Finite-volume method realizable

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

Vikas, V, Wang, Z. J. Fox, R. O. 2012 Realizable high-order finite-volume schemes for quadrature-based moment methods applied to diffusion population balance equations. Journal of Computational Physics (submitted). 

Therefore, the measurement precision of the sensor hardware has to be very good. But, even with an idealized sensor hardware, the extinction cross sections of particles are determined with uncertainties of at least 2 % with data block sizes of 10,000 values [4]. This is caused by the statistical uncertainty, due to the finite number of values, which are used to calculate the mean value and the root mean square deviation of the transmission signal. Therefore, a determination of particle size distributions with an advanced SE-Method is fairly difficult. However, a reliable process monitoring with the SE-Method is possible, which enables the detection of relative changes of the mean particle sizes (see also measurement results in the next section). Therefore, the pinhole diameters, as well as the length of the measurement volume, have to be adjusted to realize a mean value of the transmission and a theoretical relative extinction cross section within the optimal range. 

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