Fixed pivot method

The most important full CLD methods are the extended methods of moments, such as the probabihty generating function method and coarse-graining-based techniques the fixed pivot method and the discrete weighted Galerkin formulation. In what follows, the most important aspects of these methods are highlighted. 

In the fixed pivot method, which is a population balance method, the chain length domain is divided into Aimax (typically 50) intervals, which are fixed during the integration but can be... 

Division of the chain length range for the fixed pivot method each interval can be of a different size (zl. ) but the sizes are fixed during the integration typically a logarithmic scale is used g ay... 

Mh chain length domain for the fixed pivot method... 

The current section of the chapter on numerical methods is devoted to an outline of the most frequently used numerical methods for solving the population balance equation either for the particle number distribution function or for a few moments of the number density function. The methods considered are the standard method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the sectional quadrature method of moments (SQMOM), the discrete fixed pivot method, the finite volume method, and the family of spectral weighted residual methods with emphasis on the least squares method. 

The discrete fixed pivot method proposed by Kumar and Ramkrishna [112] is one of the most widely used sectional method due to its generality and robustness [52, 114]. Kumar and Warnecke [114] did analyzed the performance of several sectional methods and concluded that the fixed pivot technique predicts the first two moments of the distribution very accurately. However, the method consistently over-predicts the number density as well as its higher moments. 

Following the notation of Ramkrishna [186] the method is called the discrete fixed pivot method in which derivatives and integrals are represented by some finite difference approximations. Such discretizations were considered of coarse nature thus the population balance under consideration was essentially macroscopic in particle state space. The wide use of the finite difference schemes to solve PBEs is mainly due to the simple construction of these schemes. 

In the discrete fixed pivot method, the dispersed phase size domain is divided into size intervals or sections. If the inner coordinate is the particle diameter the continuous size domain 2 e [ min. maxl is divided into a number of sections ... 

The closed equation (12.383) does not necessary conserve the moments of the distribution due to the macroscopic or finite grid resolution employed in the size domain, thus some sort of ad hoc numerical correction must be induced to enforce the conservative moment properties. It is noted that it is mainly at this point in the formulation of the numerical algorithms that the class method of Hounslow et al. [88], the discrete fixed pivot method of Kumar and Ramkrishna [112] and the multi-group approach used by Carrica et al. [30], among others, differs to some extent. The problem in question is related to the birth terms only. Following the discrete fixed pivot method of Kumar and Ramkrishna [112], the formation of a particle of size in size range... 

Kumar J, Warneche G (2008) Convergence analysis of sectional methods for solving breakage population balance equations—1 the fixed pivot technique. Numer Math 111 81-108... 

Fluorescent receptor ligands can provide a sensitive means of identifying and localizing some of the most pivotal molecules in cell biology. Many types of fluorescently labeled and unlabeled ligands exist for various cellular receptors, ion channels and ion carriers. Many of these site-selective fluorescent probes may be used on live or fixed cells, as well as in cell-free extracts. Many new dyes provide extremely sensitive detection, which enables measurement of low-abundance receptors. Various methods for further amplifying detection of these receptors have been reported. 

The limit of detection (LoD) represents the minimum concentration at which the presence of the analyte can be distinguished from its absence with a high statistical probability. There are different possible ways to define the LoD pivoting about the interpretation of significantly different from the blank. The choice of definition will influence which type of error is more likely Type I, false positives or Type II, false negatives. In addition to these considerations, the suitability of a specific LoD is of course affected by the chosen instrumentation and analytical method. Conversely, at higher levels the concentration of an analyte can be quantified. The lowest concentration that can be reliably quantified is known as the limit of quantification (LoQ). The LoQ is often set at a fixed multiple (usually between 2 and 3.3) of the LoD. 

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