Quadrature method

Tellinghuisen J A 1974 A fast quadrature method for computing diatomic RKR potential energy curves Comput. Phys. Commun. 6 221-8... 

Evaluation of the integral in Eq. (14-86) requires a knowledge of the liquid-phase bulk concentration of B as a function of y. This relationship is obtained by means of a material balance around the tower, as shown in Eq. (14-73). Numerical integration by a quadrature method such as Simpson s nrle normally will be required for this calculation. 

AN ADAPTIVE QUADRATURE METHOD FOR BRILLOUIN ZONE INTEGRATION OF KKR FUNCTIONS... 

For fluid particles that continuously coalesce and breakup and where the bubble size distributions have local variations, there is still no generally accepted model available and the existing models are contradictory [20]. A population density model is required to describe the changing bubble and drop size. Usually, it is sufficient to simulate a handful of sizes or use some quadrature model, for example, direct quadrature method of moments (DQMOM) to decrease the number of variables. 

In theory, this model can be used to fix up to three moments of the mixture fraction (e.g., (c), ( 2), and (c3)). In practice, we want to choose the CFD transport equations such that the moments computed from Eqs. (34) and (35) are exactly the same as those found by solving Eqs. (28) and (29). An elegant mathematical procedure for forcing the moments to agree is the direct quadrature method of moments (DQMOM), and is described in detail in Fox (2003). For the two-environment model, the transport equations are... 

Quadrature methods that approximate integral constraints (e.g., moments) of... 

For cases with only one internal coordinate, either approach can be implemented in a CFD code (but the computational cost for the same accuracy can be very different). However, for cases with more than one internal coordinate, only the quadrature methods are computationally tractable on current computers. Thus, in the examples below, we will describe only CFD models based on the... 

To overcome the difficulty of inverting the moment equations, Marchisio and Fox (2005) introduced the direct quadrature method of moments (DQMOM). With this approach, transport equations are derived for the weights and abscissas directly, thereby avoiding the need to invert the moment equations during the course of the CFD simulation. As shown in Marchisio and Fox (2005), the NDF for one variable with moment equations given by Eq. (121) yields two microscopic transport equations of the form... 

The quadrature method of moments (QMOM) is a presumed PDF approach that determines the unknown parameters by forcing the lower-order moments of the presumed PDF to agree with the moment transport equations (McGraw 1997 Barrett and Webb 1998 Marchisio et al. 2003a Marchisio et al. 2003b). As with the multi-environment presumed PDF method discussed in Section 5.10, the form of the presumed PDF is... 

The direct quadrature method of moments (DQMOM) begins with a closed1 joint composition PDF transport equation (see Section 6.3). For simplicity, we will consider the high-Reynolds-number form of (6.30) on p. 251 with the IEM mixing model ... 

McGraw, R. (1997). Description of aerosol dynamics by the quadrature method of moments. Aerosol Science and Technology 27, 255-265. 

Marchisio, D. L. and R. O. Fox (2003). Direct quadrature method of moments Derivation, analysis and applications. Journal of Computational Physics (in press). 

Wright, D. L., R. McGraw, and D. E. Rosner (2001). Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering particle populations. Journal of Colloid and Interface Science 236, 242-251. 

TABLE 3.1 Approximations to the Integral J j sinf-n-x) dx = 1 Using the Trapezoidal and Legendre Quadrature Methods... 

APPLIED ANALYSIS, Cornelius Lanczos. Classic work on analysis and design ol linite processes for approximating solution of analytical problems. Algebraic equations, matrices, harmonic analysis, quadrature methods, much more. 559pp. 5H x 8H. 65656-X Pa. 11.95... 

Two approximation methods are, however, peculiar to the continuous description the so-called method of moments, and the quadrature method. Some basic conceptual issues about these two methods are discussed next. 

First, pseudocomponents determined by the quadrature method may well be unrealistic ones for instance, if the label x is (proportional to) the number of carbon atoms, pseudocomponents may well correspond to noninteger jcjc values. This may be aesthetically unpleasant, but it does not represent a real problem. More seriously, the appropriate pseudocomponents obviously depend on the composition and, hence, in a repeated calculation such as is required in a distillation tower, pseudocomponents will need to be different at each step. This puts out of tilt the mass balance equations that are coupled to the equilibrium ones, and, even if this problem could be circumvented (as, at least in principle, it can), the procedure would certainly not be applicable to existing software for distillation column calculations. 

Keywords distribution shaping control, population balance modeling, method of characteristics, optimal control, quadrature method of moments. 

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