Moment equation

The transition moment (Equation 2.13) for a transition between lower and upper states with vibrational wave functions and j/[ respectively is given by... 

Properties of the distribution such as total number of crystals per unit volume, total length of crystals per unit volume, total area of crystals per unit volume, and total volume of soUds (crystals) per unit volume may be expUcitiy evaluated from the moment equations. 

The estimation of the mean and standard deviation using the moment equations as described in Appendix I gives little indication of the degree of fit of the distribution to the set of experimental data. We will next develop the concepts from which any continuous distribution can be modelled to a set of data. This ultimately provides the most suitable way of determining the distributional parameters. 

These equations are ealled the moment equations, beeause we are effeetively taking moments of the data about a point to measure the dispersion over the whole set of data. Note that in the varianee, the positive and negative deviates when squared do not eaneel eaeh other out but provide a powerful measure of dispersion whieh... 

Often, it is not necessary to know the total CSD since some mean particle characteristic and possibly its variance may suffice. In general, such characteristics of the CSD can be obtained from the moment equation, below, by... 

Partiele eharaeteristies = Moment equation + Size distribution for /ij equation for n L)... 

The solids hold-up. Mi, is related to the CSD by the third moment equation... 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

The infinite set of simultaneous ODEs has been transfonned to a set of moment equations that can be solved sequentially. The first three members of the set are ... 

Invoking the steady-state hypothesis for radicals, one can derive the following moment equations. 

We now turn to the question of developing a CFD model for fine-particle production that includes nucleation, growth, aggregation, and breakage. Applying QMOM to Eq. (114) leads to a closed set of moment equations as follows ... 

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... 

Note that the right-hand side of this expression contains the closed micromixing term for the moments (Eq. 137). To find the 3 M micromixing source terms (Amn, Bmn, Cmn) from this expression, we must choose a set of 3 M bi-variate moments. Note that because the moment equations are closed when only micromixing is considered, the chosen moments will be reproduced exactly. A convenient choice is to use the uncoupled moments m ) and mak. (Note that this same choice should then be used in Eq. (136).) This yields the linear system... 

A rational (but not necessarily easy to implement) rule for choosing the micromixing functions given e is to require that the corresponding mixture-fraction moment equation... 

As compared with the other closures discussed in this chapter, computation studies based on the presumed conditional PDF are relatively rare in the literature. This is most likely because of the difficulties of deriving and solving conditional moment equations such as (5.399). Nevertheless, for chemical systems that can exhibit multiple reaction branches for the same value of the mixture fraction,162 these methods may offer an attractive alternative to more complex models (such as transported PDF methods). Further research to extend multi-environment conditional PDF models to inhomogeneous flows should thus be pursued. 

In addition to the six moments used to find (B.42), we can choose from the third-order moments (mi, m2) = (3, 0), (2,1), (1, 2), and (0, 3). Note that only three additional moments are needed. Thus, one can either arbitrarily choose any three of the four third-order moments, or one can form three linearly independent combinations of the four moment equations. In either case, A will be full rank (provided that the points in composition space are distinct), or can be made so by perturbing the non-distinct compositions as discussed for the uni-variate case. 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

In the case of dipolar produced line width, a rigorous expression for the second moment (Equation 17) of the resonance absorption line allows one to obtain quantitative structural information (Section II,B and C). The... 

In applying the resulting state space model for control system design, the order of the state space model is important. This order is directly affected by the number of ordinary differential equations (moment equations) required to describe the population balance. From the structure of the moment equations, it follows that the dynamics of m.(t) is described by the moment equations for m (t) to m. t). Because the concentration balance contains c(t)=l-k m Vt), at I east the first four moments equations are required to close off the overall model. The final number of equations is determined by the moment m (t) in the equation for the nucleation rate (usually m (t)) and the highest moment to be controlled. 

Consider GMM estimation of a regression model as shown at the beginning of Example 18.8. Let Wj be the optimal weighting matrix based on the moment equations. Let W2 be some other positive definite matrix. Compare the asymptotic covariance matrices of the two proposed estimators. Show conclusively that the asymptotic covariance matrix of the estimator based on Wj is not larger than that based on W2. 

Clearly, the model cannot be estimated by ordinary least squares, since there is an autocorrelated disturbance and a lagged dependent variable. The parameters can be estimated consistently, but inefficiently by linear instrumental variables. The inefficiency arises from the fact that the parameters are overidentified. The linear estimator estimates seven functions of the five underlying parameters. One possibility is a GMM estimator. Let v, = g, -(y+< >)g,-i + (y< >)g, 2. Then, a GMM estimator can be defined in terms of, say, a set of moment equations of the fonn E[v,w,] = 0, where w, is current and lagged values of x and z. A minimum distance estimator could then be used for estimation. 

As a second conclusion from the moment equations we compute the autocorrelation matrix. Using (5.10a) and (5.12) one first finds... 

Note that the omission of terms of order Q 3/2 in < > and < 2) has the effect that the hierarchy of moment equations automatically breaks off. 

For describing dust formation and growth we adopt a frame of reference which is moving with the flow. Then the system of moment equations which describes these processes in a sufficient approximation reduces to... 

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