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applied for each size class. There have been numerous

models developed for one-dimensional ﬂows using such

techniques e.g. [11–18]. In order to achieve reasonably

accurate resolution and to overcome numerical diﬀusion,

however, the particle size distribution must be divided into

a large number of bins. Consequently, implementation in

CFD codes becomes prohibitive for all but the most restrictive cases [19–26].

An alternative approach is the methods of moments

(MOM) in which the moments of the particle size distribution are solved by integrating out the internal coordinate

(e.g. diameter or volume). This allows the problem to be

reconstructed in a reduced variable space and in which

each moment is a transportable scalar, however, the problem of mathematical closure arises [27,28]. Three methods

are typically employed to resolve the closure issue:

through the assumption of a closed form of the distribution function in which the properties of the distribution

are a function of the solved

models developed for one-dimensional ﬂows using such

techniques e.g. [11–18]. In order to achieve reasonably

accurate resolution and to overcome numerical diﬀusion,

however, the particle size distribution must be divided into

a large number of bins. Consequently, implementation in

CFD codes becomes prohibitive for all but the most restrictive cases [19–26].

An alternative approach is the methods of moments

(MOM) in which the moments of the particle size distribution are solved by integrating out the internal coordinate

(e.g. diameter or volume). This allows the problem to be

reconstructed in a reduced variable space and in which

each moment is a transportable scalar, however, the problem of mathematical closure arises [27,28]. Three methods

are typically employed to resolve the closure issue:

through the assumption of a closed form of the distribution function in which the properties of the distribution

are a function of the solved

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