**4. Conclusions**

The moment-generating function was used to convert the governing partial differential equation into a system of ordinary differential equations. The mathematical term for fines dissolution was incorporated in the model for improving the quality of the product. The Gaussian quadrature method was implemented to solve the complicated integrals in this research. An orthogonal polynomial of degree three, which utilizes the first six moments, was used for better accuracy of the proposed scheme. To check the significance of the scheme, two case studies were discussed. The results of the proposed scheme are in perfect agreemen<sup>t</sup> with the available analytical results. No dissipation was observed in the results. This work is extendable for batch preferential crystallization models with fines dissolution. Furthermore, the developed scheme could be applicable to solve multidimensional batch crystallization models with fines dissolution.

**Author Contributions:** Conceptualization: S.M. and I.A.; Methodology: S.M.; Software: I.A. and M.S.; Validation: S.M. and I.A.; Formal Analysis: S.M.; Investigation: S.M.; Writing-Original Draft Preparation: S.M.; Writing-Review & Editing: S.M., M.S., and I.A.

**Funding:** Safyan Mukhtar is thankful to the Deanship of Scientific Research, King Faisal University, for research gran<sup>t</sup> through the Nasher track (186122).

**Conflicts of Interest:** The authors declare no conflict of interest.
