Fixed-pivot technique

Kumar, S. and Ramkrishna, D., 1996a. On the solution of population balance by discretization I. A fixed pivot technique. Chemical Engineering Science, 51, 1311-1332. 

The fixed-pivot technique can, in general, be formulated for any grid while preserving two moments of the distribution of orders ri and r2. The resulting equation for the evolution of the number of particles of the /, interval is (Kumar Ramkrishna, 1996a)... 

Equation (7.19) was also obtained by Hounslow et al. (1988) after correcting the original equation proposed by Batterham et al. (1981). However, diis equation is nodiing odier than the fixed-pivot technique applied for a geometric grid with ratio equal to two and after superimposing the conservation of the zeroth- and first-order moments (with particle mass as internal coordinate). 

With the fixed-pivot technique, when a new particle is formed it is reassigned to the... 

As for the univariate case, when a particle is formed after an aggregation event, it has to be assigned to the neighboring cells (four in a bivariate case). When the fixed-pivot technique is used to assign the particles to cells, the following expression is obtained ... 

Vale, M. McKenna, F. 2005 Solution of the population balance equation for two-component aggregation by an extended fixed pivot technique. Industrial Engineering Chemistry Research 44, 7885-7891. 

In addition, the PSD (or MDSD) can be calculated analogously by replacing the variable / by its diameter dp or volume Pp. Most commonly, the fixed pivot technique has been applied while dividing the reactor into two compartments, an impeller and a circulation compartment. 

Kumar, S. and D. Ramkrishna, On the Solution of Population Balance Equations-I. A Fixed Pivot Technique, Chem. Eng. ScL, 8, 1311-1332 (1996a). 

The droplet volume (and hence diameter) is discretized using the fixed pivot technique. Consequently it is partitioned into grid points according to the structure v, =, ... 

The basic idea in the fixed pivot technique is that when a droplet of volume, v, is... 

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. 

These source term expressions contain the continuous particle number density function, /(, r, t). However, in order to close the equation the source terms must be expressed entirely in terms of the dependent variable j(r, f). The fixed pivot technique is based on the idea of birth modification. The pivot concentrates the particles in the interval at a single representative point. The number density distribution function/i(, r, t) can be approximated as ... 

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... 

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