**3. Results**

Test Problem 1: Aggregation with Fines Dissolution

Here, the proposed scheme is analyzed for aggregation with fines dissolution (see Figure 1) problem encountered in several particulate methods (i.e., fluidized beds, formation of rain droplets, and manufacture of dry powders). The effects of further procedures, such as breakage, growth, and nucleation, are negligible. During the aggregation process, the total mass of particles *m*1 is conserved and the amount of particles *m*0 reduces during the processing time. The aggregation kernel is held to be constant and is defined as β(*Vp*, *<sup>V</sup>p*) = β0, where β0 = 1. The exponential initial particle size distribution is given by

$$
\mu\_d \begin{pmatrix} 0 \ \dot{V}\_p \end{pmatrix} = \frac{N\_0}{V\_{P\_0}} \exp \left( -V\_p / V\_{P\_0} \right) \tag{26}
$$

where *N*0 = 1 and *Vp*0 = 1. The dissolution term *hVp* explains the dissolution of particles below certain critical size, that is, 2 × 10−4*m*3. The analytical solution in terms of the number density *ud<sup>T</sup>*, *Vp* is given by Scott [16]:

$$
\mu\_d \left( T\_\prime V\_p \right) = \frac{4N\_0}{V\_{P\_0} \left( \pi + 2 \right)^2} \exp \left( -2V\_p^\prime / \pi + 2 \right) \tag{27}
$$

where τ = *N*0β0*<sup>t</sup>* and *<sup>V</sup>p* = *<sup>V</sup>*/*Vp*0. The plot in Figure 2 shows normalized moments. The outcomes of our QMOM are in decent agreemen<sup>t</sup> with analytical outcomes. It is also observed from Figure 2 that during the aggregation process, the number of particles *<sup>m</sup>*0(*T*) decreases while the volume of the particles *<sup>m</sup>*1(*T*) remains constant.

**Figure 1.** Single batch setup process with fines dissolution.

**Figure 2.** Aggregation with fines dissolution.

Test Problem 2: Aggregation and Breakage with Fines Dissolution

In this problem, we take a batch crystallizer in which aggregation and breakage are the main occurrences and which is connected with a fines dissolver. The growth and nucleation terms are neglected in this process. The initial distribution is given by

$$
\Delta u\_d \left( 0, V\_p \right) = M\_0 \cdot \frac{M\_0}{M\_1} \exp \left( -\frac{M\_0}{M\_1} V\_p \right) \tag{28}
$$

where, *M*0 = 2*N*0 2+β0*N*0*<sup>t</sup>* and *M*1 = *<sup>N</sup>*0*Vp*0 <sup>1</sup> − 2*G*0 β0*N*0*Vp*0 ln 2 <sup>2</sup>+β0*N*0*<sup>t</sup>* represent the zero and first moments, respectively. A constant aggregation term, β(*Vp*, *<sup>V</sup>p*) = β0 = 1, a breakage kernel *<sup>b</sup>*(*Vp*, *<sup>V</sup>p*) = 2/*Vp*,, and uniform daughter distribution *S*(*Vp*) = *Vp* are taken. The analytical solution is given by Patel [17]:

$$
\mu\_d \left( T, V\_p \right) = \frac{M\_0^2}{M\_1} \exp \left( -\frac{M\_0}{M\_1} V\_p \right) \tag{29}
$$

The numerical results are displayed in Figure 3. The moments of the numerical system are in good agreemen<sup>t</sup> with those taken from the analytical solution. It is also observed from Figure 3 that during the aggregation and breakage process, the number of particles *<sup>m</sup>*0(*T*) decreases while the volume of the particles *<sup>m</sup>*1(*T*) remains constant.

**Figure 3.** Aggregation and breakage with fines dissolution.
