Open Access This article is
- freely available
Processes 2019, 7(7), 453; https://doi.org/10.3390/pr7070453
A Numerical Approach to Solve Volume-Based Batch Crystallization Model with Fines Dissolution Unit
Department of Basic Sciences, Deanship of Preparatory Year, King Faisal University, 31982 Hofuf, Al Ahsa, Saudi Arabia
Department of Mathematics & Statistics, Bacha Khan University, 24461 Palosa, Charsadda Khyber Pakhtunkhwa, Pakistan
Author to whom correspondence should be addressed.
Received: 11 June 2019 / Accepted: 12 July 2019 / Published: 15 July 2019
In this article, a numerical study of a one-dimensional, volume-based batch crystallization model (PBM) is presented that is used in numerous industries and chemical engineering sciences. A numerical approximation of the underlying model is discussed by using an alternative Quadrature Method of Moments (QMOM). Fines dissolution term is also incorporated in the governing equation for improvement of product quality and removal of undesirable particles. The moment-generating function is introduced in order to apply the QMOM. To find the quadrature abscissas, an orthogonal polynomial of degree three is derived. To verify the efficiency and accuracy of the proposed technique, two test problems are discussed. The numerical results obtained by the proposed scheme are plotted versus the analytical solutions. Thus, these findings line up well with the analytical findings.
Keywords:volume-based population balance model with fines dissolution; quadrature method of moments; orthogonal polynomials
Population balance models (PBMs) show a significant role in different 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  were first to discuss the PBMs in chemical engineering. A detailed description of these models is given in . 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-differential 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, [3,4,5,6,7,8,9,10,11].
The Quadrature Method of Moments (QMOM) for solving the governed models was first introduced by McGraw . The direct QMOM was proposed by Fan et al. . In addition, a new QMOM has been proposed by Gimbun . Qamar et al.  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.  followed the technique of  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.
2. Materials and Methods
Suppose represents the number density function, then a general PBE is given by [1,2]:where . The variable represents the time and may be size, length, or composition. In this article, it represents the particle volume. In the above equation, each term has its specific definition. These terms are given bywhereand
Here, represents the birth of particles of volume resulting for amalgamation of two particles with respective volume and where and is the death of particles, describing the decrease in particle volume by aggregation with other particles of any volume. The aggregation kernel is the rate at which aggregation of two particles and produces a particle volume .whereand
Here, represents the birth of new particles during the breakage process and the breakage function is the probability density function for the formation of particle volume from particle volume , whereas, is the death of particles, and is the selection function describing the rate at which the particles are selected to break.
represents the dissolution term, is the volume of crystallizer, is the volumetric flow rate from the crystallizer to dissolution unit, is the dissolution function describing dissolution of small particles below some critical size. The population balance equation can be simplified through introducing the moment function. The moment of the population density function is mathematically written in the form:
The first and second moments denote particle population and volume, respectively, at any instant . Multiply to left- and right-hand side of Equation (1) and at that moment integrate it over the volume , so we obtain the following equation:where
The aggregation terms and are mathematically defined aswhereas and are birth and death functions because of breakage term and are given by
Due to aggregation, the upper limit in the birth function is not infinity; therefore, we cannot apply quadrature method of moment for this integral. To apply the QMOM, we have to convert the upper limit to infinity. Here, we introduce a Heaviside step function to solve the integral such that when and otherwise. As a result, we will find the limits of integration over from to . Thus, by applying it to the birth function, we obtain the following equation:
On the right side of Equation (7), we have switched the order of integration and made the replacement Lastly, we have replaced for with no damage of generality and caught the necessary outcome for birth due to aggregation which is given by
After substituting all of the above terms in Equation (6), we get the system of differential equations:
In addition, with solid phase, the liquid phase yields ordinary differential equation for the solute mass in the form:with where is the density of crystals and is a volume shape factor. is the incoming mass flux from dissolution unit to crystallizer and outgoing mass flux from crystallizer to dissolution unit, respectively, which is mathematically defined bywhere is the mass of solvent and is the density of the solution. is the residence time in the dissolution unit, where represents volume of the pipe.
Here, we will consider the Gaussian quadrature method to solve the complicated integral terms appearing in Equations (8) and (9). Consequently, a closed-form system of moments is found. Further, we have accurately and efficiently solved the closed-form system with an ODE solver. The approximation of definite integral with the help of quadrature rule is an important numerical aspect. For this purpose, first we calculate the function at definite points and then use the formula of weight function which gives approximation of the definite integral. To approximate a definite integral given in Equations (8) and (9), first we find the values of the function at a set of equidistant points. After evaluating, the weight function is multiplied to approximate the integrals. In this rule, there is no limitation of choosing abscissas and weights. This rule also works for the points which are not likewise spread out. Let us assume the integral of the formula . We can find the weights and abscissas by approximating the definite integral aswhere provided it is exact and is smooth function. A set of orthogonal polynomials is necessary for making certain order orthogonal polynomial and it should contain only one polynomial of order for . We can define the scalar product of the two functions and over a weight function by
If we find the zero scalar product of any two functions and , then these functions are termed as orthogonal functions. It is also noted that supplementary information will be required for obtaining abscissas and weights if the classical weight function is not provided. For this purpose, we have used the moment:
In the above equation, is used as a weight function . given in Equation (13) denotes the polynomial of order. To find abscissas weights, we need moments from to . For this approximation will be exact. For simplicity and accurate approximations, we set . As a result, six-moment system for is calculated. To discuss the procedure, we consider ODEs (8) and (9). Using Equation (13) in Equations (8) and (9), we obtain the following equations:where and
Our next step is computing the quadrature points and the quadrature weights . The quadrature points are obtained from the roots of orthogonal polynomials. The construction of orthogonal polynomials is described as follows:with
Since the function is used as a weight function , so from Equation (12) we have
Using all of the above definitions, the orthogonal polynomials of any order can be obtained. Our first step is to find the roots of the order polynomial. The quadrature points are then obtained from these roots. To explain the procedure of deriving these orthogonal polynomials, we have calculated , . as
First, will be calculated, which is given byso
Again from Equations (17) and (18) we haveandso from Equation (20) we have
Proceeding in a similar way, we can calculate the polynomial of higher order. The third-order polynomial is given by the following equation:
The roots of the selected polynomial will give us the abscissas . Next, the weights will be calculated. According to Press et al. , the expression for the weight function is given bywhere N is the order of the selected polynomial. At last, the resulting system of ODEs is then solved by any standard ODE solver in MatLab.
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 is conserved and the amount of particles reduces during the processing time. The aggregation kernel is held to be constant and is defined as , where . The exponential initial particle size distribution is given bywhere and . The dissolution term explains the dissolution of particles below certain critical size, that is, . The analytical solution in terms of the number density is given by Scott :where and . The plot in Figure 2 shows normalized moments. The outcomes of our QMOM are in decent agreement with analytical outcomes. It is also observed from Figure 2 that during the aggregation process, the number of particles decreases while the volume of the particles remains constant.
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 bywhere, and represent the zero and first moments, respectively. A constant aggregation term, a breakage kernel , and uniform daughter distribution are taken. The analytical solution is given by Patel :
The numerical results are displayed in Figure 3. The moments of the numerical system are in good agreement 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 decreases while the volume of the particles remains constant.
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 agreement 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.
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.
Safyan Mukhtar is thankful to the Deanship of Scientific Research, King Faisal University, for research grant through the Nasher track (186122).
Conflicts of Interest
The authors declare no conflict of interest.
- Hulburt, H.M.; Katz, S. Some problems in particle technology. Chem. Eng. Sci. 1964, 19, 555–574. [Google Scholar] [CrossRef]
- Randolph, A.D.; Larson, M.A. Theory of Particulate Processes, 2nd ed.; Academic Press: Cambridge, MA, USA, 1988. [Google Scholar]
- Barrett, J.C.; Jheeta, J.S. Improving the accuracy of the moments method for solving the aerosol general dynamic equation. J. Aerosol Sci. 1996, 27, 1135–1142. [Google Scholar] [CrossRef]
- Fan, R.; Marchisio, D.L.; Fox, R.O. Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds. J. Powder Technol. 2004, 139, 7–20. [Google Scholar] [CrossRef]
- Gimbbun, J.; Nagy, Z.K.; Rielly, C.D. Simultaneous quadrature method of moments for the solution of population balance equations using a differential algebraic framework. Ind. Chem. Eng. 2009, 48, 7798–7812. [Google Scholar] [CrossRef]
- Gunawan, R.; Fusman, I.; Braatz, R.D. High resolution algorithms for multidimensional population balance equations. AICHE J. 2004, 50, 2738–2749. [Google Scholar] [CrossRef]
- Kumar, S.; Ramkrishna, D. On the solution of population balance equations by discretization-I. A fixed pivot technique. Chem. Eng. Sci. 1996, 51, 1311–1332. [Google Scholar] [CrossRef]
- Lim, Y.I.; Lann, J.-M.L.; Meyer, L.M.; Joulia, L.; Lee, G.; Yoon, E.S. On the solution of population balance equation (PBE) with accurate front tracking method in practical crystallization processes. Chem. Eng. Sci. 2002, 57, 3715–3732. [Google Scholar] [CrossRef]
- Madras, G.; McCoy, B.J. Reversible crystal growth-dissolution and aggregation breakage: Numerical and moment solutions for population balance equations. J. Powder Technol. 2004, 143, 297–307. [Google Scholar] [CrossRef]
- Marchisio, D.L.; Vigil, R.D.; Fox, R.O. Quadrature method of moments for aggregation-breakage processes. J. Colloid Interface Sci. 2003, 258, 322–334. [Google Scholar] [CrossRef]
- Gahn, C.; Mersmann, A. Brittle fracture in crystallization processes Part A. Attrition and abrasion of brittle solids. Chem. Eng. Sci. 1999, 54, 1273–1282. [Google Scholar] [CrossRef]
- McGraw, R. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Tech. 1997, 27, 255–265. [Google Scholar] [CrossRef]
- Qamar, S.; Mukhtar, S.; Seidel-Morgenstern, A.; Elsner, M.P. An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates. Chem. Eng. Sci. 2009, 64, 3659–3667. [Google Scholar] [CrossRef]
- Mukhtar, S.; Hussain, I.; Ali, A. Quadrature method of moments for solving volume-based population balance models. World Appl. Sci. J. 2012, 20, 1574–1583. [Google Scholar]
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes: The Art of Scientific Computing, 3rd ed.; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Scott, W.T. Analytic studies of cloud droplet coalescence I. J. Atmos. Sci. 1968, 25, 54–65. [Google Scholar] [CrossRef]
- Patel, D.P.; Andrews, J.R.G. An analytical solution to continuous population balance model describing coalescence and breakage. J. Chem. Eng. Sci. 1998, 53, 599–601. [Google Scholar] [CrossRef]
Figure 1. Single batch setup process with fines dissolution.
Figure 2. Aggregation with fines dissolution.
Figure 3. Aggregation and breakage with fines dissolution.
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).