Use of orthogonal collocation methods for the modeling of catalyst particles—I. Analysis of the multiplicity of the solutions

Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

“…A less formal method formulation of the least-squares method is available to obtain the generalized form of the algebraic equation system (24)- (27). The less formal mathematical approach is outlined in the Appendix.…”

“…In the equation system Af = F that is defined by Equations (25)- (27), it is noticed that a quadrature rule is required. In the equation system Af = F that is defined by Equations (25)- (27), it is noticed that a quadrature rule is required.…”

“…Implementation. In the equation system Af = F that is defined by Equations (25)- (27), it is noticed that a quadrature rule is required. Thus, adoption of the quadrature rule (8), the equation system is presented by:…”

“…Equation (91) is identical to Equation (23) as obtained based on variational analysis. Hence, the system of equations on the form Af = F can be presented by Equations (25)- (27). The method derivation provided in this section is not rigorous because ordinary calculus principles for continuous functions have been applied to the norm equivalent functional (10).…”

Application of the Lagrangian Meshfree Approach to Modelling of Batch Crystallisation: Part II—An Efficient Solution of Integrated CFD and Population Balance Equations

“…These approaches can be categorised in five main classes: a) Standard method of moments (SMOM) (Randolph and Larson, 1971), b) Numerical non-linear model reduction approaches: method of characteristics (MOCH) (Hounslow and Reynolds, 2006), quadrature method of moments (QMOM) (McGraw, 1997;Marchisio et al, 2003), fixed quadrature method of moments (FQMOM) (Alopaeus, 2006), Jacobian matrix transformation (JMT) (McGraw and Wright, 2003) and direct quadrature method of moments (DQMOM) (Fan et al, 2004), c) Direct numerical solution approaches involving finite-element or finite-volume discretisation of the partial differential equation (discretised population balances, DPB) (LeVeque, 2002;Gunawan et al, 2004;Costa et al, 2007), d) various methods in the weighted residuals framework such as the least squares, orthogonal collocation and Galerkin methods (Singh and Ramkrishna, 1977;Sporleder et al, 2011;van den Bosch and Padmanabhan, 1974;Nigam and Nigam, 1980;Roussos et al, 2005), and e) Dynamic Monte Carlo simulation (DMC) (Haseltine and Rawlings, 2005;Rosner et al, 2003). The two most often used techniques are the standard method of moments and the quadrature method of moments.…”