box 2 representing surface water in equilibrium with the atmosphere. At the lake surface " Hei = " Hcequ and Hei = Hcequ, implying X2 = 0. Because Heequ varies only very little with temperature Heequ is approximately constant. Then FqHe.sed = Vq/Ao A Hci/xi with A Hci and x being volume weighted mean values. 

The volume-surface mean is also known as the Sauter mean. This defines the average particle size based on the specific surface area per unit volume or per unit weight, as shown in Eq. (69) ... 

The harmonic mean is actually related to the average spherical particle corresponding to the particle surface per unit weight. Mathematically, the harmonic mean is similar to the volume-surface mean or Sauter mean. 

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