**1. Introduction**

Population balance models (PBMs) show a significant role in di fferent areas of science and engineering. These models have numerous applications in high-energy physics, geophysics, biophysics, meteorology, pharmacy, food science, chromatography, chemical engineering, civil engineering, and environmental engineering. These models are used in the process of cell dynamics, polymerization, cloud formation, and crystallization. In biophysics, these models are concerned with population of various kinds of cells. Population balance models are also used in the formation of ceramics mixtures and nanoparticles which have a lot of applications.

In 1964, Hulburt & Katz [1] were first to discuss the PBMs in chemical engineering. A detailed description of these models is given in [2]. The main components in the models are the process of nucleation, growth, aggregation, breakage, inlet, outlet, growth, and dissolution. The mathematical model of population balance equations are partial integro-di fferential equations (PIDEs). Analytical solutions of these PIDEs are very rare except for a few simple cases. Therefore, researchers are interested in developing numerical solutions for these equations. Numerous numerical methods are accessible in the literature to solve certain kinds of PBMs; see, for example, [3–11].

The Quadrature Method of Moments (QMOM) for solving the governed models was first introduced by McGraw [12]. The direct QMOM was proposed by Fan et al. [4]. In addition, a new QMOM has been proposed by Gimbun [5]. Qamar et al. [13] introduced an alternative QMOM for solving length-based batch crystallization models telling crystals nucleation, size-dependent growth, aggregation, breakage, and dissolution of small nuclei below certain critical size. Safyan et al. [14] followed the technique of [13] for solving volume-based batch crystallization model nucleation, size-dependent growth, aggregation, breakage. In this article, the QMOM is used to solve volume-based

batch crystallization models with fines dissolution. The fines dissolution unit improves the product quality and removes unwanted particles.

In the proposed method, orthogonal polynomials, taken from the lower-order moments, are used to find the quadrature points and weights. For confirming accuracy of the scheme, a third-order orthogonal polynomial, employing the first six moments, was chosen to estimate the quadrature points (abscissas) and equivalent quadrature weights. Hence, it was essential to solve at least a six-moment system (i.e., 0, ... , 5). This type of polynomial provides a three-point Gaussian quadrature rule which usually produces precise output for polynomials of degree five or less. The expression for the preferred orthogonal polynomial can be derived analytically and the mandatory number of moments rises with order of polynomial. It is important to mention that the calculation of six moments to generate each time the third-order polynomial is compulsory. However, the technique itself is not limited to the calculation of every listed number of moments. The calculation of additional moments is conducted by adding ordinary differential equation.

This article is divided into two parts: In the first part, the mathematical model of volume-based Population Balance Equation (PBE) with fines dissolution is presented. In the second part, the proposed QMOM is derived for PBEs. Furthermore, the mathematical outcomes of QMOM are compared with the analytical outcomes that are accessible in literature.
