Sieves transformation equations

The definition of the surface-volume mean diamefer given by equation 1.11 must be modified for use wifh dafa from a sieve analysis. By assuming that the shape and density of fhe particles are constant for all size fractions, a number distribution can be transformed fo a mass distribution (Smith, 2003) and therefore the surface-volume diameter becomes... 

Ring transformation of pyrido[2,3-f]furoxan 676 with 1,3-diketones or /3-keto esters was catalyzed by silica gel, alumina, or molecular sieves to give mainly the corresponding pyrido[2,3- ]pyrazine 1,4-dioxides 677 in addition to pyrido[2,3- ]pyrazine 1-oxides 678 in a low yield (Equation 57) <1997H(45)107>. 

Equations (31) and (32) can be used to analyze impedance spectra without knowledge of structural electrode parameters (thickness, density, etc). However, we need this information in order to transform the ohmic parameters obtained by a fit into specific electrochemical parameters. In particular, this information can be used to calculate the effective surface area of the particles. Particles used in practical batteries can usually be treated either as thin plates (Levi and Aurbach [1997]) or as pseudospherical in shape (Barsoukov [2003]), and have a narrow size distribution due to sieving. Particle size values are provided by material manufacturers. The number of particles in a given volume can be estimated from the ratio of their crystallographic density of particles, Op, to the density of the composite-electrode film, a. This allows one to calculate the electrochemically active surface area for a composite electrode for thin-plate particles as 5 = xAdalUOp] and for spherical particles as 5 = 3xAdal[rCp. Here x is the fraction of active material in the composite A is the geometric area of the electrode d is the thickness of the composite electrode <7 is the density of the composite electrode Op is the true density of particles and I and r are the thickness of the plate and radius of spherical particles, respectively. 

Equations 2.34 and 2.35 allow for transformations between mesh number and aperture size. Both features of sieves are listed in standard screen scales. Standardized sieve apertures were first proposed by Rittinger (1867). Modem standards are based on either a-Jl or progression. In the United States, the series of sieves with standard opening sizes are called "Tyler" sieves and the openings of successive sieves are based onayfz progression starting at 45 pm. 

The process dynamic model of a batch crystalliser is straightforward, fully described by the energy, mass and population balances. However, the dynamic of the crystal size distribution can be ignored if a batch is initially fed with seeds closely sized between two adjacent sieve sizes. General equations and constraints are developed for anhydrous salts. Additional equations are required to describe other transformations as in the case of hydrates and organic compounds. The subscript f and the superscript in the following equations denotes feed and saturation, respectively. The rate change in ... 

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