Method of Moment

It is also helpful to define moments for the hulk polymer the total polymer in the system including live radicals [Eq. (65)]. 

With [D ] [P ] there is little difference in magnitude between fij and Cj. Its introduction, however, eliminates the moment closure problem created by the LCB mechanism [40, 81, 85]. Many of the moments have precise physical meanings. 

The zeroth Uve moment, Aq, is the concentration of polymer radicals in the system (denoted by [Pm] in Section 4.3), and the first live moment, Ai, is the concentration of monomer tmits contained in all growing radicals. Similarly, is the concentration of all polymer chains in the system, and //j is the concentration of monomer units bound in all polymer chains. These moment definitions collapse the infinite set of equations for polymeric species into a manageable subset used to calculate MW averages, where is the molecular weight of the monomeric repeat unit [Eqs. (66)]. 

For this example, equations for the kinetic expressions for and /12 will be 

The first step is to formulate balances for Uve radicals, dead chains, and total chains of length n, accounting for all of the consumption and generation terms from the kinetic mechanisms [Eqs. (67)-(69)]. 

The method of moments (MoM) [4] is also frequently employed in transient electromagnetic computations. This method is based on an electric-field integral equation in either frequency or time domain, which relates the induced current on a conductor to the incident electric field. Only the conducting structure to be analyzed needs to be represented as a combination of short cylindrical segments. 

Advantages of this method are summarized as follows (1) it is computationally more efficient than the FDTD method (2) it requires no absorbing boundary condition (3) it can represent oblique conductors easily without any staircase approximation (4) it is capable of considering dispersive materials in the frequency-domain MoM and (5) it is capable of incorporating nonlinear effects and components in the time-domain MoM. 

In the following, an electric-field integral equation in the frequency domain is described first. Then, the corresponding electric-field integral equation in the time domain is explained. 

The electric field at e space point r is generally expressed in the frequency domain as 

The inversion of to the 9 domain is quite difficult. Statistical methods (the method of moments) are therefore used to obtain an expression for the elution profile (the mobile phase concentration distribution). 

Every statistical distribution can be described by its moments. If the distribution is defined by a polynomial expansion, then the coefficients of the polynomial are related to the moments. The peak-like form of the concentration profile suggests that we can define it by its moments. The zeroth moment measures the area under the curve, the first moment gives the mean residence angle of the solute sample and the second moment gives the variance of die peak. The higher moments gives the skewness and flatness. If concentration is denoted by C 9,z) then the moments about the origin of 9 are defined by 

The Laplace transformation may therefore be regarded as a moment-generating function. The coefficients in this expansion can be evaluated by 

The moments of Eq. (14) are evaluated at the outlet of the column z = 1. The first four moments of the mobile phase concentration distribution are given below. 

These moments describe the properties of the mobile phase concentration distribution at the unit outlet To obtain the elution profile, and to study the effect of parameters, an analytical expression is required. The mobile phase concentration distribution is represented as an expansion using Hermite polynomials. The Hermite polynomials are used for the series expansion because the z variable has a domain of [—oo, oo]. The Hermite polynomials cover this domain and the zeros of the polynomial can be obtained over the entire domain. The Hermite polynomial expansion is given as 

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Method of Moments The first step in the analysis of chromatographic systems is often a characterization of the column response to sm l pulse injections of a solute under trace conditions in the Henry s law limit. For such conditions, the statistical moments of the response peak are used to characterize the chromatographic behavior. Such an approach is generally preferable to other descriptions of peak properties which are specific to Gaussian behavior, since the statisfical moments are directly correlated to eqmlibrium and dispersion parameters. Useful references are Schneider and Smith [AJChP J., 14, 762 (1968)], Suzuki and Smith [Chem. Eng. ScL, 26, 221 (1971)], and Carbonell et al. [Chem. Eng. Sci., 9, 115 (1975) 16, 221 (1978)]. 

Research looking into tolerance allocation in assembly stacks is by no means new. A current theme is towards an optimization approach using complex routines and/ or cost models (Lin et al., 1997 Jeang, 1995). Advanced methods are also available, such as Monte Carlo Simulation and Method of Moment. ) (Chase and Parkinson, 1991 Wu et al., 1988). The approach presented here is based on empirical process capability measures using simple tolerance models, cost analogies and optimization... 

The method of moments reduees the eomputational problem to solution of a set of ordinary differential equations and thus solves for the average properties of the distribution. 

Estimation of model parameters is frequently accomplished by the method of moments. For example, for the uniform distribution, the mean is... 

In calculations the moments can be treated as concentrations. Kinetic simulation of radical polymerization to evaluate dispcrsitics typically involves evaluation of the moments rather than the complete distribution. This method of moments is accurate as long as the kinetics are independent of chain length. 

Bulman RA (1978) Chemistry of Plutonium and the Transuranics in the Biosphere. 34 39-77 Bulman RA (1987) The Chemistry of Chelating Agents in Medical Sciences. 67 91-141 Burdett JK (1987) Some Structural Problems Examined Using the Method of Moments. 65 29-90... 

The present section analyzes the above concepts in detail. There are many different mathematical methods for analyzing molecular weight distributions. The method of moments is particularly easy when applied to a living pol5mer polymerization. Equation (13.30) shows the propagation reaction, each step of which consumes one monomer molecule. Assume equal reactivity. Then for a batch polymerization. 

Example 13.8 Apply the method of moments to an anionic polymerization in a CSTR. 

Application of the Method of Mcanents. In order to apply the method of moments (6,7), the pseudo-kinetic rate constant for the crosslinking reaction should be defined as follows. 

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. 

The effects of various pore-size distributions, including Gaussian, rectangular distributions, and continuous power-law, coupled with an assumption of cylindrical pores and mass transfer resistance on chromatographic behavior, have been developed by Goto and McCoy [139]. This study utilized the method of moments to determine the effects of the various distributions on mean retention and band spreading in size exclusion chromatography. 

Burden, J. K. Some Structural Problems Examined Using the Method of Moments. Vol. 65, pp. 29-90. 

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... 

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. 

APPENDIX. CALCULATION OF THE DENSITY OF ELECTRONIC STATES WITHIN THE TIGHT BINDING THEORY BY THE METHOD OF MOMENTS... 

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