Mesoscale variables for particle size

As a matter of practice, it is useful to limit the number of internal coordinates required to describe a particle s size. Thus, in this section, we review the available descriptions of the particle size that use either one or two internal coordinates. 

When the elements of the disperse phase can be classified as equidimensional, namely they have nearly the same size or spread in multiple directions, and have constant material density, typically a single internal coordinate is used to identify the size of the elements. This could be particle mass (or volume), particle surface area or particle length. In fact, in the case of equidimensional particles these quantities are all related to each other. For example, in the trivial cases of spherical or cubic particles, particle volume and particle surface area can be easily written as Vp = k d and Ap = k d, or, in other words, as functions of a characteristic length, d (i.e. the diameter for the sphere and the edge for the cube), a volume shape factor, k, and a surface-area shape factor, k. For equidimensional objects the choice of the characteristic length is straightforward and the ratio between kp, and k is always equal to six. The approach can, however, be extended also to non-equidimensional objects. In this context, the extension turns out to be very useful only if 

Under these hypotheses and if particle size is the (first) internal coordinate (i.e. = dp) 

The rate of change of particle size is also indicated as Gp and can be positive, in the case of growing particles, or negative, in the case of shrinking particles. This is probably the most popular way to indicate the rate of phase-space advection due to mass exchange, perhaps because it is quite easy to measure the change in particle size at different instants, for example by simple imaging techniques. If the internal coordinate is instead particle volume (i.e. = Vp), the definition becomes 

Since M , Pp, kp, and ks are typically constants, the only relevant mesoscale dependences are those with respect to and with respect to the molar flux T(U, U, from the 

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