Inhomogeneous PBE

The realizability condition on A/ is thus the same as in Eq. (8.52). In summary, the four-stage time-stepping procedure starting from the transported moment set M is as follows  

In this section, we consider the advection and diffusion of a univariafe NDF n(t, x, ), where R+ is a passive scalar (i.e. the velocity u t, x) does not depend on ). For example, could denote the particle mass for fine particles in a liquid with very small Stokes number. The diffusion coefficient, F(4, is assumed to be a function of For example, using the Stokes-Einstein formula for diffusivity reported in Eq. (5.116) on page 187, F( ) = Fq/, where is the particle mass. The PBE for this case reads 

Moreover, employing the Stokes-Einstein formula, we can write f f r(f)ndf = Fom/i-i, which is essentially the form that will be used in the numerical examples in Section 8.3.4. In summary, we will consider two variations of the moment-transport equations in the numerical examples in Section 8.3.4. The first example will use a closed moment system wherein the diffusivity does not depend on f  

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In this appendix we discuss the probiem of moment conservation with the DQMOM, and present a variation caiied DQMOM fuiiy conservative (DQMOM-FC) that can be empioyed to partiaiiy overcome it. To iiiustrate the issue, we consider a one-dimensional spatially inhomogeneous PBE for the univariate NDF n(t, x, with constant velocity and constant diffusion coefficient as described in Section 8.3. However, as will become clearer below, the extension to other, more complex, problems is straightforward. 

In the more sophisticated treatments, spatial inhomogeneities are modeled by connected zones or regions of Space that are assumed to be homogeneous. The resulting population-balance equation (PBE) does not explicitly account for local variations in die flow field. 

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